1. Ray Tracing Geometry
CSE 681

2. Camera Setup
Need to establish viewing geometry from camera’s point of view
And make a ‘virtual’ array of pixels in front of the camera
Rays will go through pixel centers
Need view direction, angle of view, and which way is up
CSE 681

3. Camera Setup v n w u “up” vector View direction
‘u’ can be formed as cross product of view vector and up vector
Use info about camera to construct 3D camera coordinate system
CSE 681

4. Camera Setup =? Tan(q/2) = yres*pixelHeight/2D
Side view of camera
Given theta (either half angle or full angle of view can be specified), D
Compute extent of virtual frame buffer
Then using pixel resolution, computer pixel height and width
Tan(q/2) = yres*pixelHeight/2D
pixelHeight = 2*Tan(qy/2) *D/yres =D
pixelWidth = 2*Tan(qx/2) *D/xres
CSE 681

5. Screen Placement
How do images differ if the resolution doesn’t change?
Q: if same pixel resolution, what’s different about the images generated when using different distances for virtual frame buffer?
A: nothing
So use D = 1 to simplify computations
CSE 681

6. Pixel Calculation Tan(q/2) = yres*pixelHeight/2D
pixelHeight = 2*Tan(qy/2) *D/yres
pixelWidth = 2*Tan(qx/2) *D/xres
Pixel AspectRatio = pixelWidth/pixelHeight
Aspect ratio is width to height ratio
Coordinate (in u,v,n space) of upper left corner of screen = ?
CSE 681

7. Pixel Calculation
Coordinate (in u,v,n space) of upper left corner of screen = ? v u w
Tan(q/2) = yres*pixelHeight/2
pixelHeight = 2*Tan(qy/2) /yres
pixelWidth = 2*Tan(qx/2) /xres
Using camera coordinate system vectors, form location of upper left corner of screen in world space
Assume virtual screen is one unit away (D=1) in w direction
Eye + w - (xres/2)*PixelWidth*u + (yres/2)*PixelHeight *v
CSE 681

8. Pixel Calculation
Coordinate (in u,v,n space) of upper left pixel center = ?
Then back off half a pixel width and height to compute pixel center
Eye + w - (xres/2)*PixelWidth*u + (yres/2)*PixelHeight *v
+ (pixelWidth/2)*u - (pixelHeight/2)*v
CSE 681

9. Interate through pixel Centers
scanlineStart = Eye +
w -
(xres/2)*PixelWidth*u +
(yres/2)*PixelHeight *v +
(pixelWidth/2)*u -
(pixelHeight/2)*v
Update by pixel width and pixel height to go to adjacent pixel on scanline and next scanline, respectively
There will
pixelCenter += pixelWidth * u
scanlineStart -= pixelHeight * v
CSE 681

10. Pixel loops ScenlineStart = [from previous slide] For each scanline {
pixelCenter = scanlineStart
For each pixel across {
form ray from camera through pixel ….
pixelCenter += pixelWidth*u }
scanlineStart -= pixelHeight*v
Update by pixel width and pixel height to go to adjacent pixel on scanline and next scanline, respectively
There will
CSE 681

11. Process Objects For each pixel { Form ray from eye through pixel
distancemin = infinity
For each object {
If (distance=intersect(ray,object)) {
If (distance< distancemin) {
closestObject = object
distancemin = distance }
Color pixel according to intersection information
Keep closest intersection
Need intersection function for each object type processed
Later, we’ll build ‘intersection structure’
CSE 681

12. After all objects are tested
If (distancemin > infinityThreshold) {
pixelColor = background color
else
pixelColor = color of object at distancemin along ray d
ray
object
eye
CSE 681

13. Ray-Sphere Intersection - geometric
Knowns
C, r
Eye
Ray
t= |C-eye| k
d+k = (C-eye) · Ray s
t2= (k+d) 2 + s2 d r C
r2 = k2+ s2
If s > r, then no intersection
If s == r, then ray just grazes intersection
eye t
d = (k+d) - k
CSE 681

14. Ray-Sphere Intersection - algebraic
x2 + y2 + z2 = r2
P(u) = eye + u*Ray
Substitute definition of p into first equation:
(eye.x+ u *ray.x) 2 + (eye.y+ u *ray.y)2 + (eye.z+ u *ray.z) 2 = r2
Analytic equation of sphere and parametric form for ray
Use quadratic equation to solve for u
If term inside radical is <0, no intersection
If term == 0, ray just grazes sphere
Expand squared terms and collect terms based on powers of u:
A* u 2 + B* u + C = 0
CSE 681

15. Determine Color Use z-component of normalized normal vector FOR LAB #1
Clamp to [ ]
objectColor*Nz
ray
What’s the normal at a point on the sphere?
Simple illumination example
Just use z coordinate for normalized normal vector, clamped from below to 0.3,
eye
CSE 681