1. Ray Tracing Geometry
CSE 681

2. The Camera Model Based on a simpile pin-hole camera model
Simplest lens model
Pure geometric optics – based on similar triangles
Perfect image if hole infinitely small
Inverted image
pin-hole camera
simplified pin-hole camera
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3. Basic Ray Tracing Algorithm
for every pixel {
cast a ray from the eye
for every object in the scene
find intersections with the ray
keep it if closest }
compute color at the intersection point
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4. Construct a Ray 3D parametric line r(t) = eye + t (p-eye)
r(t): ray equation
eye: eye (camera) position
p: pixel position
t: ray parameter
r(t) p
t=0
Question: How to calculate the pixel position P?
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5. What are given? Camera (eye) position
View direction or center of interest
Camera orientation (which way is up?)
specified by an “up” vector
Field of view + aspect ration
Distance to the image plane
Pixel resolutions in x and y
“up” vector
eye
ray
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6. Camera Setup v n w u We need to have a ‘view
“up” vector v n w
View direction u
We need to have a ‘view
coordinate system, i.e., compute
the u, v, w vectors
u v w
w image plane
Eye + w goes through the image plane center
Use info about camera to construct 3D camera coordinate system
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7. Camera Setup v n w u w: known u = w x “up” v = u x w
“up” vector v n w
View direction u
w: known
u = w x “up”
v = u x w
“up” may not be perpendicular to w
x: cross product
CSE 681
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8. Pixel Calculation
Coordinate (in u,v,n space) of upper left corner of screen
Eye + w v
yres u w
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

9. Pixel Calculation
Coordinate (in u,v,n space) of upper left corner of screen
Eye + w v
yres u w
xres
How do we calculate
PixelWidth and PixelHeight?
Assume virtual screen is one unit away (D=1) in w direction
Eye + w - (xres/2)*PixelWidth*u + (yres/2)*PixelHeight *v
CSE 681
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10. Camera Setup (yres/2) * pixelHeight 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
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11. 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
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12. 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 = ?
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13. 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

14. 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
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15. 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
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16. 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
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17. 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’
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18. After all objects are tested
If (distancemin > infinityThreshold) {
pixelColor = background color
else
pixelColor = color of object at distancemin along ray d
ray
object
eye
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19. 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
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20. Ray-Sphere Intersection - algebraic
x2 + y2 + z2 = r2
P(t) = eye + t*Ray
Substitute definition of p into first equation:
(eye.x+ t *ray.x) 2 + (eye.y+ t *ray.y)2 + (eye.z+ t *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* t 2 + B* t + C = 0
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21. Ray-Sphere Intersection (cont’d)
For a sphere with its center at c
A sphere with center c = (xc,yc,zc) and radius R can be represented as:
(x-xc)2 + (y-yc)2 + (z-zc)2 – R2 = 0
For a point p on the sphere, we can write the above in vector form:
(p-c).(p-c) – R2 = 0 (note ‘.’ is a dot product)
Solve p similarly
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22. Quadratic Equation When solving a quadratic equation at2 + bt + c = 0
Discriminant:
And Solution:
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23. Ray-Sphere Intersection
b2 – 4ac <0 : No intersection
b2 – 4ac >0 : Two solutions (enter and exit)
b2 – 4ac = 0 : One solution (ray grazes the sphere)
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24. 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
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