Building a Ray Tracer

Not done yet!!!!

Vectors
Ray tracing
Path tracing

Vectors

Normalize

\hat{v} = \frac{\vec{v}}{\|\vec{v}\|}

​ v ​^​ ​ = \frac{​ ​ v ​ ⃗ ​ ​ ​ ​}{​ ∥ ​ v ​ ⃗ ​ ​ ∥ ​}

Dot Product

\vec{v} \cdot \vec{w} = \|\vec{v}\| \|\vec{w}\|\cos{\theta}

​ v ​ ⃗ ​ ​ \cdot ​ w ​ ⃗ ​ ​ = ∥ ​ v ​ ⃗ ​ ​ ∥ ∥ ​ w ​ ⃗ ​ ​ ∥ cos θ

\vec{v} \cdot \vec{w} = v_x w_x + v_y w_y + v_z w_z

​ v ​ ⃗ ​ ​ \cdot ​ w ​ ⃗ ​ ​ = v ​ x ​ ​ w ​ x ​ ​ + v ​ y ​ ​ w ​ y ​ ​ + v ​ z ​ ​ w ​ z ​ ​

\rightarrow \vec{v} \cdot \vec{v} = \|\vec{v}\|

\to ​ v ​ ⃗ ​ ​ \cdot ​ v ​ ⃗ ​ ​ = ∥ ​ v ​ ⃗ ​ ​ ∥

Special Cases

\vec{v}

​ v ​ ⃗ ​ ​

\vec{w} = 90^\circ \rightarrow \vec{v} \cdot \vec{w} = 0

​ w ​ ⃗ ​ ​ = 9 0 ​ \circ ​ ​ \to ​ v ​ ⃗ ​ ​ \cdot ​ w ​ ⃗ ​ ​ = 0

The angle between

and

The dot product of a vector and itself

The dot product of a unit vector and itself

\rightarrow \vec{v} \cdot \vec{v} = 1

\to ​ v ​ ⃗ ​ ​ \cdot ​ v ​ ⃗ ​ ​ = 1

Cross Product

\|\vec{v} \times \vec{w}\| = \|\vec{v}\| \|\vec{w}\| \sin{\theta}

∥ ​ v ​ ⃗ ​ ​ \times ​ w ​ ⃗ ​ ​ ∥ = ∥ ​ v ​ ⃗ ​ ​ ∥ ∥ ​ w ​ ⃗ ​ ​ ∥ sin θ

\vec{v} \times \vec{w} = (v_y w_z - v_z w_y)\vec{x} + (v_z w_x - v_x w_z)\vec{y} + (v_x w_y - v_y w_x)\vec{z}

​ v ​ ⃗ ​ ​ \times ​ w ​ ⃗ ​ ​ = (v ​ y ​ ​ w ​ z ​ ​ - v ​ z ​ ​ w ​ y ​ ​) ​ x ​ ⃗ ​ ​ + (v ​ z ​ ​ w ​ x ​ ​ - v ​ x ​ ​ w ​ z ​ ​) ​ y ​ ⃗ ​ ​ + (v ​ x ​ ​ w ​ y ​ ​ - v ​ y ​ ​ w ​ x ​ ​) ​ z ​ ⃗ ​ ​

Ray tracing

Summary

Algorithm
Ray-Scene Intersection
Ray Intersections
Lighting
Optimizations

General Algorithm

Image RayCast(Camera camera, Scene scene, int width, int height) {
  Image img = new Image(width, height);
  for (int i = 0; i < width; i++) {
    for (int j = 0; j < height; j++) {
      Ray ray = ConstructRayThroughPixel(camera, i, j);
      Intersection hit = FindIntersection(ray, scene);
      img[i][j] = GetColor(hit);
    }
  }
  return img;
}

\text{Ray} = P_0 + tV

R a y = P ​ 0 ​ ​ + t V

Ray Intersections

Intersection FindIntersection(Ray ray, Scene scene) {
  min_t = inf;
  min_shape = NULL;
  for each primitive in scene {
    t = Intersect(ray, primitive);
    if (t > 0 and t < min_t) {
      min_shape = primitive;
      min_t = t;
    }
  }
  return Intersection(min_t, min_shape);
}