add targeting probability density function

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

@@ -35,6 +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);
@@ -212,6 +213,38 @@ public class Examples {
         );
     }
 
+    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 aluminum = new MetallicMaterial(new Color(0.8, 0.85, 0.88));
+        var glass = new DielectricMaterial(1.5);
+
+        return new Example(
+                new Scene(
+                        new Box(
+                                new AABB(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 Box(
+                                new AABB(new Vec3(0, 0, 0), new Vec3(165, 330, 165)),
+                                white, white, white, white, white, aluminum
+                        ).rotateY(Math.toRadians(15)).translate(new Vec3(265, 0, 295)),
+                        new Sphere(new Vec3(190, 90, 190), 90, glass)
+                ),
+                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 getDiagramm(int height) {
         if (height <= 0) height = 450;
 

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

@@ -3,7 +3,7 @@ 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.HittableProbabilityDensityFunction;
+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;
@@ -137,9 +137,6 @@ public final class SimpleRenderer implements Renderer {
             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()) break;
 
             switch (result.get()) {
@@ -148,8 +145,16 @@ public final class SimpleRenderer implements Renderer {
                     ray = scattered;
                 }
                 case Material.PdfScatterResult(var a, var pdf) -> {
-                    attenuation = Color.multiply(attenuation, a);
+                    if (scene.getLights() == null) {
-                    ray = new Ray(hit.position(), pdf.generate(random));
+                        attenuation = Color.multiply(attenuation, a);
+                        ray = new Ray(hit.position(), pdf.generate(random));
+                    } else {
+                        var mixed = new MixtureProbabilityDensityFunction(new TargetingProbabilityDensityFunction(hit.position(), scene.getLights()), pdf);
+                        var direction = mixed.generate(random);
+                        var factor = pdf.value(direction) / mixed.value(direction);
+                        attenuation = Color.multiply(attenuation, Color.multiply(a, factor));
+                        ray = new Ray(hit.position(), direction);
+                    }
                 }
             }
         }

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

@@ -0,0 +1,28 @@
+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.Objects;
+import java.util.random.RandomGenerator;
+
+/**
+ * A probability density function targeting a target.
+ */
+public record TargetingProbabilityDensityFunction(@NotNull Vec3 origin, @NotNull Target target) implements ProbabilityDensityFunction {
+    public TargetingProbabilityDensityFunction {
+        Objects.requireNonNull(origin, "origin");
+        Objects.requireNonNull(target, "target");
+    }
+
+    @Override
+    public double value(@NotNull Vec3 direction) {
+        return target.getProbabilityDensity(origin, direction);
+    }
+
+    @Override
+    public @NotNull Vec3 generate(@NotNull RandomGenerator random) {
+        return target.getTargetingDirection(origin, random);
+    }
+}

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

@@ -26,6 +26,10 @@ 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());
     }

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

@@ -19,6 +19,9 @@ public interface 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) {

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

@@ -6,6 +6,7 @@ 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.List;
 import java.util.Objects;
@@ -14,6 +15,8 @@ public final class Scene extends HittableCollection {
     private final @NotNull HittableCollection objects;
     private final @NotNull SkyBox background;
 
+    private final @Nullable Target light;
+
     public Scene(@NotNull List<? extends @NotNull Hittable> objects) {
         this(Color.BLACK, objects);
     }
@@ -25,6 +28,8 @@ public final class Scene extends HittableCollection {
     public Scene(@NotNull SkyBox background, @NotNull List<? extends @NotNull Hittable> objects) {
         this.objects = new HittableBinaryTree(objects);
         this.background = Objects.requireNonNull(background);
+
+        this.light = (Target) objects.get(1);
     }
 
     public Scene(@NotNull Hittable @NotNull... objects) {
@@ -49,6 +54,10 @@ public final class Scene extends HittableCollection {
         return objects.getBoundingBox();
     }
 
+    public @Nullable Target getLights() {
+        return light;
+    }
+
     public @NotNull Color getBackgroundColor(@NotNull Ray ray) {
         return background.getColor(ray);
     }

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/Parallelogram.java

@@ -1,11 +1,17 @@
 package eu.jonahbauer.raytracing.scene.hittable2d;
 
 import eu.jonahbauer.raytracing.math.AABB;
+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.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 +28,24 @@ public final class Parallelogram extends Hittable2D {
     public @NotNull AABB getBoundingBox() {
         return bbox;
     }
+
+    @Override
+    public double getProbabilityDensity(@NotNull Vec3 origin, @NotNull Vec3 direction) {
+        var result = hit(new Ray(origin, direction), new Range(0.001, Double.POSITIVE_INFINITY));
+        if (result.isEmpty()) return 0;
+
+        var a = this.origin;
+        var b = this.origin.plus(u);
+        var c = this.origin.plus(v);
+        var d = b.plus(v);
+        var angle = PdfUtil.getSolidAngle(origin, a, b, d) + PdfUtil.getSolidAngle(origin, 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 this.origin.plus(u.times(alpha)).plus(v.times(beta)).minus(origin);
+    }
 }

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

@@ -7,13 +7,16 @@ 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 final class Box implements Hittable {
+public final class Box implements Hittable, Target {
     private final @NotNull AABB box;
     private final @Nullable Material @NotNull[] materials;
 
@@ -115,6 +118,61 @@ public final class Box implements Hittable {
         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), new Range(0.001, Double.POSITIVE_INFINITY)).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, true);
+
+        // 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();
+    }
+
     private enum Side {
         NEG_X(Vec3.UNIT_X.neg()),
         NEG_Y(Vec3.UNIT_Y.neg()),
@@ -155,5 +213,78 @@ public final class Box implements Hittable {
                 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);
+            var b = get(box, 0, 1);
+            var c = get(box, 1, 1);
+            var d = get(box, 1, 0);
+
+            return PdfUtil.getSolidAngle(pos, a, b, d) + PdfUtil.getSolidAngle(pos, c, b, d);
+        }
     }
 }

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

@@ -7,12 +7,14 @@ 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;
@@ -74,4 +76,25 @@ public final class Sphere implements Hittable {
     public @NotNull AABB getBoundingBox() {
         return bbox;
     }
+
+    @Override
+    public double getProbabilityDensity(@NotNull Vec3 origin, @NotNull Vec3 direction) {
+        if (hit(new Ray(origin, direction), new Range(0.001, Double.POSITIVE_INFINITY)).isEmpty()) 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);
+
+        Vec3 target;
+        do {
+            target = Vec3.random(random, true);
+        } while (target.times(direction) >= 0);
+
+        return target.times(radius).plus(center).minus(origin);
+    }
 }

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

@@ -5,8 +5,11 @@ 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.random.RandomGenerator;
+
 public final class RotateY extends Transform {
     private final double cos;
     private final double sin;
@@ -49,41 +52,44 @@ public final class RotateY extends Transform {
         var origin = ray.origin();
         var direction = ray.direction();
 
-        var newOrigin = new Vec3(
+        var newOrigin = transform(origin);
-                cos * origin.x() - sin * origin.z(),
+        var newDirection = transform(direction);
-                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()
-        );
-
         return new Ray(newOrigin, newDirection);
     }
 
     @Override
     protected @NotNull HitResult transform(@NotNull HitResult result) {
         var position = result.position();
-        var newPosition = new Vec3(
+        var newPosition = untransform(position);
-                cos * position.x() + sin * position.z(),
-                position.y(),
-                - sin * position.x() + cos * position.z()
-        );
 
         var normal = result.normal();
-        var newNormal = new Vec3(
+        var newNormal = untransform(normal);
-                cos * normal.x() + sin * normal.z(),
-                normal.y(),
-                -sin * normal.x() + cos * normal.z()
-        );
 
         return result.withPositionAndNormal(newPosition, newNormal);
     }
 
+    private @NotNull Vec3 transform(@NotNull Vec3 vec) {
+        return new Vec3(cos * vec.x() - sin * vec.z(), vec.y(), sin * vec.x() + cos * vec.z());
+    }
+
+    private @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 double getProbabilityDensity(@NotNull Vec3 origin, @NotNull Vec3 direction) {
+        if (!(object instanceof Target target)) throw new UnsupportedOperationException();
+        return target.getProbabilityDensity(transform(origin), transform(direction));
+    }
+
+    @Override
+    public @NotNull Vec3 getTargetingDirection(@NotNull Vec3 origin, @NotNull RandomGenerator random) {
+        if (!(object instanceof Target target)) throw new UnsupportedOperationException();
+        return untransform(target.getTargetingDirection(transform(origin), random));
+    }
 }

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

@@ -2,14 +2,17 @@ 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 {
+public abstract class Transform implements Hittable, Target {
     protected final @NotNull Hittable object;
 
     protected Transform(@NotNull Hittable object) {

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

@@ -5,8 +5,11 @@ 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.random.RandomGenerator;
+
 public final class Translate extends Transform {
     private final @NotNull Vec3 offset;
     private final @NotNull AABB bbox;
@@ -36,4 +39,16 @@ public final class Translate extends Transform {
     public @NotNull AABB getBoundingBox() {
         return bbox;
     }
+
+    @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 +1,18 @@
+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();
+    }
+
+    public static double getSolidAngle(@NotNull Vec3 o, @NotNull Vec3 a, @NotNull Vec3 b, @NotNull Vec3 c) {
+        var i = a.minus(o).unit();
+        var j = b.minus(o).unit();
+        var k = c.minus(o).unit();
+
+        return 2 * Math.atan(Math.abs(i.times(j.cross(k))) / (1 + i.times(j) + j.times(k) + k.times(i)));
+    }
+}