fix preview of final scene

The previous image was generated with a version of this code that did not apply gamma to textures which resulted in incorrect colors of the globe.

README.md

@@ -47,3 +47,11 @@ java -jar raytracing.jar --samples 50000 --height 600 CORNELL_SMOKE
 ```
 java -jar raytracing.jar --samples 1000 --height 1080 DIAGRAMM
 ```
+
+### a little bit of everything
+
+![](./docs/final.png)
+
+```
+java -jar raytracing.jar --samples 10000 --height 1200 FINAL
+```

src/main/java/eu/jonahbauer/raytracing/Examples.java

@@ -1,5 +1,6 @@
 package eu.jonahbauer.raytracing;
 
+import eu.jonahbauer.raytracing.math.AABB;
 import eu.jonahbauer.raytracing.math.Vec3;
 import eu.jonahbauer.raytracing.render.texture.CheckerTexture;
 import eu.jonahbauer.raytracing.render.texture.Color;
@@ -34,6 +35,7 @@ public class Examples {
         register("LIGHT", Examples::getLight);
         register("CORNELL", Examples::getCornellBox);
         register("CORNELL_SMOKE", Examples::getCornellBoxSmoke);
+        register("CORNELL_SPHERE", Examples::getCornellBoxSphere);
         register("DIAGRAMM", Examples::getDiagramm);
         register("EARTH", Examples::getEarth);
         register("PERLIN", Examples::getPerlin);
@@ -49,14 +51,13 @@ public class Examples {
     public static @NotNull Example getSimpleScene(int height) {
         if (height <= 0) height = 675;
         return new Example(
-                new Scene(
-                        getSkyBox(),
+                new Scene(getSkyBox(), List.of(
                         new Sphere(new Vec3(0, -100.5, -1.0), 100.0, new LambertianMaterial(new Color(0.8, 0.8, 0.0))),
                         new Sphere(new Vec3(0, 0, -1.2), 0.5, new LambertianMaterial(new Color(0.1, 0.2, 0.5))),
                         new Sphere(new Vec3(-1.0, 0, -1.2), 0.5, new DielectricMaterial(1.5)),
                         new Sphere(new Vec3(-1.0, 0, -1.2), 0.4, new DielectricMaterial(1 / 1.5)),
                         new Sphere(new Vec3(1.0, 0, -1.2), 0.5, new MetallicMaterial(new Color(0.8, 0.6, 0.2), 0.0))
-                ),
+                )),
                 SimpleCamera.builder()
                         .withImage(height * 16 / 9, height)
                         .build()
@@ -117,14 +118,13 @@ public class Examples {
     public static @NotNull Example getSquares(int height) {
         if (height <= 0) height = 600;
         return new Example(
-                new Scene(
-                        getSkyBox(),
+                new Scene(getSkyBox(), List.of(
                         new Parallelogram(new Vec3(-3, -2, 5), new Vec3(0, 0, -4), new Vec3(0, 4, 0), new LambertianMaterial(new Color(1.0, 0.2, 0.2))),
                         new Parallelogram(new Vec3(-2, -2, 0), new Vec3(4, 0, 0), new Vec3(0, 4, 0), new LambertianMaterial(new Color(0.2, 1.0, 0.2))),
                         new Parallelogram(new Vec3(3, -2, 1), new Vec3(0, 0, 4), new Vec3(0, 4, 0), new LambertianMaterial(new Color(0.2, 0.2, 1.0))),
                         new Parallelogram(new Vec3(-2, 3, 1), new Vec3(4, 0, 0), new Vec3(0, 0, 4), new LambertianMaterial(new Color(1.0, 0.5, 0.0))),
                         new Parallelogram(new Vec3(-2, -3, 5), new Vec3(4, 0, 0), new Vec3(0, 0, -4), new LambertianMaterial(new Color(0.2, 0.8, 0.8)))
-                ),
+                )),
                 SimpleCamera.builder()
                         .withImage(height, height)
                         .withFieldOfView(Math.toRadians(80))
@@ -137,12 +137,12 @@ public class Examples {
     public static @NotNull Example getLight(int height) {
         if (height <= 0) height = 225;
         return new Example(
-                new Scene(
+                new Scene(List.of(
                         new Sphere(new Vec3(0, -1000, 0), 1000, new LambertianMaterial(new Color(0.2, 0.2, 0.2))),
                         new Sphere(new Vec3(0, 2, 0), 2, new LambertianMaterial(new Color(0.2, 0.2, 0.2))),
                         new Parallelogram(new Vec3(3, 1, -2), new Vec3(2, 0, 0), new Vec3(0, 2, 0), new DiffuseLight(new Color(4.0, 4.0, 4.0))),
                         new Sphere(new Vec3(0, 7, 0), 2, new DiffuseLight(new Color(4.0, 4.0, 4.0)))
-                ),
+                )),
                 SimpleCamera.builder()
                         .withImage(height * 16 / 9, height)
                         .withFieldOfView(Math.toRadians(20))
@@ -161,16 +161,16 @@ public class Examples {
         var light = new DiffuseLight(new Color(15.0, 15.0, 15.0));
 
         return new Example(
-                new Scene(
-                        new Parallelogram(new Vec3(555, 0, 0), new Vec3(0, 555, 0), new Vec3(0, 0, 555), green),
-                        new Parallelogram(new Vec3(0, 0, 0), new Vec3(0, 555, 0), new Vec3(0, 0, 555), red),
+                new Scene(List.of(
+                        new Box(new Vec3(0, 0, 0), new Vec3(555, 555, 555), white, white, red, green, white, null),
                         new Parallelogram(new Vec3(343, 554, 332), new Vec3(-130, 0, 0), new Vec3(0, 0, -105), light),
-                        new Parallelogram(new Vec3(0, 0, 0), new Vec3(555, 0, 0), new Vec3(0, 0, 555), white),
-                        new Parallelogram(new Vec3(555, 555, 555), new Vec3(-555, 0, 0), new Vec3(0, 0, -555), white),
-                        new Parallelogram(new Vec3(0, 0, 555), new Vec3(555, 0, 0), new Vec3(0, 555, 0), white),
-                        new Box(new Vec3(0, 0, 0), new Vec3(165, 330, 165), white).rotateY(Math.toRadians(15)).translate(new Vec3(265, 0, 295)),
-                        new Box(new Vec3(0, 0, 0), new Vec3(165, 165, 165), white).rotateY(Math.toRadians(-18)).translate(new Vec3(130, 0, 65))
-                ),
+                        new Box(new Vec3(0, 0, 0), new Vec3(165, 330, 165), white)
+                                .rotateY(Math.toRadians(15))
+                                .translate(new Vec3(265, 0, 295)),
+                        new Box(new Vec3(0, 0, 0), new Vec3(165, 165, 165), white)
+                                .rotateY(Math.toRadians(-18))
+                                .translate(new Vec3(130, 0, 65))
+                )),
                 SimpleCamera.builder()
                         .withImage(height, height)
                         .withFieldOfView(Math.toRadians(40))
@@ -188,22 +188,49 @@ public class Examples {
         var light = new DiffuseLight(new Color(7.0, 7.0, 7.0));
 
         return new Example(
-                new Scene(
-                        new Parallelogram(new Vec3(555, 0, 0), new Vec3(0, 555, 0), new Vec3(0, 0, 555), green),
-                        new Parallelogram(new Vec3(0, 0, 0), new Vec3(0, 555, 0), new Vec3(0, 0, 555), red),
+                new Scene(List.of(
+                        new Box(new Vec3(0, 0, 0), new Vec3(555, 555, 555), white, white, red, green, white, null),
                         new Parallelogram(new Vec3(113, 554, 127), new Vec3(330, 0, 0), new Vec3(0, 0, 305), light),
-                        new Parallelogram(new Vec3(0, 0, 0), new Vec3(555, 0, 0), new Vec3(0, 0, 555), white),
-                        new Parallelogram(new Vec3(555, 555, 555), new Vec3(-555, 0, 0), new Vec3(0, 0, -555), white),
-                        new Parallelogram(new Vec3(0, 0, 555), new Vec3(555, 0, 0), new Vec3(0, 555, 0), white),
                         new ConstantMedium(
-                                new Box(new Vec3(0, 0, 0), new Vec3(165, 330, 165), white).rotateY(Math.toRadians(15)).translate(new Vec3(265, 0, 295)),
+                                new Box(new Vec3(0, 0, 0), new Vec3(165, 330, 165), white)
+                                        .rotateY(Math.toRadians(15))
+                                        .translate(new Vec3(265, 0, 295)),
                                 0.01, new IsotropicMaterial(Color.BLACK)
                         ),
                         new ConstantMedium(
-                                new Box(new Vec3(0, 0, 0), new Vec3(165, 165, 165), white).rotateY(Math.toRadians(-18)).translate(new Vec3(130, 0, 65)),
+                                new Box(new Vec3(0, 0, 0), new Vec3(165, 165, 165), white)
+                                        .rotateY(Math.toRadians(-18))
+                                        .translate(new Vec3(130, 0, 65)),
                                 0.01, new IsotropicMaterial(Color.WHITE)
                         )
-                ),
+                )),
+                SimpleCamera.builder()
+                        .withImage(height, height)
+                        .withFieldOfView(Math.toRadians(40))
+                        .withPosition(new Vec3(278, 278, -800))
+                        .withTarget(new Vec3(278, 278, 0))
+                        .build()
+        );
+    }
+
+    public static @NotNull Example getCornellBoxSphere(int height) {
+        if (height <= 0) height = 600;
+
+        var red = new LambertianMaterial(new Color(.65, .05, .05));
+        var white = new LambertianMaterial(new Color(.73, .73, .73));
+        var green = new LambertianMaterial(new Color(.12, .45, .15));
+        var light = new DiffuseLight(new Color(7.0, 7.0, 7.0));
+        var glass = new DielectricMaterial(1.5);
+
+        var room = new Box(new Vec3(0, 0, 0), new Vec3(555, 555, 555), white, white, red, green, white, null);
+        var lamp = new Parallelogram(new Vec3(343, 554, 332), new Vec3(-130, 0, 0), new Vec3(0, 0, -105), light);
+        var box = new Box(new Vec3(0, 0, 0), new Vec3(165, 330, 165), white)
+                .rotateY(Math.toRadians(15))
+                .translate(new Vec3(265, 0, 295));
+        var sphere = new Sphere(new Vec3(190, 90, 190), 90, glass);
+
+        return new Example(
+                new Scene(List.of(room, box), List.of(lamp, sphere)),
                 SimpleCamera.builder()
                         .withImage(height, height)
                         .withFieldOfView(Math.toRadians(40))
@@ -265,10 +292,9 @@ public class Examples {
         if (height <= 0) height = 450;
 
         return new Example(
-                new Scene(
-                        getSkyBox(),
+                new Scene(getSkyBox(), List.of(
                         new Sphere(Vec3.ZERO, 2, new LambertianMaterial(new ImageTexture("/earthmap.jpg")))
-                ),
+                )),
                 SimpleCamera.builder()
                         .withImage(height * 16 / 9, height)
                         .withFieldOfView(Math.toRadians(20))
@@ -284,11 +310,10 @@ public class Examples {
         var material = new LambertianMaterial(new PerlinTexture(4));
 
         return new Example(
-                new Scene(
-                        getSkyBox(),
+                new Scene(getSkyBox(), List.of(
                         new Sphere(new Vec3(0, -1000, 0), 1000, material),
                         new Sphere(new Vec3(0, 2, 0), 2, material)
-                ),
+                )),
                 SimpleCamera.builder()
                         .withImage(height * 16 / 9, height)
                         .withFieldOfView(Math.toRadians(20))

src/main/java/eu/jonahbauer/raytracing/Main.java

@@ -13,6 +13,8 @@ import java.nio.file.Path;
 import java.util.function.IntFunction;
 
 public class Main {
+    public static final boolean DEBUG = false;
+
     public static void main(String[] args) throws IOException {
         var config = Config.parse(args);
         var example = config.example;

src/main/java/eu/jonahbauer/raytracing/math/AABB.java

@@ -35,26 +35,49 @@ public record AABB(@NotNull Vec3 min, @NotNull Vec3 max) {
         return Optional.ofNullable(bbox);
     }
 
+    /**
+     * {@return the range of x values}
+     */
     public @NotNull Range x() {
         return new Range(min.x(), max.x());
     }
 
+    /**
+     * {@return the range of y values}
+     */
     public @NotNull Range y() {
         return new Range(min.y(), max.y());
     }
 
+    /**
+     * {@return the range of z values}
+     */
     public @NotNull Range z() {
         return new Range(min.z(), max.z());
     }
 
+    /**
+     * {@return the center of this bounding box}
+     */
     public @NotNull Vec3 center() {
         return Vec3.average(min, max, 2);
     }
 
+    /**
+     * Expands this bounding box to include the other bounding box.
+     * @param box a bounding box
+     * @return the expanded bounding box
+     */
     public @NotNull AABB expand(@NotNull AABB box) {
         return new AABB(Vec3.min(this.min, box.min), Vec3.max(this.max, box.max));
     }
 
+    /**
+     * Tests whether the {@code ray} intersects this bounding box withing the {@code range}
+     * @param ray a ray
+     * @param range a range of valid {@code t}s
+     * @return {@code true} iff the ray intersects this bounding box, {@code false} otherwise
+     */
     public boolean hit(@NotNull Ray ray, @NotNull Range range) {
         var origin = ray.origin();
         var direction = ray.direction();
@@ -85,6 +108,13 @@ public record AABB(@NotNull Vec3 min, @NotNull Vec3 max) {
         return tlmax < tumin && tumin >= range.min() && tlmax <= range.max();
     }
 
+    /**
+     * Computes the {@code t} values of the intersections of a ray with the axis-aligned planes through a point.
+     * @param corner the point
+     * @param origin the origin point of the ray
+     * @param invDirection the {@linkplain Vec3#inv() inverted} direction of the ray
+     * @return a three-element array of the {@code t} values of the intersection with the yz-, xz- and xy-plane through {@code corner}
+     */
     public static double @NotNull[] intersect(@NotNull Vec3 corner, @NotNull Vec3 origin, @NotNull Vec3 invDirection) {
         return new double[] {
                 (corner.x() - origin.x()) * invDirection.x(),

src/main/java/eu/jonahbauer/raytracing/math/Ray.java

@@ -11,16 +11,6 @@ public record Ray(@NotNull Vec3 origin, @NotNull Vec3 direction) {
     }
 
     public @NotNull Vec3 at(double t) {
-        return new Vec3(
-                Math.fma(t, direction.x(), origin.x()),
-                Math.fma(t, direction.y(), origin.y()),
-                Math.fma(t, direction.z(), origin.z())
-        );
-    }
-
-    public int vmask() {
-        return (direction().x() < 0 ? 1 : 0)
-                | (direction().y() < 0 ? 2 : 0)
-                | (direction().z() < 0 ? 4 : 0);
+        return Vec3.fma(t, direction, origin);
     }
 }

src/main/java/eu/jonahbauer/raytracing/math/Vec3.java

@@ -5,6 +5,8 @@ import org.jetbrains.annotations.NotNull;
 import java.util.Optional;
 import java.util.random.RandomGenerator;
 
+import static eu.jonahbauer.raytracing.Main.DEBUG;
+
 public record Vec3(double x, double y, double z) {
     public static final Vec3 ZERO = new Vec3(0, 0, 0);
     public static final Vec3 MAX = new Vec3(Double.MAX_VALUE, Double.MAX_VALUE, Double.MAX_VALUE);
@@ -14,31 +16,63 @@ public record Vec3(double x, double y, double z) {
     public static final Vec3 UNIT_Z = new Vec3(0, 0, 1);
 
     public Vec3 {
-        assert Double.isFinite(x) && Double.isFinite(y) && Double.isFinite(z) : "x, y and z must be finite";
+        if (DEBUG) {
+            if (!Double.isFinite(x) || !Double.isFinite(y) || !Double.isFinite(z)) {
+                throw new IllegalArgumentException("x, y and z must be finite");
+            }
+        }
     }
 
+    /**
+     * {@return a uniformly random unit vector}
+     */
     public static @NotNull Vec3 random(@NotNull RandomGenerator random) {
-        return random(random, false);
+        double x, y, z;
+        double squared;
+        do {
+            x = Math.fma(2, random.nextDouble(), -1);
+            y = Math.fma(2, random.nextDouble(), -1);
+            z = Math.fma(2, random.nextDouble(), -1);
+            squared = x * x + y * y + z * z;
+        } while (squared > 1);
+        var factor = 1 / Math.sqrt(squared);
+        return new Vec3(x * factor, y * factor, z * factor);
     }
 
-    public static @NotNull Vec3 random(@NotNull RandomGenerator random, boolean unit) {
-        var vec = new Vec3(
-                Math.fma(2, random.nextDouble(), -1),
-                Math.fma(2, random.nextDouble(), -1),
-                Math.fma(2, random.nextDouble(), -1)
-        );
-        return unit ? vec.unit() : vec;
+    /**
+     * {@return a uniformly random unit vector on the opposite hemisphere of the given <code>direction</code>}
+     */
+    public static @NotNull Vec3 randomOppositeHemisphere(@NotNull RandomGenerator random, @NotNull Vec3 direction) {
+        double x, y, z;
+        double squared;
+        do {
+            x = Math.fma(2, random.nextDouble(), -1);
+            y = Math.fma(2, random.nextDouble(), -1);
+            z = Math.fma(2, random.nextDouble(), -1);
+            squared = x * x + y * y + z * z;
+        } while (squared > 1 || direction.x() * x + direction.y() * y + direction.z() * z >= 0);
+        var factor = 1 / Math.sqrt(squared);
+        return new Vec3(x * factor, y * factor, z * factor);
     }
 
+    /**
+     * Reflects a vector on the given {@code normal} vector.
+     * @param vec a vector
+     * @param normal the surface normal (must be a unit vector)
+     * @return the reflected vector
+     */
     public static @NotNull Vec3 reflect(@NotNull Vec3 vec, @NotNull Vec3 normal) {
         var factor = - 2 * normal.times(vec);
-        return new Vec3(
-                Math.fma(factor, normal.x(), vec.x()),
-                Math.fma(factor, normal.y(), vec.y()),
-                Math.fma(factor, normal.z(), vec.z())
-        );
+        return Vec3.fma(factor, normal, vec);
     }
 
+    /**
+     * Refracts a vector on the given {@code normal} vector.
+     * @param vec a vector
+     * @param normal the surface normal (must be a unit vector)
+     * @param ri the refractive index
+     * @return the refracted vector
+     */
     public static @NotNull Optional<Vec3> refract(@NotNull Vec3 vec, @NotNull Vec3 normal, double ri) {
         vec = vec.unit();
         var cosTheta = Math.min(- vec.times(normal), 1.0);
@@ -50,16 +84,35 @@ public record Vec3(double x, double y, double z) {
         return Optional.of(rOutPerp.plus(rOutParallel));
     }
 
+    /**
+     * Rotates a vector around an {@code axis}.
+     * @param vec a vector
+     * @param axis the rotation axis
+     * @param angle the angle in radians
+     * @return the rotated vector
+     */
     public static @NotNull Vec3 rotate(@NotNull Vec3 vec, @NotNull Vec3 axis, double angle) {
         Vec3 vxp = axis.cross(vec);
         Vec3 vxvxp = axis.cross(vxp);
         return vec.plus(vxp.times(Math.sin(angle))).plus(vxvxp.times(1 - Math.cos(angle)));
     }
 
+    /**
+     * {@return the euclidean distance between two vectors}
+     * @param a a vector
+     * @param b another vector
+     */
     public static double distance(@NotNull Vec3 a, @NotNull Vec3 b) {
         return a.minus(b).length();
     }
 
+    /**
+     * Computes a running average of vectors.
+     * @param current the current running average
+     * @param next the next vector
+     * @param index the one-based index of the next vector
+     * @return the new running average
+     */
     public static @NotNull Vec3 average(@NotNull Vec3 current, @NotNull Vec3 next, int index) {
         var factor = 1d / index;
         return new Vec3(
@@ -69,6 +122,11 @@ public record Vec3(double x, double y, double z) {
         );
     }
 
+    /**
+     * {@return a component-wise maximum vector}
+     * @param a a vector
+     * @param b another vector
+     */
     public static @NotNull Vec3 max(@NotNull Vec3 a, @NotNull Vec3 b) {
         return new Vec3(
                 Math.max(a.x(), b.x()),
@@ -77,6 +135,11 @@ public record Vec3(double x, double y, double z) {
         );
     }
 
+    /**
+     * {@return a component-wise minimum vector}
+     * @param a a vector
+     * @param b another vector
+     */
     public static @NotNull Vec3 min(@NotNull Vec3 a, @NotNull Vec3 b) {
         return new Vec3(
                 Math.min(a.x(), b.x()),
@@ -85,6 +148,24 @@ public record Vec3(double x, double y, double z) {
         );
     }
 
+    /**
+     * {@return <code>a * b + c</code>}
+     * @param a scalar
+     * @param b a vector
+     * @param c another vector
+     */
+    public static @NotNull Vec3 fma(double a, @NotNull Vec3 b, @NotNull Vec3 c) {
+        return new Vec3(
+                Math.fma(a, b.x(), c.x()),
+                Math.fma(a, b.y(), c.y()),
+                Math.fma(a, b.z(), c.z())
+        );
+    }
+
+    public static double tripleProduct(@NotNull Vec3 a, @NotNull Vec3 b, @NotNull Vec3 c) {
+        return a.x * b.y * c.z + a.y * b.z * c.x + a.z * b.x * c.y - c.x * b.y * a.z - c.y * b.z * a.x - c.z * b.x * a.y;
+    }
+
     public @NotNull Vec3 plus(double x, double y, double z) {
         return new Vec3(this.x + x, this.y + y, this.z + z);
     }
@@ -93,59 +174,115 @@ public record Vec3(double x, double y, double z) {
         return new Vec3(this.x - x, this.y - y, this.z - z);
     }
 
-    public @NotNull Vec3 plus(@NotNull Vec3 b) {
-        return new Vec3(this.x + b.x, this.y + b.y, this.z + b.z);
+    /**
+     * Adds a vector to this vector
+     * @param other a vector
+     * @return the sum of this and the other vector
+     */
+    public @NotNull Vec3 plus(@NotNull Vec3 other) {
+        return new Vec3(this.x + other.x, this.y + other.y, this.z + other.z);
     }
 
-    public @NotNull Vec3 minus(@NotNull Vec3 b) {
-        return new Vec3(this.x - b.x, this.y - b.y, this.z - b.z);
+    /**
+     * Subtracts a vector from this vector
+     * @param other a vector
+     * @return the difference of this and the other vector
+     */
+    public @NotNull Vec3 minus(@NotNull Vec3 other) {
+        return new Vec3(this.x - other.x, this.y - other.y, this.z - other.z);
     }
 
-    public double times(@NotNull Vec3 b) {
-        return this.x * b.x + this.y * b.y + this.z * b.z;
+    /**
+     * Computes the scalar product of this and another vector
+     * @param other a vector
+     * @return the scalar product
+     */
+    public double times(@NotNull Vec3 other) {
+        return this.x * other.x + this.y * other.y + this.z * other.z;
     }
 
-    public @NotNull Vec3 times(double b) {
-        return new Vec3(this.x * b, this.y * b, this.z * b);
+    /**
+     * Multiplies this vector with a scalar
+     * @param t a scalar
+     * @return the product of this vector and the scalar
+     */
+    public @NotNull Vec3 times(double t) {
+        return new Vec3(this.x * t, this.y * t, this.z * t);
     }
 
+    /**
+     * Negates this vector.
+     * {@return the negated vector}
+     */
     public @NotNull Vec3 neg() {
         return new Vec3(-x, -y, -z);
     }
 
+    /**
+     * Inverts each component of this vector.
+     * @return the inverted vector.
+     */
     public @NotNull Vec3 inv() {
         return new Vec3(1 / x, 1 / y, 1 / z);
     }
 
-    public @NotNull Vec3 cross(@NotNull Vec3 b) {
+    /**
+     * Computes the cross-product of this and another vector
+     * @param other a vector
+     * @return the cross-product
+     */
+    public @NotNull Vec3 cross(@NotNull Vec3 other) {
         return new Vec3(
-                this.y() * b.z() - b.y() * this.z(),
-                this.z() * b.x() - b.z() * this.x(),
-                this.x() * b.y() - b.x() * this.y()
+                Math.fma(this.y, other.z, - other.y * this.z),
+                Math.fma(this.z, other.x, - other.z * this.x),
+                Math.fma(this.x, other.y, - other.x * this.y)
         );
     }
 
-    public @NotNull Vec3 div(double b) {
-        return new Vec3(this.x / b, this.y / b, this.z / b);
+    /**
+     * Divides this vector by a scalar
+     * @param t a scalar
+     * @return this vector divided by the scalar
+     */
+    public @NotNull Vec3 div(double t) {
+        return new Vec3(this.x / t, this.y / t, this.z / t);
     }
 
+    /**
+     * {@return the squared length of this vector}
+     */
     public double squared() {
         return this.x * this.x + this.y * this.y + this.z * this.z;
     }
 
+    /**
+     * {@return the length of this vector}
+     */
     public double length() {
         return Math.sqrt(squared());
     }
 
+    /**
+     * {@return whether this vector is near zero}
+     */
     public boolean isNearZero() {
         var s = 1e-8;
         return Math.abs(x) < s && Math.abs(y) < s && Math.abs(z) < s;
     }
 
+    /**
+     * {@return a unit vector with the same direction as this vector}
+     */
     public @NotNull Vec3 unit() {
-        return div(length());
+        var squared = squared();
+        if (squared == 1) return this;
+        return div(Math.sqrt(squared));
     }
 
+    /**
+     * {@return the n-th component of this vector}
+     * @param axis the component index
+     */
     public double get(int axis) {
         return switch (axis) {
             case 0 -> x;
@@ -166,4 +303,9 @@ public record Vec3(double x, double y, double z) {
     public @NotNull Vec3 withZ(double z) {
         return new Vec3(x, y, z);
     }
+
+    @Override
+    public @NotNull String toString() {
+        return "(" + x + "," + y + "," + z + ")";
+    }
 }

src/main/java/eu/jonahbauer/raytracing/render/camera/Camera.java

@@ -18,7 +18,13 @@ public interface Camera {
 
     /**
      * Casts a ray through the given pixel.
+     * @param x the pixel x coordinate
+     * @param y the pixel y coordinate
+     * @param i the subpixel x coordinate
+     * @param j the subpixel y coordinate
+     * @param n the subpixel count (per side)
+     * @param random a random number generator
      * @return a new ray
      */
-    @NotNull Ray cast(int x, int y, @NotNull RandomGenerator random);
+    @NotNull Ray cast(int x, int y, int i, int j, int n, @NotNull RandomGenerator random);
 }

src/main/java/eu/jonahbauer/raytracing/render/camera/SimpleCamera.java

@@ -60,7 +60,7 @@ public final class SimpleCamera implements Camera {
 
         this.pixel00 = origin.plus(direction.times(builder.focusDistance))
                 .minus(u.times(0.5 * viewportWidth)).minus(v.times(0.5 * viewportHeight))
-                .plus(pixelU.div(2)).plus(pixelV.div(2));
+                .plus(pixelU.times(.5)).plus(pixelV.times(.5));
     }
 
     /**
@@ -79,13 +79,20 @@ public final class SimpleCamera implements Camera {
 
     /**
      * {@inheritDoc}
+     * @param x {@inheritDoc}
+     * @param y {@inheritDoc}
+     * @param i {@inheritDoc}
+     * @param j {@inheritDoc}
+     * @param n {@inheritDoc}
+     * @param random {@inheritDoc}
      */
-    public @NotNull Ray cast(int x, int y, @NotNull RandomGenerator random) {
+    @Override
+    public @NotNull Ray cast(int x, int y, int i, int j, int n, @NotNull RandomGenerator random) {
         Objects.checkIndex(x, width);
         Objects.checkIndex(y, height);
 
         var origin = getRayOrigin(random);
-        var target = getRayTarget(x, y, random);
+        var target = getRayTarget(x, y, i, j, n, random);
         return new Ray(origin, target.minus(origin));
     }
 
@@ -98,8 +105,8 @@ public final class SimpleCamera implements Camera {
         if (blurRadius <= 0) return origin;
 
         while (true) {
-            var du = 2 * random.nextDouble() - 1;
-            var dv = 2 * random.nextDouble() - 1;
+            var du = Math.fma(2, random.nextDouble(), -1);
+            var dv = Math.fma(2, random.nextDouble(), -1);
             if (du * du + dv * dv >= 1) continue;
 
             var ru = blurRadius * du;
@@ -116,9 +123,10 @@ public final class SimpleCamera implements Camera {
     /**
      * {@return the target vector for a ray through the given pixel} The position is randomized within the pixel.
      */
-    private @NotNull Vec3 getRayTarget(int x, int y, @NotNull RandomGenerator random) {
-        double dx = x + random.nextDouble() - 0.5;
-        double dy = y + random.nextDouble() - 0.5;
+    private @NotNull Vec3 getRayTarget(int x, int y, int i, int j, int n, @NotNull RandomGenerator random) {
+        var factor = 1d / n;
+        var dx = x + Math.fma(factor, i + random.nextDouble(), -0.5);
+        var dy = y + Math.fma(factor, j + random.nextDouble(), -0.5);
         return pixel00.plus(pixelU.times(dx)).plus(pixelV.times(dy));
     }
 

src/main/java/eu/jonahbauer/raytracing/render/material/DielectricMaterial.java

@@ -22,7 +22,7 @@ public record DielectricMaterial(double refractionIndex, @NotNull Texture textur
 
     @Override
     public @NotNull Optional<ScatterResult> scatter(@NotNull Ray ray, @NotNull HitResult hit, @NotNull RandomGenerator random) {
-        var ri = hit.frontFace() ? (1 / refractionIndex) : refractionIndex;
+        var ri = hit.isFrontFace() ? (1 / refractionIndex) : refractionIndex;
 
         var cosTheta = Math.min(- ray.direction().unit().times(hit.normal()), 1.0);
         var reflectance = reflectance(cosTheta);
@@ -32,7 +32,7 @@ public record DielectricMaterial(double refractionIndex, @NotNull Texture textur
                 .orElseGet(() -> Vec3.reflect(ray.direction(), hit.normal()));
 
         var attenuation = texture.get(hit);
-        return Optional.of(new ScatterResult(new Ray(hit.position(), newDirection), attenuation));
+        return Optional.of(new SpecularScatterResult(attenuation, new Ray(hit.position(), newDirection)));
     }
 
     private double reflectance(double cos) {

src/main/java/eu/jonahbauer/raytracing/render/material/DirectionalMaterial.java

@@ -0,0 +1,68 @@
+package eu.jonahbauer.raytracing.render.material;
+
+import eu.jonahbauer.raytracing.math.Ray;
+import eu.jonahbauer.raytracing.math.Vec3;
+import eu.jonahbauer.raytracing.render.texture.Color;
+import eu.jonahbauer.raytracing.render.texture.Texture;
+import eu.jonahbauer.raytracing.scene.HitResult;
+import org.jetbrains.annotations.NotNull;
+import org.jetbrains.annotations.Nullable;
+
+import java.util.Optional;
+import java.util.random.RandomGenerator;
+
+public final class DirectionalMaterial implements Material {
+    private final @Nullable Material front;
+    private final @Nullable Material back;
+
+    private final @NotNull Texture texture;
+
+    public DirectionalMaterial(@Nullable Material front, @Nullable Material back) {
+        if (front == null && back == null) throw new IllegalArgumentException("front and back must not both be null");
+        this.front = front;
+        this.back = back;
+        this.texture = new DirectionalTexture(
+                front != null ? front.texture() : null,
+                back != null ? back.texture() : null
+        );
+    }
+
+    @Override
+    public @NotNull Texture texture() {
+        return texture;
+    }
+
+    @Override
+    public @NotNull Optional<ScatterResult> scatter(@NotNull Ray ray, @NotNull HitResult hit, @NotNull RandomGenerator random) {
+        if (hit.isFrontFace()) {
+            if (front != null) return front.scatter(ray, hit, random);
+        } else {
+            if (back != null) return back.scatter(ray, hit, random);
+        }
+        // let the ray pass through without obstruction
+        return Optional.of(new SpecularScatterResult(Color.WHITE, new Ray(ray.at(hit.t()), ray.direction())));
+    }
+
+    @Override
+    public @NotNull Color emitted(@NotNull HitResult hit) {
+        if (hit.isFrontFace()) {
+            if (front != null) return front.emitted(hit);
+        } else {
+            if (back != null) return back.emitted(hit);
+        }
+        return Material.super.emitted(hit);
+    }
+
+    private record DirectionalTexture(@Nullable Texture front, @Nullable Texture back) implements Texture {
+
+        @Override
+        public @NotNull Color get(double u, double v, @NotNull Vec3 p) {
+            throw new UnsupportedOperationException();
+        }
+
+        @Override
+        public boolean isUVRequired() {
+            return front() != null && front().isUVRequired() || back() != null && back().isUVRequired();
+        }
+    }
+}

src/main/java/eu/jonahbauer/raytracing/render/material/IsotropicMaterial.java

@@ -1,7 +1,7 @@
 package eu.jonahbauer.raytracing.render.material;
 
 import eu.jonahbauer.raytracing.math.Ray;
-import eu.jonahbauer.raytracing.math.Vec3;
+import eu.jonahbauer.raytracing.render.renderer.pdf.SphereProbabilityDensityFunction;
 import eu.jonahbauer.raytracing.render.texture.Color;
 import eu.jonahbauer.raytracing.render.texture.Texture;
 import eu.jonahbauer.raytracing.scene.HitResult;
@@ -13,7 +13,7 @@ import java.util.random.RandomGenerator;
 public record IsotropicMaterial(@NotNull Color albedo) implements Material {
     @Override
     public @NotNull Optional<ScatterResult> scatter(@NotNull Ray ray, @NotNull HitResult hit, @NotNull RandomGenerator random) {
-        return Optional.of(new ScatterResult(new Ray(hit.position(), Vec3.random(random, true)), albedo()));
+        return Optional.of(new PdfScatterResult(albedo(), new SphereProbabilityDensityFunction()));
     }
 
     @Override

src/main/java/eu/jonahbauer/raytracing/render/material/LambertianMaterial.java

@@ -1,7 +1,7 @@
 package eu.jonahbauer.raytracing.render.material;
 
 import eu.jonahbauer.raytracing.math.Ray;
-import eu.jonahbauer.raytracing.math.Vec3;
+import eu.jonahbauer.raytracing.render.renderer.pdf.CosineProbabilityDensityFunction;
 import eu.jonahbauer.raytracing.render.texture.Texture;
 import eu.jonahbauer.raytracing.scene.HitResult;
 import org.jetbrains.annotations.NotNull;
@@ -17,11 +17,7 @@ public record LambertianMaterial(@NotNull Texture texture) implements Material {
 
     @Override
     public @NotNull Optional<ScatterResult> scatter(@NotNull Ray ray, @NotNull HitResult hit, @NotNull RandomGenerator random) {
-        var newDirection = hit.normal().plus(Vec3.random(random, true));
-        if (newDirection.isNearZero()) newDirection = hit.normal();
-
         var attenuation = texture.get(hit);
-        var scattered = new Ray(hit.position(), newDirection);
-        return Optional.of(new ScatterResult(scattered, attenuation));
+        return Optional.of(new PdfScatterResult(attenuation, new CosineProbabilityDensityFunction(hit.normal())));
     }
 }

src/main/java/eu/jonahbauer/raytracing/render/material/Material.java

@@ -1,6 +1,7 @@
 package eu.jonahbauer.raytracing.render.material;
 
 import eu.jonahbauer.raytracing.math.Ray;
+import eu.jonahbauer.raytracing.render.renderer.pdf.ProbabilityDensityFunction;
 import eu.jonahbauer.raytracing.render.texture.Color;
 import eu.jonahbauer.raytracing.render.texture.Texture;
 import eu.jonahbauer.raytracing.scene.HitResult;
@@ -12,18 +13,59 @@ import java.util.random.RandomGenerator;
 
 public interface Material {
 
+    /**
+     * {@return the texture associated with this material}
+     */
     @NotNull Texture texture();
 
+    /**
+     * Scatters a light ray after it hit a surface.
+     * @param ray the incoming light ray
+     * @param hit information about the light ray hitting some object
+     * @param random a random number generator
+     * @return a {@code ScatterResult} if the ray is scattered or an {@linkplain Optional#empty() empty optional} if the
+     * ray is absorbed.
+     */
     @NotNull Optional<ScatterResult> scatter(@NotNull Ray ray, @NotNull HitResult hit, @NotNull RandomGenerator random);
 
+    /**
+     * {@return the color emitted for a given hit}
+     * @implSpec the default implementation returns {@linkplain Color#BLACK black}, i.e. no emission
+     */
     default @NotNull Color emitted(@NotNull HitResult hit) {
         return Color.BLACK;
     }
 
-    record ScatterResult(@NotNull Ray ray, @NotNull Color attenuation) {
-        public ScatterResult {
-            Objects.requireNonNull(ray, "ray");
+    /**
+     * The result of a {@linkplain Material#scatter(Ray, HitResult, RandomGenerator) scattering operation}.
+     */
+    sealed interface ScatterResult {}
+
+    /**
+     * The result of a specular {@linkplain #scatter(Ray, HitResult, RandomGenerator) scattering operation}. A
+     * specular is a scattering operation with a very small number of possible scattered rays (like a
+     * perfect reflection which only has one possible scattered ray).
+     * @param attenuation the attenuation of the scattered light ray
+     * @param ray the scattered light ray
+     */
+    record SpecularScatterResult(@NotNull Color attenuation, @NotNull Ray ray) implements ScatterResult {
+        public SpecularScatterResult {
             Objects.requireNonNull(attenuation, "attenuation");
+            Objects.requireNonNull(ray, "ray");
+        }
+    }
+
+    /**
+     * The result of a probability density function based
+     * {@linkplain #scatter(Ray, HitResult, RandomGenerator) scattering operation}. A probability density function
+     * based scattering operation uses a probability density function to determine the scatter direction.
+     * @param attenuation the attenuation of the scattered light ray
+     * @param pdf the probability density function
+     */
+    record PdfScatterResult(@NotNull Color attenuation, @NotNull ProbabilityDensityFunction pdf) implements ScatterResult {
+        public PdfScatterResult {
+            Objects.requireNonNull(attenuation, "attenuation");
+            Objects.requireNonNull(pdf, "pdf");
         }
     }
 }

src/main/java/eu/jonahbauer/raytracing/render/material/MetallicMaterial.java

@@ -25,9 +25,9 @@ public record MetallicMaterial(@NotNull Texture texture, double fuzz) implements
     public @NotNull Optional<ScatterResult> scatter(@NotNull Ray ray, @NotNull HitResult hit, @NotNull RandomGenerator random) {
         var newDirection = Vec3.reflect(ray.direction(), hit.normal());
         if (fuzz > 0) {
-            newDirection = newDirection.unit().plus(Vec3.random(random, true).times(fuzz));
+            newDirection = Vec3.fma(fuzz, Vec3.random(random), newDirection.unit());
         }
         var attenuation = texture.get(hit);
-        return Optional.of(new ScatterResult(new Ray(hit.position(), newDirection), attenuation));
+        return Optional.of(new SpecularScatterResult(attenuation, new Ray(hit.position(), newDirection)));
     }
 }

src/main/java/eu/jonahbauer/raytracing/render/renderer/SimpleRenderer.java

@@ -2,6 +2,9 @@ package eu.jonahbauer.raytracing.render.renderer;
 
 import eu.jonahbauer.raytracing.math.Range;
 import eu.jonahbauer.raytracing.math.Ray;
+import eu.jonahbauer.raytracing.render.material.Material;
+import eu.jonahbauer.raytracing.render.renderer.pdf.TargetingProbabilityDensityFunction;
+import eu.jonahbauer.raytracing.render.renderer.pdf.MixtureProbabilityDensityFunction;
 import eu.jonahbauer.raytracing.render.texture.Color;
 import eu.jonahbauer.raytracing.render.camera.Camera;
 import eu.jonahbauer.raytracing.render.canvas.Canvas;
@@ -13,8 +16,10 @@ import java.util.SplittableRandom;
 import java.util.random.RandomGenerator;
 import java.util.stream.IntStream;
 
+import static eu.jonahbauer.raytracing.Main.DEBUG;
+
 public final class SimpleRenderer implements Renderer {
-    private final int samplesPerPixel;
+    private final int sqrtSamplesPerPixel;
     private final int maxDepth;
     private final double gamma;
 
@@ -30,7 +35,7 @@ public final class SimpleRenderer implements Renderer {
     }
 
     private SimpleRenderer(@NotNull Builder builder) {
-        this.samplesPerPixel = builder.samplesPerPixel;
+        this.sqrtSamplesPerPixel = (int) Math.sqrt(builder.samplesPerPixel);
         this.maxDepth = builder.maxDepth;
         this.gamma = builder.gamma;
 
@@ -38,6 +43,9 @@ public final class SimpleRenderer implements Renderer {
         this.iterative = builder.iterative;
     }
 
+    /**
+     * {@inheritDoc}
+     */
     @Override
     public void render(@NotNull Camera camera, @NotNull Scene scene, @NotNull Canvas canvas) {
         if (canvas.getWidth() != camera.getWidth() || canvas.getHeight() != camera.getHeight()) {
@@ -45,42 +53,69 @@ public final class SimpleRenderer implements Renderer {
         }
 
         if (iterative) {
-            var random = new Random();
+            renderIterative(camera, scene, canvas);
+        } else {
+            renderNonIterative(camera, scene, canvas);
+        }
+    }
 
-            // render one sample after the other
-            for (int i = 1 ; i <= samplesPerPixel; i++) {
-                var sample = i;
+    /**
+     * Renders the {@code scene} as seen by the {@code camera} to the {@code canvas}, taking one sample per pixel at
+     * a time and updating the canvas after each sample.
+     */
+    private void renderIterative(@NotNull Camera camera, @NotNull Scene scene, @NotNull Canvas canvas) {
+        var random = new Random(0);
+
+        // render one sample after the other
+        int i = 0;
+        for (int sj = 0; sj < sqrtSamplesPerPixel; sj++) {
+            for (int si = 0; si < sqrtSamplesPerPixel; si++) {
+                var sample = ++i;
+                var sif = si;
+                var sjf = sj;
                 getScanlineStream(camera.getHeight(), parallel).forEach(y -> {
                     for (int x = 0; x < camera.getWidth(); x++) {
-                        var ray = camera.cast(x, y, random);
+                        var ray = camera.cast(x, y, sif, sjf, sqrtSamplesPerPixel, random);
                         var c = getColor(scene, ray, random);
                         canvas.set(x, y, Color.average(canvas.get(x, y), c, sample));
                     }
                 });
             }
-
-            // apply gamma correction
-            getScanlineStream(camera.getHeight(), parallel).forEach(y -> {
-                for (int x = 0; x < camera.getWidth(); x++) {
-                    canvas.set(x, y, Color.gamma(canvas.get(x, y), gamma));
-                }
-            });
-        } else {
-            var splittable = new SplittableRandom();
-            // render one pixel after the other
-            getScanlineStream(camera.getHeight(), parallel).forEach(y -> {
-                var random = splittable.split();
-                for (int x = 0; x < camera.getWidth(); x++) {
-                    var color = Color.BLACK;
-                    for (int i = 1; i <= samplesPerPixel; i++) {
-                        var ray = camera.cast(x, y, random);
-                        var c = getColor(scene, ray, random);
-                        color = Color.average(color, c, i);
-                    }
-                    canvas.set(x, y, Color.gamma(color, gamma));
-                }
-            });
         }
+
+        // apply gamma correction
+        getScanlineStream(camera.getHeight(), parallel).forEach(y -> {
+            for (int x = 0; x < camera.getWidth(); x++) {
+                canvas.set(x, y, Color.gamma(canvas.get(x, y), gamma));
+            }
+        });
+    }
+
+    /**
+     * Renders the {@code scene} as seen by the {@code camera} to the {@code canvas}, taking some amount of samples
+     * per pixel and updating the canvas after each pixel.
+     */
+    private void renderNonIterative(@NotNull Camera camera, @NotNull Scene scene, @NotNull Canvas canvas) {
+        var splittable = new SplittableRandom(0);
+        // render one pixel after the other
+        getScanlineStream(camera.getHeight(), parallel).forEach(y -> {
+            var random = splittable.split();
+            for (int x = 0; x < camera.getWidth(); x++) {
+                var color = Color.BLACK;
+                int i = 0;
+                for (int sj = 0; sj < sqrtSamplesPerPixel; sj++) {
+                    for (int si = 0; si < sqrtSamplesPerPixel; si++) {
+                        var ray = camera.cast(x, y, si, sj, sqrtSamplesPerPixel, random);
+                        if (DEBUG) {
+                            System.out.println("Casting ray " + ray + " through pixel (" + x + "," + y + ") at subpixel (" + si + "," + sj + ")...");
+                        }
+                        var c = getColor(scene, ray, random);
+                        color = Color.average(color, c, ++i);
+                    }
+                }
+                canvas.set(x, y, Color.gamma(color, gamma));
+            }
+        });
     }
 
     /**
@@ -95,21 +130,82 @@ public final class SimpleRenderer implements Renderer {
         var attenuation = Color.WHITE;
 
         while (depth-- > 0) {
-            var optional = scene.hit(ray, new Range(0.001, Double.POSITIVE_INFINITY));
+            var optional = scene.hit(ray);
             if (optional.isEmpty()) {
-                color = Color.add(color, Color.multiply(attenuation, scene.getBackgroundColor(ray)));
+                var background = scene.getBackgroundColor(ray);
+                color = Color.add(color, Color.multiply(attenuation, background));
+                if (DEBUG) {
+                    System.out.println("  Hit background: " + background);
+                }
                 break;
             }
 
             var hit = optional.get();
+            if (DEBUG) {
+                System.out.println("  Hit " + hit.target() + " at t=" + hit.t() + " (" + hit.position() + ")");
+            }
             var material = hit.material();
             var emitted = material.emitted(hit);
-            var scatter = material.scatter(ray, hit, random);
+            if (DEBUG && !Color.BLACK.equals(emitted)) {
+                System.out.println("  Emitted: " + emitted);
+            }
+
+            var result = material.scatter(ray, hit, random);
             color = Color.add(color, Color.multiply(attenuation, emitted));
 
-            if (scatter.isEmpty()) break;
-            attenuation = Color.multiply(attenuation, scatter.get().attenuation());
-            ray = scatter.get().ray();
+            if (result.isEmpty()) {
+                if (DEBUG) {
+                    System.out.println("  Absorbed");
+                }
+                break;
+            }
+
+            switch (result.get()) {
+                case Material.SpecularScatterResult(var a, var scattered) -> {
+                    attenuation = Color.multiply(attenuation, a);
+                    ray = scattered;
+
+                    if (DEBUG) {
+                        System.out.println("  Specular scattering with albedo " + a);
+                    }
+                }
+                case Material.PdfScatterResult(var a, var pdf) -> {
+                    if (scene.getTargets() == null) {
+                        attenuation = Color.multiply(attenuation, a);
+                        ray = new Ray(hit.position(), pdf.generate(random));
+
+                        if (DEBUG) {
+                            System.out.println("  Pdf scattering with albedo " + a);
+                        }
+                    } else {
+                        var mixed = new MixtureProbabilityDensityFunction(new TargetingProbabilityDensityFunction(hit.position(), scene.getTargets()), pdf, 0.5);
+                        var direction = mixed.generate(random).unit();
+
+                        var idealPdf = pdf.value(direction);
+                        var actualPdf = mixed.value(direction);
+                        if (actualPdf == 0) break; // when actualPdf is 0, the ray should have never been generated by mixed.generate
+
+                        var factor =  idealPdf / actualPdf;
+
+                        attenuation = Color.multiply(attenuation, Color.multiply(a, factor));
+                        ray = new Ray(hit.position(), direction);
+
+                        if (DEBUG) {
+                            System.out.println("  Pdf scattering with albedo " + a + " and factor " + factor);
+                        }
+                    }
+                }
+            }
+
+            if (DEBUG) {
+                System.out.println("  Combined color is " + color);
+                System.out.println("  Combined attenuation is " + attenuation);
+                System.out.println("  New ray is " + ray);
+            }
+        }
+
+        if (DEBUG) {
+            System.out.println("  Final color is " + color);
         }
 
         return color;

src/main/java/eu/jonahbauer/raytracing/render/renderer/pdf/CosineProbabilityDensityFunction.java

@@ -0,0 +1,27 @@
+package eu.jonahbauer.raytracing.render.renderer.pdf;
+
+import eu.jonahbauer.raytracing.math.Vec3;
+import org.jetbrains.annotations.NotNull;
+
+import java.util.Objects;
+import java.util.random.RandomGenerator;
+
+public record CosineProbabilityDensityFunction(@NotNull Vec3 normal) implements ProbabilityDensityFunction {
+
+    public CosineProbabilityDensityFunction {
+        Objects.requireNonNull(normal, "normal");
+        normal = normal.unit();
+    }
+
+    @Override
+    public double value(@NotNull Vec3 direction) {
+        var cos = normal.times(direction);
+        return Math.max(0, cos / Math.PI);
+    }
+
+    @Override
+    public @NotNull Vec3 generate(@NotNull RandomGenerator random) {
+        var out = normal().plus(Vec3.random(random));
+        return out.isNearZero() ? normal() : out;
+    }
+}

src/main/java/eu/jonahbauer/raytracing/render/renderer/pdf/MixtureProbabilityDensityFunction.java

@@ -0,0 +1,46 @@
+package eu.jonahbauer.raytracing.render.renderer.pdf;
+
+import eu.jonahbauer.raytracing.math.Vec3;
+import org.jetbrains.annotations.NotNull;
+
+import java.util.Objects;
+import java.util.random.RandomGenerator;
+
+/**
+ * Mixes between two probability density functions (pdf) using a weight. When the weight is closer to zero, the
+ * influence of the second pdf is stronger. When the weight is closer to one, the influence of the first pdf is stronger.
+ * @param a the first probability density function
+ * @param b the second probability density function
+ * @param weight a weight in the range [0, 1]
+ */
+public record MixtureProbabilityDensityFunction(
+        @NotNull ProbabilityDensityFunction a,
+        @NotNull ProbabilityDensityFunction b,
+        double weight
+) implements ProbabilityDensityFunction {
+    public MixtureProbabilityDensityFunction(@NotNull ProbabilityDensityFunction a, @NotNull ProbabilityDensityFunction b) {
+        this(a, b, 0.5);
+    }
+
+    public MixtureProbabilityDensityFunction {
+        Objects.requireNonNull(a);
+        Objects.requireNonNull(b);
+        weight = Math.clamp(weight, 0, 1);
+    }
+
+    @Override
+    public double value(@NotNull Vec3 direction) {
+        var v = a.value(direction);
+        var w = b.value(direction);
+        return Math.fma(weight, v, Math.fma(-weight, w, w));
+    }
+
+    @Override
+    public @NotNull Vec3 generate(@NotNull RandomGenerator random) {
+        if (random.nextDouble() < weight) {
+            return a.generate(random);
+        } else {
+            return b.generate(random);
+        }
+    }
+}

src/main/java/eu/jonahbauer/raytracing/render/renderer/pdf/ProbabilityDensityFunction.java

@@ -0,0 +1,25 @@
+package eu.jonahbauer.raytracing.render.renderer.pdf;
+
+import eu.jonahbauer.raytracing.math.Vec3;
+import org.jetbrains.annotations.NotNull;
+
+import java.util.random.RandomGenerator;
+
+/**
+ * A probability density function used for sampling random directions when scattering a ray.
+ */
+public interface ProbabilityDensityFunction {
+
+    /**
+     * {@return the value of this probability density function at the given point}
+     * @param direction the direction
+     */
+    double value(@NotNull Vec3 direction);
+
+    /**
+     * Generates a random direction based on this probability density function.
+     * @param random a random number generator
+     * @return the random direction
+     */
+    @NotNull Vec3 generate(@NotNull RandomGenerator random);
+}

src/main/java/eu/jonahbauer/raytracing/render/renderer/pdf/SphereProbabilityDensityFunction.java

@@ -0,0 +1,22 @@
+package eu.jonahbauer.raytracing.render.renderer.pdf;
+
+import eu.jonahbauer.raytracing.math.Vec3;
+import org.jetbrains.annotations.NotNull;
+
+import java.util.random.RandomGenerator;
+
+/**
+ * A probability density function sampling the sphere uniformly.
+ */
+public record SphereProbabilityDensityFunction() implements ProbabilityDensityFunction {
+
+    @Override
+    public double value(@NotNull Vec3 direction) {
+        return 1 / (4 * Math.PI);
+    }
+
+    @Override
+    public @NotNull Vec3 generate(@NotNull RandomGenerator random) {
+        return Vec3.random(random);
+    }
+}

src/main/java/eu/jonahbauer/raytracing/render/renderer/pdf/TargetingProbabilityDensityFunction.java

@@ -0,0 +1,41 @@
+package eu.jonahbauer.raytracing.render.renderer.pdf;
+
+import eu.jonahbauer.raytracing.math.Vec3;
+import eu.jonahbauer.raytracing.scene.Target;
+import org.jetbrains.annotations.NotNull;
+
+import java.util.ArrayList;
+import java.util.List;
+import java.util.Objects;
+import java.util.random.RandomGenerator;
+
+/**
+ * A probability density function targeting a target.
+ * @see Target
+ */
+public final class TargetingProbabilityDensityFunction implements ProbabilityDensityFunction {
+    private final @NotNull Vec3 origin;
+    private final @NotNull List<@NotNull Target> targets;
+
+    public TargetingProbabilityDensityFunction(@NotNull Vec3 origin, @NotNull List<@NotNull Target> targets) {
+        this.origin = Objects.requireNonNull(origin, "origin");
+        this.targets = new ArrayList<>(targets);
+    }
+
+    @Override
+    public double value(@NotNull Vec3 direction) {
+        var weight = 1d / targets.size();
+        var sum = 0.0;
+
+        for (var target : targets) {
+            sum = Math.fma(weight, target.getProbabilityDensity(origin, direction), sum);
+        }
+
+        return sum;
+    }
+
+    @Override
+    public @NotNull Vec3 generate(@NotNull RandomGenerator random) {
+        return targets.get(random.nextInt(targets.size())).getTargetingDirection(origin, random);
+    }
+}

src/main/java/eu/jonahbauer/raytracing/render/texture/CheckerTexture.java

@@ -5,7 +5,6 @@ import org.jetbrains.annotations.NotNull;
 
 public record CheckerTexture(double scale, @NotNull Texture even, @NotNull Texture odd) implements Texture {
 
-
     @Override
     public @NotNull Color get(double u, double v, @NotNull Vec3 p) {
         var x = (int) Math.floor(p.x() / scale);

src/main/java/eu/jonahbauer/raytracing/render/texture/Color.java

@@ -1,11 +1,15 @@
 package eu.jonahbauer.raytracing.render.texture;
 
+import eu.jonahbauer.raytracing.math.Ray;
 import eu.jonahbauer.raytracing.math.Vec3;
+import eu.jonahbauer.raytracing.scene.SkyBox;
 import org.jetbrains.annotations.NotNull;
 
 import java.util.Random;
 
-public record Color(double r, double g, double b) implements Texture {
+import static eu.jonahbauer.raytracing.Main.DEBUG;
+
+public record Color(double r, double g, double b) implements Texture, SkyBox {
     public static final @NotNull Color BLACK = new Color(0.0, 0.0, 0.0);
     public static final @NotNull Color WHITE = new Color(1.0, 1.0, 1.0);
     public static final @NotNull Color RED = new Color(1.0, 0.0, 0.0);
@@ -26,6 +30,10 @@ public record Color(double r, double g, double b) implements Texture {
         return new Color(a.r() * b.r(), a.g() * b.g(), a.b() * b.b());
     }
 
+    public static @NotNull Color multiply(@NotNull Color a, double b) {
+        return new Color(a.r() * b, a.g() * b, a.b() * b);
+    }
+
     public static @NotNull Color add(@NotNull Color a, @NotNull Color b) {
         return new Color(a.r() + b.r(), a.g() + b.g(), a.b() + b.b());
     }
@@ -44,10 +52,11 @@ public record Color(double r, double g, double b) implements Texture {
     }
 
     public static @NotNull Color average(@NotNull Color current, @NotNull Color next, int index) {
+        var factor = 1d / index;
         return new Color(
-                current.r() + (next.r() - current.r()) / index,
-                current.g() + (next.g() - current.g()) / index,
-                current.b() + (next.b() - current.b()) / index
+                Math.fma(factor, next.r() - current.r(), current.r()),
+                Math.fma(factor, next.g() - current.g(), current.g()),
+                Math.fma(factor, next.b() - current.b(), current.b())
         );
     }
 
@@ -77,6 +86,17 @@ public record Color(double r, double g, double b) implements Texture {
         this(red / 255f, green / 255f, blue / 255f);
     }
 
+    public Color {
+        if (DEBUG) {
+            if (!Double.isFinite(r) || !Double.isFinite(g) || !Double.isFinite(b)) {
+                throw new IllegalArgumentException("r, g and b must be finite");
+            }
+            if (r < 0 || g < 0 || b < 0) {
+                throw new IllegalArgumentException("r, g and b must be non-negative");
+            }
+        }
+    }
+
     public int red() {
         return toInt(r);
     }
@@ -94,6 +114,11 @@ public record Color(double r, double g, double b) implements Texture {
         return this;
     }
 
+    @Override
+    public @NotNull Color getColor(@NotNull Ray ray) {
+        return this;
+    }
+
     @Override
     public boolean isUVRequired() {
         return false;

src/main/java/eu/jonahbauer/raytracing/render/texture/PerlinTexture.java

@@ -53,7 +53,7 @@ public final class PerlinTexture implements Texture {
         this.mask = count - 1;
         this.randvec = new Vec3[count];
         for (int i = 0; i < count; i++) {
-            this.randvec[i] = Vec3.random(random, true);
+            this.randvec[i] = Vec3.random(random);
         }
         this.permX = generatePerm(count, random);
         this.permY = generatePerm(count, random);

src/main/java/eu/jonahbauer/raytracing/render/texture/Texture.java

@@ -5,12 +5,27 @@ import eu.jonahbauer.raytracing.scene.HitResult;
 import org.jetbrains.annotations.NotNull;
 
 public interface Texture {
+    /**
+     * {@return the color of <code>this</code> texture for a hit}
+     */
     default @NotNull Color get(@NotNull HitResult hit) {
         return get(hit.u(), hit.v(), hit.position());
     }
 
+    /**
+     * {@return the color of <code>this</code> texture at the specified position}
+     * @param u the texture u coordinate
+     * @param v the texture v coordinate
+     * @param p the position
+     */
     @NotNull Color get(double u, double v, @NotNull Vec3 p);
 
+    /**
+     * Returns whether {@link #get(double, double, Vec3)} uses the {@code u} and/or {@code v} parameters.
+     * When a texture indicates that the {@code u} and {@code v} coordinates are not required, the calculation may be
+     * skipped and {@link Double#NaN} will be passed.
+     * @return whether {@link #get(double, double, Vec3)} uses the {@code u} and/or {@code v} parameters
+     */
     default boolean isUVRequired() {
         return true;
     }

src/main/java/eu/jonahbauer/raytracing/scene/HitResult.java

@@ -1,14 +1,28 @@
 package eu.jonahbauer.raytracing.scene;
 
+import eu.jonahbauer.raytracing.math.Range;
+import eu.jonahbauer.raytracing.math.Ray;
 import eu.jonahbauer.raytracing.math.Vec3;
 import eu.jonahbauer.raytracing.render.material.Material;
+import eu.jonahbauer.raytracing.render.texture.Texture;
 import org.jetbrains.annotations.NotNull;
 
 import java.util.Objects;
 
+/**
+ * The result of a {@linkplain Hittable#hit(Ray, Range) hit}.
+ * @param t the {@code t} value at which the hit occurs
+ * @param position the position of the hit
+ * @param normal the surface normal at the hit position
+ * @param target the hit target (for debug purposes only)
+ * @param material the material of the surface
+ * @param u the texture u coordinate (or {@code Double.NaN} if the {@linkplain Material#texture() material's texture} does {@linkplain Texture#isUVRequired() not depend} on the uv-coordinates)
+ * @param v the texture v coordinate (or {@code Double.NaN} if the {@linkplain Material#texture() material's texture} does {@linkplain Texture#isUVRequired() not depend} on the uv-coordinates)
+ * @param isFrontFace whether the front or the back of the surface was it
+ */
 public record HitResult(
-        double t, @NotNull Vec3 position, @NotNull Vec3 normal,
-        @NotNull Material material, double u, double v, boolean frontFace
+        double t, @NotNull Vec3 position, @NotNull Vec3 normal, @NotNull Hittable target,
+        @NotNull Material material, double u, double v, boolean isFrontFace
 ) implements Comparable<HitResult> {
     public HitResult {
         Objects.requireNonNull(position, "position");
@@ -16,7 +30,7 @@ public record HitResult(
     }
 
     public @NotNull HitResult withPositionAndNormal(@NotNull Vec3 position, @NotNull Vec3 normal) {
-        return new HitResult(t, position, normal, material, u, v, frontFace);
+        return new HitResult(t, position, normal, target, material, u, v, isFrontFace);
     }
 
     @Override

src/main/java/eu/jonahbauer/raytracing/scene/Hittable.java

@@ -11,21 +11,38 @@ import org.jetbrains.annotations.NotNull;
 import java.util.Optional;
 
 public interface Hittable {
+    @NotNull Range FORWARD = new Range(0.001, Double.POSITIVE_INFINITY);
 
     /**
-     * {@return the value <code>t</code> such that <code>ray.at(t)</code> is the intersection of this shaped closest to
-     * the ray origin, or <code>Double.NaN</code> if the ray does not intersect this shape}
+     * @see #hit(Ray, Range)
+     */
+    default @NotNull Optional<HitResult> hit(@NotNull Ray ray) {
+        return hit(ray, FORWARD);
+    }
+
+    /**
+     * Tests whether the {@code ray} intersects {@code this} hittable.
+     * <p>
+     * The second parameter {@code range} allows the implementation to skip unnecessary calculations if it can
+     * determine that a hit (if any) will fall outside the valid range of {@code t}s. The returned hit may still be
+     * outside the valid range and has to be checked by the caller.
      * @param ray a ray
+     * @param range the range of valid {@code t}s
+     * @return the result of the hit test, containing (among others) the value {@code t} such that {@code ray.at(t)} is
+     *         a point on {@code this} hittable
      */
     @NotNull Optional<HitResult> hit(@NotNull Ray ray, @NotNull Range range);
 
+    /**
+     * {@return the axis-aligned bounding box of this hittable}
+     */
     @NotNull AABB getBoundingBox();
 
     default @NotNull Hittable translate(@NotNull Vec3 offset) {
-        return new Translate(this, offset);
+        return Translate.create(this, offset);
     }
 
     default @NotNull Hittable rotateY(double angle) {
-        return new RotateY(this, angle);
+        return RotateY.create(this, angle);
     }
 }

src/main/java/eu/jonahbauer/raytracing/scene/Scene.java

@@ -6,7 +6,9 @@ import eu.jonahbauer.raytracing.render.texture.Color;
 import eu.jonahbauer.raytracing.scene.util.HittableBinaryTree;
 import eu.jonahbauer.raytracing.scene.util.HittableCollection;
 import org.jetbrains.annotations.NotNull;
+import org.jetbrains.annotations.Nullable;
 
+import java.util.ArrayList;
 import java.util.List;
 import java.util.Objects;
 
@@ -14,29 +16,28 @@ public final class Scene extends HittableCollection {
     private final @NotNull HittableCollection objects;
     private final @NotNull SkyBox background;
 
+    private final @Nullable List<@NotNull Target> targets;
+
     public Scene(@NotNull List<? extends @NotNull Hittable> objects) {
-        this(Color.BLACK, objects);
+        this(objects, null);
     }
 
-    public Scene(@NotNull Color background, @NotNull List<? extends @NotNull Hittable> objects) {
-        this(SkyBox.solid(background), objects);
+    public Scene(@NotNull List<? extends @NotNull Hittable> objects, @Nullable List<? extends @NotNull Target> targets) {
+        this(Color.BLACK, objects, targets);
     }
 
     public Scene(@NotNull SkyBox background, @NotNull List<? extends @NotNull Hittable> objects) {
-        this.objects = new HittableBinaryTree(objects);
+        this(background, objects, null);
+    }
+
+    public Scene(@NotNull SkyBox background, @NotNull List<? extends @NotNull Hittable> objects, @Nullable List<? extends @NotNull Target> targets) {
+        var list = new ArrayList<Hittable>(objects.size() + (targets != null ? targets.size() : 0));
+        list.addAll(objects);
+        if (targets != null) list.addAll(targets);
+        this.objects = new HittableBinaryTree(list);
         this.background = Objects.requireNonNull(background);
-    }
 
-    public Scene(@NotNull Hittable @NotNull... objects) {
-        this(List.of(objects));
-    }
-
-    public Scene(@NotNull Color background, @NotNull Hittable @NotNull... objects) {
-        this(background, List.of(objects));
-    }
-
-    public Scene(@NotNull SkyBox background, @NotNull Hittable @NotNull... objects) {
-        this(background, List.of(objects));
+        this.targets = targets != null ? List.copyOf(targets) : null;
     }
 
     @Override
@@ -49,6 +50,10 @@ public final class Scene extends HittableCollection {
         return objects.getBoundingBox();
     }
 
+    public @Nullable List<@NotNull Target> getTargets() {
+        return targets;
+    }
+
     public @NotNull Color getBackgroundColor(@NotNull Ray ray) {
         return background.getColor(ray);
     }

src/main/java/eu/jonahbauer/raytracing/scene/SkyBox.java

@@ -18,8 +18,4 @@ public interface SkyBox {
             return Color.lerp(bottom, top, alt / Math.PI + 0.5);
         };
     }
-
-    static @NotNull SkyBox solid(@NotNull Color color) {
-        return _ -> color;
-    }
 }

src/main/java/eu/jonahbauer/raytracing/scene/Target.java

@@ -0,0 +1,39 @@
+package eu.jonahbauer.raytracing.scene;
+
+import eu.jonahbauer.raytracing.math.Vec3;
+import eu.jonahbauer.raytracing.render.renderer.pdf.TargetingProbabilityDensityFunction;
+import org.jetbrains.annotations.NotNull;
+
+import java.util.random.RandomGenerator;
+
+/**
+ * An interface for objects that can be targeted. A target can construct randomly distributed directions in which
+ * it will be hit from a given origin.
+ * @see TargetingProbabilityDensityFunction
+ */
+public interface Target extends Hittable {
+    /**
+     * Returns the probability density for a direction as sampled by {@link #getTargetingDirection(Vec3, RandomGenerator)}.
+     * @param origin the origin
+     * @param direction the direction
+     * @return the probability density for a direction as sampled by {@link #getTargetingDirection(Vec3, RandomGenerator)}
+     */
+    double getProbabilityDensity(@NotNull Vec3 origin, @NotNull Vec3 direction);
+
+    /**
+     * {@return a vector targeting this hittable from the <code>origin</code>} The vector is chosen randomly.
+     * @param origin the origin
+     * @param random a random number generator
+     */
+    @NotNull Vec3 getTargetingDirection(@NotNull Vec3 origin, @NotNull RandomGenerator random);
+
+    @Override
+    default @NotNull Target translate(@NotNull Vec3 offset) {
+        return (Target) Hittable.super.translate(offset);
+    }
+
+    @Override
+    default @NotNull Target rotateY(double angle) {
+        return (Target) Hittable.super.rotateY(angle);
+    }
+}

src/main/java/eu/jonahbauer/raytracing/scene/hittable2d/Hittable2D.java

@@ -52,10 +52,40 @@ public abstract class Hittable2D implements Hittable {
 
         var frontFace = denominator < 0;
         return Optional.of(new HitResult(
-                t, position, frontFace ? normal : normal.neg(),
+                t, position, frontFace ? normal : normal.neg(), this,
                 material, alpha, beta, frontFace
         ));
     }
 
+    protected double hit0(@NotNull Ray ray, @NotNull Range range) {
+        var denominator = ray.direction().times(normal);
+        if (Math.abs(denominator) < 1e-8) return Double.NaN; // parallel
+
+        var t = (d - ray.origin().times(normal)) / denominator;
+        if (!range.surrounds(t)) return Double.NaN;
+
+        var position = ray.at(t);
+        var p = position.minus(origin);
+
+        var alpha = Vec3.tripleProduct(w, p, v);
+        var beta = Vec3.tripleProduct(w, u, p);
+        if (!isInterior(alpha, beta)) return Double.NaN;
+
+        return t;
+    }
+
+    protected @NotNull Vec3 get(double alpha, double beta) {
+        return new Vec3(
+                Math.fma(beta, v.x(), Math.fma(alpha, u.x(), origin.x())),
+                Math.fma(beta, v.y(), Math.fma(alpha, u.y(), origin.y())),
+                Math.fma(beta, v.z(), Math.fma(alpha, u.z(), origin.z()))
+        );
+    }
+
     protected abstract boolean isInterior(double alpha, double beta);
+
+    @Override
+    public @NotNull String toString() {
+        return this.getClass().getSimpleName() + "(origin=" + origin + ", u=" + u + ", v=" + v + ")";
+    }
 }

src/main/java/eu/jonahbauer/raytracing/scene/hittable2d/Parallelogram.java

@@ -1,11 +1,16 @@
 package eu.jonahbauer.raytracing.scene.hittable2d;
 
 import eu.jonahbauer.raytracing.math.AABB;
+import eu.jonahbauer.raytracing.math.Ray;
 import eu.jonahbauer.raytracing.math.Vec3;
 import eu.jonahbauer.raytracing.render.material.Material;
+import eu.jonahbauer.raytracing.scene.Target;
+import eu.jonahbauer.raytracing.scene.util.PdfUtil;
 import org.jetbrains.annotations.NotNull;
 
-public final class Parallelogram extends Hittable2D {
+import java.util.random.RandomGenerator;
+
+public final class Parallelogram extends Hittable2D implements Target {
     private final @NotNull AABB bbox;
 
     public Parallelogram(@NotNull Vec3 origin, @NotNull Vec3 u, @NotNull Vec3 v, @NotNull Material material) {
@@ -22,4 +27,24 @@ public final class Parallelogram extends Hittable2D {
     public @NotNull AABB getBoundingBox() {
         return bbox;
     }
+
+    @Override
+    public double getProbabilityDensity(@NotNull Vec3 origin, @NotNull Vec3 direction) {
+        if (Double.isNaN(hit0(new Ray(origin, direction), FORWARD))) return 0;
+
+        var o = this.origin.minus(origin);
+        var a = o.unit();
+        var b = o.plus(u).unit();
+        var c = o.plus(v).unit();
+        var d = o.plus(u).plus(v).unit();
+        var angle = PdfUtil.getSolidAngle(a, b, d) + PdfUtil.getSolidAngle(c, b, d);
+        return 1 / angle;
+    }
+
+    @Override
+    public @NotNull Vec3 getTargetingDirection(@NotNull Vec3 origin, @NotNull RandomGenerator random) {
+        var alpha = random.nextDouble();
+        var beta = random.nextDouble();
+        return get(alpha, beta).minus(origin);
+    }
 }

src/main/java/eu/jonahbauer/raytracing/scene/hittable3d/Box.java

@@ -7,22 +7,46 @@ import eu.jonahbauer.raytracing.math.Vec3;
 import eu.jonahbauer.raytracing.render.material.Material;
 import eu.jonahbauer.raytracing.scene.HitResult;
 import eu.jonahbauer.raytracing.scene.Hittable;
+import eu.jonahbauer.raytracing.scene.Target;
+import eu.jonahbauer.raytracing.scene.util.PdfUtil;
 import org.jetbrains.annotations.NotNull;
+import org.jetbrains.annotations.Nullable;
 
 import java.util.Objects;
 import java.util.Optional;
+import java.util.random.RandomGenerator;
 
-public record Box(@NotNull AABB box, @NotNull Material material) implements Hittable {
-
-    public Box {
-        Objects.requireNonNull(box, "box");
-        Objects.requireNonNull(material, "material");
-    }
+public final class Box implements Hittable, Target {
+    private final @NotNull AABB box;
+    private final @Nullable Material @NotNull[] materials;
 
     public Box(@NotNull Vec3 a, @NotNull Vec3 b, @NotNull Material material) {
         this(new AABB(a, b), material);
     }
 
+    public Box(@NotNull AABB box, @NotNull Material material) {
+        this(box, Objects.requireNonNull(material, "material"), material, material, material, material, material);
+    }
+
+    public Box(
+            @NotNull Vec3 a, @NotNull Vec3 b,
+            @Nullable Material top, @Nullable Material bottom,
+            @Nullable Material left, @Nullable Material right,
+            @Nullable Material front, @Nullable Material back
+    ) {
+        this(new AABB(a, b), top, bottom, left, right, front, back);
+    }
+
+    public Box(
+            @NotNull AABB box,
+            @Nullable Material top, @Nullable Material bottom,
+            @Nullable Material left, @Nullable Material right,
+            @Nullable Material front, @Nullable Material back
+    ) {
+        this.box = Objects.requireNonNull(box, "box");
+        this.materials = new Material[] { left, bottom, back, right, top, front };
+    }
+
     @Override
     public @NotNull Optional<HitResult> hit(@NotNull Ray ray, @NotNull Range range) {
         // based on AABB#hit with additional detection of the side hit
@@ -70,25 +94,32 @@ public record Box(@NotNull AABB box, @NotNull Material material) implements Hitt
     }
 
     private @NotNull Optional<HitResult> hit0(double tmin, double tmax, @NotNull Side entry, @NotNull Side exit, @NotNull Ray ray, @NotNull Range range) {
-        boolean frontFace;
         double t;
-        if (range.surrounds(tmin)) {
-            frontFace = true;
+        Side side;
+        boolean frontFace;
+        Material material;
+        Vec3 normal;
+        if (range.surrounds(tmin) && materials[entry.ordinal()] != null) {
             t = tmin;
-        } else if (range.surrounds(tmax)) {
-            frontFace = false;
+            side = entry;
+            frontFace = true;
+            material = materials[side.ordinal()];
+            normal = side.normal;
+        } else if (range.surrounds(tmax) && materials[exit.ordinal()] != null) {
             t = tmax;
+            side = exit;
+            frontFace = false;
+            material = materials[side.ordinal()];
+            normal = side.normal.neg();
         } else {
             return Optional.empty();
         }
 
-        var side = frontFace ? entry : exit;
-        var normal = frontFace ? side.normal : side.normal.neg();
         var position = ray.at(t);
-        var uv = material().texture().isUVRequired();
+        var uv = material.texture().isUVRequired();
         var u = uv ? side.getTextureU(box, position) : Double.NaN;
         var v = uv ? side.getTextureV(box, position) : Double.NaN;
-        return Optional.of(new HitResult(t, position, normal, material, u, v, frontFace));
+        return Optional.of(new HitResult(t, position, normal, this, material, u, v, frontFace));
     }
 
     @Override
@@ -96,6 +127,66 @@ public record Box(@NotNull AABB box, @NotNull Material material) implements Hitt
         return box;
     }
 
+    @Override
+    public double getProbabilityDensity(@NotNull Vec3 origin, @NotNull Vec3 direction) {
+        if (contains(origin)) return 1 / (4 * Math.PI);
+        if (hit(new Ray(origin, direction)).isEmpty()) return 0;
+
+        var solidAngle = 0d;
+        for (var s : Side.values()) {
+            if (!s.isExterior(box, origin)) continue;
+            solidAngle += s.getSolidAngle(box, origin);
+        }
+
+        return 1 / solidAngle;
+    }
+
+    @Override
+    public @NotNull Vec3 getTargetingDirection(@NotNull Vec3 origin, @NotNull RandomGenerator random) {
+        if (contains(origin)) return Vec3.random(random);
+
+        // determine sides facing the origin and their solid angles
+        int visible = 0;
+
+        // at most three faces are visible
+        Side[] sides = new Side[3];
+        double[] solidAngle = new double[3];
+
+        double accumSolidAngle = 0;
+        for (var s : Side.values()) {
+            if (!s.isExterior(box, origin)) continue;
+
+            var sa = s.getSolidAngle(box, origin);
+            accumSolidAngle += sa;
+            sides[visible] = s;
+            solidAngle[visible] = accumSolidAngle;
+            visible++;
+        }
+
+        // choose a random side facing the origin based on their relative solid angles
+        var r = random.nextDouble() * solidAngle[visible - 1];
+        for (int j = 0; j < visible; j++) {
+            if (r < solidAngle[j]) {
+                // choose a random point on that side
+                var target = sides[j].random(box, random);
+                return target.minus(origin);
+            }
+        }
+
+        throw new AssertionError();
+    }
+
+    private boolean contains(@NotNull Vec3 point) {
+        return box.min().x() < point.x() && point.x() < box.max().x()
+               && box.min().y() < point.y() && point.y() < box.max().y()
+               && box.min().z() < point.z() && point.z() < box.max().z();
+    }
+
+    @Override
+    public @NotNull String toString() {
+        return "Box(min=" + box.min() + ", max=" + box.max() + ")";
+    }
+
     private enum Side {
         NEG_X(Vec3.UNIT_X.neg()),
         NEG_Y(Vec3.UNIT_Y.neg()),
@@ -136,5 +227,78 @@ public record Box(@NotNull AABB box, @NotNull Material material) implements Hitt
                 case POS_Y -> (box.max().z() - pos.z()) / (box.max().z() - box.min().z());
             };
         }
+
+        /**
+         * {@return whether the given position is outside of the box only considering <code>this</code> side}
+         */
+        public boolean isExterior(@NotNull AABB box, @NotNull Vec3 pos) {
+            return switch (this) {
+                case NEG_X -> pos.x() < box.min().x();
+                case NEG_Y -> pos.y() < box.min().y();
+                case NEG_Z -> pos.z() < box.min().z();
+                case POS_X -> pos.x() > box.max().x();
+                case POS_Y -> pos.y() > box.max().y();
+                case POS_Z -> pos.z() > box.max().z();
+            };
+        }
+
+        /**
+         * {@return the point on <code>this</code> side of the <code>box</code> with the given <code>u</code>-<code>v</code>-coordinates}
+         */
+        public @NotNull Vec3 get(@NotNull AABB box, double u, double v) {
+            return switch (this) {
+                case NEG_X -> new Vec3(
+                        box.min().x(),
+                        Math.fma(v, box.max().y() - box.min().y(), box.min().y()),
+                        Math.fma(u, box.max().z() - box.min().z(), box.min().z())
+                );
+                case NEG_Y -> new Vec3(
+                        Math.fma(u, box.max().x() - box.min().x(), box.min().x()),
+                        box.min().y(),
+                        Math.fma(v, box.max().z() - box.min().z(), box.min().z())
+                );
+                case NEG_Z -> new Vec3(
+                        Math.fma(u, box.min().x() - box.max().x(), box.max().x()),
+                        Math.fma(v, box.max().y() - box.min().y(), box.min().y()),
+                        box.min().z()
+                );
+                case POS_X -> new Vec3(
+                        box.max().x(),
+                        Math.fma(v, box.max().y() - box.min().y(), box.min().y()),
+                        Math.fma(u, box.min().z() - box.max().z(), box.max().z())
+                );
+                case POS_Y -> new Vec3(
+                        Math.fma(u, box.max().x() - box.min().x(), box.min().x()),
+                        box.max().y(),
+                        Math.fma(v, box.min().z() - box.max().z(), box.max().z())
+                );
+                case POS_Z -> new Vec3(
+                        Math.fma(u, box.max().x() - box.min().x(), box.min().x()),
+                        Math.fma(v, box.max().y() - box.min().y(), box.min().y()),
+                        box.max().z()
+                );
+            };
+        }
+
+        /**
+         * {@return a random point on <code>this</code> side of the <code>box</code>}
+         */
+        public @NotNull Vec3 random(@NotNull AABB box, @NotNull RandomGenerator random) {
+            var u = random.nextDouble();
+            var v = random.nextDouble();
+            return get(box, u, v);
+        }
+
+        /**
+         * {@return the solid angle covered by <code>this</code> side of the <code>box</code> when viewed from <code>pos</code>}
+         */
+        public double getSolidAngle(@NotNull AABB box, @NotNull Vec3 pos) {
+            var a = get(box, 0, 0).minus(pos).unit();
+            var b = get(box, 0, 1).minus(pos).unit();
+            var c = get(box, 1, 1).minus(pos).unit();
+            var d = get(box, 1, 0).minus(pos).unit();
+
+            return PdfUtil.getSolidAngle(a, b, d) + PdfUtil.getSolidAngle(c, b, d);
+        }
     }
 }

src/main/java/eu/jonahbauer/raytracing/scene/hittable3d/ConstantMedium.java

@@ -32,7 +32,7 @@ public record ConstantMedium(@NotNull Hittable boundary, double density, @NotNul
         if (hitDistance > distance) return Optional.empty();
 
         var t = tmin + hitDistance / length;
-        return Optional.of(new HitResult(t, ray.at(t), Vec3.UNIT_X, material, 0, 0, true)); // arbitrary normal and frontFace
+        return Optional.of(new HitResult(t, ray.at(t), Vec3.UNIT_X, this, material, 0, 0, true)); // arbitrary normal, u, v and isFrontFace
     }
 
     @Override

src/main/java/eu/jonahbauer/raytracing/scene/hittable3d/Sphere.java

@@ -7,24 +7,32 @@ import eu.jonahbauer.raytracing.math.Ray;
 import eu.jonahbauer.raytracing.math.Vec3;
 import eu.jonahbauer.raytracing.scene.HitResult;
 import eu.jonahbauer.raytracing.scene.Hittable;
+import eu.jonahbauer.raytracing.scene.Target;
 import org.jetbrains.annotations.NotNull;
 
 import java.util.Objects;
 import java.util.Optional;
+import java.util.random.RandomGenerator;
 
-public final class Sphere implements Hittable {
+public final class Sphere implements Hittable, Target {
     private final @NotNull Vec3 center;
     private final double radius;
     private final @NotNull Material material;
 
     private final @NotNull AABB bbox;
 
+    private final @NotNull Vec3 normalizedCenter;
+    private final double invRadius;
+
     public Sphere(@NotNull Vec3 center, double radius, @NotNull Material material) {
         this.center = Objects.requireNonNull(center, "center");
         this.material = Objects.requireNonNull(material, "material");
         if (radius <= 0 || !Double.isFinite(radius)) throw new IllegalArgumentException("radius must be positive");
         this.radius = radius;
 
+        this.invRadius = 1 / radius;
+        this.normalizedCenter = this.center.times(-this.invRadius);
+
         this.bbox = new AABB(
                 center.minus(radius, radius, radius),
                 center.plus(radius, radius, radius)
@@ -33,23 +41,11 @@ public final class Sphere implements Hittable {
 
     @Override
     public @NotNull Optional<HitResult> hit(@NotNull Ray ray, @NotNull Range range) {
-        var oc = ray.origin().minus(center);
-
-        var a = ray.direction().squared();
-        var h = ray.direction().times(oc);
-        var c = oc.squared() - radius * radius;
-
-        var discriminant = h * h - a * c;
-        if (discriminant < 0) return Optional.empty();
-
-        var sd = Math.sqrt(discriminant);
-
-        double t = (- h - sd) / a;
-        if (!range.surrounds(t)) t = (- h + sd) / a;
-        if (!range.surrounds(t)) return Optional.empty();
+        var t = hit0(ray, range);
+        if (Double.isNaN(t)) return Optional.empty();
 
         var position = ray.at(t);
-        var normal = position.minus(center).div(radius);
+        var normal = Vec3.fma(invRadius, position, normalizedCenter);
         var frontFace = normal.times(ray.direction()) < 0;
 
         double u;
@@ -65,13 +61,57 @@ public final class Sphere implements Hittable {
         }
 
         return Optional.of(new HitResult(
-                t, position, frontFace ? normal : normal.neg(),
+                t, position, frontFace ? normal : normal.neg(), this,
                 material, u, v, frontFace
         ));
     }
 
+    private double hit0(@NotNull Ray ray, @NotNull Range range) {
+        var oc = ray.origin().minus(center);
+
+        var a = ray.direction().squared();
+        var h = ray.direction().times(oc);
+        var c = oc.squared() - radius * radius;
+
+        var discriminant = h * h - a * c;
+        if (discriminant < 0) return Double.NaN;
+
+        var sd = Math.sqrt(discriminant);
+
+        double t = (- h - sd) / a;
+        if (!range.surrounds(t)) t = (- h + sd) / a;
+        if (!range.surrounds(t)) return Double.NaN;
+
+        return t;
+    }
+
     @Override
     public @NotNull AABB getBoundingBox() {
         return bbox;
     }
+
+    @Override
+    public double getProbabilityDensity(@NotNull Vec3 origin, @NotNull Vec3 direction) {
+        if (Double.isNaN(hit0(new Ray(origin, direction), FORWARD))) return 0;
+
+        var cos = Math.sqrt(1 - radius * radius / (center.minus(origin).squared()));
+        var solidAngle = 2 * Math.PI * (1 - cos);
+        return 1 / solidAngle;
+    }
+
+    @Override
+    public @NotNull Vec3 getTargetingDirection(@NotNull Vec3 origin, @NotNull RandomGenerator random) {
+        var direction = center.minus(origin);
+        var out = Vec3.randomOppositeHemisphere(random, direction);
+        return new Vec3(
+                Math.fma(radius, out.x(), center.x() - origin.x()),
+                Math.fma(radius, out.y(), center.y() - origin.y()),
+                Math.fma(radius, out.z(), center.z() - origin.z())
+        );
+    }
+
+    @Override
+    public @NotNull String toString() {
+        return "Sphere(center=" + center + ", radius=" + radius + ")";
+    }
 }

src/main/java/eu/jonahbauer/raytracing/scene/transform/RotateY.java

@@ -5,15 +5,26 @@ import eu.jonahbauer.raytracing.math.Ray;
 import eu.jonahbauer.raytracing.math.Vec3;
 import eu.jonahbauer.raytracing.scene.HitResult;
 import eu.jonahbauer.raytracing.scene.Hittable;
+import eu.jonahbauer.raytracing.scene.Target;
 import org.jetbrains.annotations.NotNull;
 
-public final class RotateY extends Transform {
+import java.util.random.RandomGenerator;
+
+public sealed class RotateY extends Transform {
     private final double cos;
     private final double sin;
 
     private final @NotNull AABB bbox;
 
-    public RotateY(@NotNull Hittable object, double angle) {
+    public static @NotNull RotateY create(@NotNull Hittable object, double angle) {
+        if (object instanceof Target) {
+            return new RotateYTarget(object, angle);
+        } else {
+            return new RotateY(object, angle);
+        }
+    }
+
+    private RotateY(@NotNull Hittable object, double angle) {
         super(object);
         this.cos = Math.cos(angle);
         this.sin = Math.sin(angle);
@@ -45,45 +56,58 @@ public final class RotateY extends Transform {
     }
 
     @Override
-    protected @NotNull Ray transform(@NotNull Ray ray) {
+    protected final @NotNull Ray transform(@NotNull Ray ray) {
         var origin = ray.origin();
         var direction = ray.direction();
 
-        var newOrigin = new Vec3(
-                cos * origin.x() - sin * origin.z(),
-                origin.y(),
-                sin * origin.x() + cos * origin.z()
-        );
-        var newDirection = new Vec3(
-                cos * direction.x() - sin * direction.z(),
-                direction.y(),
-                sin * direction.x() + cos * direction.z()
-        );
-
+        var newOrigin = transform(origin);
+        var newDirection = transform(direction);
         return new Ray(newOrigin, newDirection);
     }
 
     @Override
-    protected @NotNull HitResult transform(@NotNull HitResult result) {
+    protected final @NotNull HitResult transform(@NotNull HitResult result) {
         var position = result.position();
-        var newPosition = new Vec3(
-                cos * position.x() + sin * position.z(),
-                position.y(),
-                - sin * position.x() + cos * position.z()
-        );
+        var newPosition = untransform(position);
 
         var normal = result.normal();
-        var newNormal = new Vec3(
-                cos * normal.x() + sin * normal.z(),
-                normal.y(),
-                -sin * normal.x() + cos * normal.z()
-        );
+        var newNormal = untransform(normal);
 
         return result.withPositionAndNormal(newPosition, newNormal);
     }
 
+    protected final @NotNull Vec3 transform(@NotNull Vec3 vec) {
+        return new Vec3(cos * vec.x() - sin * vec.z(), vec.y(), sin * vec.x() + cos * vec.z());
+    }
+
+    protected final @NotNull Vec3 untransform(@NotNull Vec3 vec) {
+        return new Vec3(cos * vec.x() + sin * vec.z(), vec.y(), - sin * vec.x() + cos * vec.z());
+    }
+
     @Override
     public @NotNull AABB getBoundingBox() {
         return bbox;
     }
+
+    @Override
+    public @NotNull String toString() {
+        return object + " rotated by " + Math.toDegrees(Math.atan2(sin, cos)) + "° around the y axis";
+    }
+
+    private static final class RotateYTarget extends RotateY implements Target {
+
+        private RotateYTarget(@NotNull Hittable object, double angle) {
+            super(object, angle);
+        }
+
+        @Override
+        public double getProbabilityDensity(@NotNull Vec3 origin, @NotNull Vec3 direction) {
+            return ((Target) object).getProbabilityDensity(transform(origin), transform(direction));
+        }
+
+        @Override
+        public @NotNull Vec3 getTargetingDirection(@NotNull Vec3 origin, @NotNull RandomGenerator random) {
+            return untransform(((Target) object).getTargetingDirection(transform(origin), random));
+        }
+    }
 }

src/main/java/eu/jonahbauer/raytracing/scene/transform/Transform.java

@@ -2,12 +2,15 @@ package eu.jonahbauer.raytracing.scene.transform;
 
 import eu.jonahbauer.raytracing.math.Range;
 import eu.jonahbauer.raytracing.math.Ray;
+import eu.jonahbauer.raytracing.math.Vec3;
 import eu.jonahbauer.raytracing.scene.HitResult;
 import eu.jonahbauer.raytracing.scene.Hittable;
+import eu.jonahbauer.raytracing.scene.Target;
 import org.jetbrains.annotations.NotNull;
 
 import java.util.Objects;
 import java.util.Optional;
+import java.util.random.RandomGenerator;
 
 public abstract class Transform implements Hittable {
     protected final @NotNull Hittable object;

src/main/java/eu/jonahbauer/raytracing/scene/transform/Translate.java

@@ -5,13 +5,24 @@ import eu.jonahbauer.raytracing.math.Ray;
 import eu.jonahbauer.raytracing.math.Vec3;
 import eu.jonahbauer.raytracing.scene.HitResult;
 import eu.jonahbauer.raytracing.scene.Hittable;
+import eu.jonahbauer.raytracing.scene.Target;
 import org.jetbrains.annotations.NotNull;
 
-public final class Translate extends Transform {
-    private final @NotNull Vec3 offset;
+import java.util.random.RandomGenerator;
+
+public sealed class Translate extends Transform {
+    protected final @NotNull Vec3 offset;
     private final @NotNull AABB bbox;
 
-    public Translate(@NotNull Hittable object, @NotNull Vec3 offset) {
+    public static @NotNull Translate create(@NotNull Hittable object, @NotNull Vec3 offset) {
+        if (object instanceof Target) {
+            return new TranslateTarget(object, offset);
+        } else {
+            return new Translate(object, offset);
+        }
+    }
+
+    private Translate(@NotNull Hittable object, @NotNull Vec3 offset) {
         super(object);
         this.offset = offset;
 
@@ -23,17 +34,40 @@ public final class Translate extends Transform {
     }
 
     @Override
-    protected @NotNull Ray transform(@NotNull Ray ray) {
+    protected final @NotNull Ray transform(@NotNull Ray ray) {
         return new Ray(ray.origin().minus(offset), ray.direction());
     }
 
     @Override
-    protected @NotNull HitResult transform(@NotNull HitResult result) {
+    protected final @NotNull HitResult transform(@NotNull HitResult result) {
         return result.withPositionAndNormal(result.position().plus(offset), result.normal());
     }
 
     @Override
-    public @NotNull AABB getBoundingBox() {
+    public final @NotNull AABB getBoundingBox() {
         return bbox;
     }
+
+    @Override
+    public @NotNull String toString() {
+        return object + " translated by " + offset;
+    }
+
+    private static final class TranslateTarget extends Translate implements Target {
+        private TranslateTarget(@NotNull Hittable object, @NotNull Vec3 offset) {
+            super(object, offset);
+        }
+
+        @Override
+        public double getProbabilityDensity(@NotNull Vec3 origin, @NotNull Vec3 direction) {
+            if (!(object instanceof Target target)) throw new UnsupportedOperationException();
+            return target.getProbabilityDensity(origin.minus(offset), direction);
+        }
+
+        @Override
+        public @NotNull Vec3 getTargetingDirection(@NotNull Vec3 origin, @NotNull RandomGenerator random) {
+            if (!(object instanceof Target target)) throw new UnsupportedOperationException();
+            return target.getTargetingDirection(origin.minus(offset), random);
+        }
+    }
 }

src/main/java/eu/jonahbauer/raytracing/scene/util/PdfUtil.java

@@ -0,0 +1,19 @@
+package eu.jonahbauer.raytracing.scene.util;
+
+import eu.jonahbauer.raytracing.math.Vec3;
+import org.jetbrains.annotations.NotNull;
+
+public final class PdfUtil {
+    private PdfUtil() {
+        throw new UnsupportedOperationException();
+    }
+
+    /**
+     * {@return the solid angle of the triangle abc as seen from the origin} The vectors {@code a}, {@code b} and {@code c}
+     * must be unit vectors.
+     */
+    public static double getSolidAngle(@NotNull Vec3 a, @NotNull Vec3 b, @NotNull Vec3 c) {
+        var angle = 2 * Math.atan(Math.abs(Vec3.tripleProduct(a, b, c)) / (1 + a.times(b) + b.times(c) + c.times(a)));
+        return angle < 0 ? 2 * Math.PI + angle : angle;
+    }
+}