1. Ray Tracing

2. Ray Tracing What is ray tracing? Trace the path of a ray of light.
Model its interactions with the scene.
When a ray intersects an object, send off secondary rays (reflection, shadow, transmission) and determine how they interact with the scene.

3. Ray Tracing Capabilities
Basic algorithm allows for:
Hidden surface removal (like z-buffering)
Multiple light sources
Reflections
Transparent refractions
Hard shadows
Extensions can achieve:
Soft shadows
Motion blur
Blurred reflections (glossiness)
Depth of field (finite apertures)
Translucent refractions and more

4. Ray Tracing Produces realistic images Strengths: Weaknesses:
Specular reflections
Transparency
Weaknesses:
colour bleeding (diffuse reflections)
Time consuming
References:
“An Improved Illumination Model for Shaded Display,” Turner Whitted, CACM, June 1980.
“Distributed Ray Tracing,” Cook, Porter, and Carpenter, Computer Graphics, July 1984, pp
Green, S.A., Paddon, D.J. Exploiting Coherence for “Multiprocessor Ray Tracing”. IEEE Computer Graphics Journal, 9,6, Nov 1989, pp

5. Ray Traced Images

7. Ray Tracing “Backward” ray tracing:
Traces the ray forward (in time) from the light source through potentially many scene interactions
Problem: most rays will never even get close to the eye
Very inefficient since it computes
many rays that are never seen
Light
Eye
Image plane

8. Ray Tracing “Forward” ray tracing:
Traces the ray backward (in time) from the eye, through a point on the screen
More efficient: computes only
visible rays (since we start at eye)
Generally, ray tracing refers to
forward ray tracing.
Light
Eye
Image plane

9. Ray Tracing
Ray tracing is an image- precision algorithm: Visibility determined on a per-pixel basis
Trace one (or more) rays per pixel
Compute closest object (triangle,
sphere, etc.) for each ray
Produces realistic results
Computationally expensive

10. Minimal Ray Tracer
A basic (minimal) ray tracer is simple to implement:
The code can even fit on a 3×5 card (code courtesy of Paul Heckbert):
typedef struct{double x,y,z}vec;vec U,black,amb={.02,.02,.02};struct sphere{
vec cen,colour;double rad,kd,ks,kt,kl,ir}*s,*best,sph[]={0.,6.,.5,1.,1.,1.,.9,
.05,.2,.85,0.,1.7,-1.,8.,-.5,1.,.5,.2,1.,.7,.3,0.,.05,1.2,1.,8.,-.5,.1,.8,.8,
1.,.3,.7,0.,0.,1.2,3.,-6.,15.,1.,.8,1.,7.,0.,0.,0.,.6,1.5,-3.,-3.,12.,.8,1.,
1.,5.,0.,0.,0.,.5,1.5,};yx;double u,b,tmin,sqrt(),tan();double vdot(A,B)vec A
,B;{return A.x*B.x+A.y*B.y+A.z*B.z;}vec vcomb(a,A,B)double a;vec A,B;{B.x+=a*
A.x;B.y+=a*A.y;B.z+=a*A.z;return B;}vec vunit(A)vec A;{return vcomb(1./sqrt(
vdot(A,A)),A,black);}struct sphere*intersect(P,D)vec P,D;{best=0;tmin=1e30;s=
sph+5;while(s-->sph)b=vdot(D,U=vcomb(-1.,P,s->cen)),u=b*b-vdot(U,U)+s->rad*s
->rad,u=u>0?sqrt(u):1e31,u=b-u>1e-7?b-u:b+u,tmin=u>=1e-7&&u<tmin?best=s,u:
tmin;return best;}vec trace(level,P,D)vec P,D;{double d,eta,e;vec N,colour;
struct sphere*s,*l;if(!level--)return black;if(s=intersect(P,D));else return
amb;colour=amb;eta=s->ir;d= -vdot(D,N=vunit(vcomb(-1.,P=vcomb(tmin,D,P),s->cen
)));if(d<0)N=vcomb(-1.,N,black),eta=1/eta,d= -d;l=sph+5;while(l-->sph)if((e=l
->kl*vdot(N,U=vunit(vcomb(-1.,P,l->cen))))>0&&intersect(P,U)==l)colour=vcomb(e
,l->colour,colour);U=s->colour;colour.x*=U.x;colour.y*=U.y;colour.z*=U.z;e=1-eta*
eta*(1-d*d);return vcomb(s->kt,e>0?trace(level,P,vcomb(eta,D,vcomb(eta*d-sqrt
(e),N,black))):black,vcomb(s->ks,trace(level,P,vcomb(2*d,N,D)),vcomb(s->kd,
colour,vcomb(s->kl,U,black))));}main(){puts(“P3\n32 32\n255”);while(yx<32*32)
U.x=yx%32-32/2,U.z=32/2-yx++/32,U.y=32/2/tan(25/ ),U=vcomb(255.,
trace(3,black,vunit(U)),black),printf("%.0f %.0f %.0f\n",U);}/*minray!*/

11. Minimal Ray Tracer This code implements: Multiple spheres (with
different properties)
Multiple levels of
recursion:
Reflections
Transparency:
Refraction
One point light source:
Hard shadows
Hidden surface removal
Phong illumination model
typedef struct{double x,y,z}vec;vec U,black,amb={.02,.02,.02};struct sphere{
vec cen,colour;double rad,kd,ks,kt,kl,ir}*s,*best,sph[]={0.,6.,.5,1.,1.,1.,.9,
.05,.2,.85,0.,1.7,-1.,8.,-.5,1.,.5,.2,1.,.7,.3,0.,.05,1.2,1.,8.,-.5,.1,.8,.8,
1.,.3,.7,0.,0.,1.2,3.,-6.,15.,1.,.8,1.,7.,0.,0.,0.,.6,1.5,-3.,-3.,12.,.8,1.,
1.,5.,0.,0.,0.,.5,1.5,};yx;double u,b,tmin,sqrt(),tan();double vdot(A,B)vec A
,B;{return A.x*B.x+A.y*B.y+A.z*B.z;}vec vcomb(a,A,B)double a;vec A,B;{B.x+=a*
A.x;B.y+=a*A.y;B.z+=a*A.z;return B;}vec vunit(A)vec A;{return vcomb(1./sqrt(
vdot(A,A)),A,black);}struct sphere*intersect(P,D)vec P,D;{best=0;tmin=1e30;s=
sph+5;while(s-->sph)b=vdot(D,U=vcomb(-1.,P,s->cen)),u=b*b-vdot(U,U)+s->rad*s
->rad,u=u>0?sqrt(u):1e31,u=b-u>1e-7?b-u:b+u,tmin=u>=1e-7&&u<tmin?best=s,u:
tmin;return best;}vec trace(level,P,D)vec P,D;{double d,eta,e;vec N,colour;
struct sphere*s,*l;if(!level--)return black;if(s=intersect(P,D));else return
amb;colour=amb;eta=s->ir;d= -vdot(D,N=vunit(vcomb(-1.,P=vcomb(tmin,D,P),s->cen
)));if(d<0)N=vcomb(-1.,N,black),eta=1/eta,d= -d;l=sph+5;while(l-->sph)if((e=l
->kl*vdot(N,U=vunit(vcomb(-1.,P,l->cen))))>0&&intersect(P,U)==l)colour=vcomb(e
,l->colour,colour);U=s->colour;colour.x*=U.x;colour.y*=U.y;colour.z*=U.z;e=1-eta*
eta*(1-d*d);return vcomb(s->kt,e>0?trace(level,P,vcomb(eta,D,vcomb(eta*d-sqrt
(e),N,black))):black,vcomb(s->ks,trace(level,P,vcomb(2*d,N,D)),vcomb(s->kd,
colour,vcomb(s->kl,U,black))));}main(){puts(“P3\n32 32\n255”);while(yx<32*32)
U.x=yx%32-32/2,U.z=32/2-yx++/32,U.y=32/2/tan(25/ ),U=vcomb(255.,
trace(3,black,vunit(U)),black),printf("%.0f %.0f %.0f\n",U);}/*minray!*/

12. Ray Tracing: Types of Rays
Primary rays:
Sent from the eye, through the image plane, and into the scene
May or may not intersect an
object in the scene.
No intersection:
set pixel to background colour
Intersects object: send out
secondary rays and compute
lighting model
Light
Opaque object
Transparent object P1 P2
Eye

13. Ray Tracing: Types of Rays
Secondary Rays:
Sent from the point at which the
ray intersects an object
Multiple types:
Light
Transmission (T): sent in the direction of refraction S3 S2 T2 R2
Reflection (R): sent in the direction of reflection, and used in the Phong illumination model S1 T1 R3 R1
Opaque object
Transparent object
Shadow (S): sent toward a light source to determine if point is in shadow or not. P
Eye

14. Ray Tracing: Types of Rays
Light
P = Primary rays
R = Reflected rays
T = Transmitted rays
S = Shadow rays S2 S3 R2 T2 S1 T1 R1 R3
Opaque object
Transparent
object P
Eye

15. Ray Tracing: Ray Tree Each intersection may spawn secondary rays:
Rays form a ray tree.
Nodes are the intersection points.
Edges are the secondary rays.
Rays are recursively spawned until:
Ray does not intersect any object.
Tree reaches a maximum depth.
Light reaches some minimum value.
Shadow rays are sent from every intersection point (to determine if point is in shadow), but they do not spawn additional rays.

16. Note only ambient shade on mirror and teapot
Test Scene.
Ray tree depth 1.
Note only ambient shade on mirror and teapot
28/03/2017

17. Note only ambient shade on reflection of mirror and teapot.
Test Scene.
Ray tree depth 2.
Note only ambient shade on reflection of mirror and teapot.
28/03/2017

18. Note only ambient shade on reflection of mirror in teapot.
Test Scene.
Ray tree depth 3.
Note only ambient shade on reflection of mirror in teapot.
28/03/2017

19. Test Scene. Ray tree depth 4.
Note ambient shade on reflection of teapot in reflection of mirror in teapot.
28/03/2017

20. Test Scene.
Ray tree depth 5.
28/03/2017

21. Test Scene.
Ray tree depth 6.
28/03/2017

22. Test Scene.
Ray tree depth 7.
28/03/2017

23. Ray Tracing: Ray Tree Example
Eye P O1
Light S3 S2 O3 T2 R2 R1 T1 O1 O2 S1 T1 O2 R3 R1 O1
Opaque object R2 BG R3 T2 O1 BG
Transparent object P
Eye
Ray tree is evaluated from bottom up:
Depth-first traversal
The node colour is computed based on its children’s colours.

24. Basic Ray Tracing Algorithm
Generate one ray per pixel.
For each ray:
Find the first object the ray intersects.
Compute colour for the intersection point using an illumination model.
If the surface is reflective, trace a reflection ray.
If the surface is transparent, trace a transmission ray.
Trace shadow ray.
Combine results of the intensity computation, reflection, transmission, and shadow information.
If the ray misses all objects, set to the background colour.

25. Tracing Rays Basic (non-recursive) ray tracing algorithm:
1. Send a ray from the eye through the screen
2. Determine which object that ray first intersects
3. Compute pixel colour
Most (approx. 75%) of the time in step 2:
Simple method:
Compare every ray against every object and determine the closest object hit by each ray.
Very time consuming:
Several optimizations possible.

26. Viewing Ray Which coordinate space?
The primary ray (or viewing ray) for a point s on the view plane (i.e., screen) is computed as:
Origin: ro = eye
Direction: rd = s – eye
Which coordinate space?
Want to define rays in terms world-space coordinates (x, y, z)
Eye is already in specified in terms of (x, y, z) position
Screen point s is easiest to define in terms of where it is on the window in viewing- space coordinates (u, v, w) s
rd = s – ro
ro = eye
Window

27. Viewing Ray: Screen Point
Given:
Our scene in world-coordinates
A camera position in world-coordinates (x, y, z)
A pixel (i, j) in the viewport
We need to:
Compute the point on the view plane window that corresponds to the (i, j) point in the viewport
Transform that point into world-coordinates

28. View-reference coordinatesy
LookAt
point
LookFrom
point w x u z
World coordinates
View reference coordinates

29. Transform the Ray pWS = M pOS pOS = M-1 pWS World Space Object Space
Move the ray from World Space to Object Space
r minor
r major
(x,y)
r = 1
World Space
(0,0)
Object Space
pWS = M pOS
pOS = M-1 pWS

30. Transform Ray World Space Object Space New origin: New direction:
originOS = M-1 originWS
directionOS = M-1 (originWS + 1 * directionWS) M-1 originWS
directionOS = M-1 directionWS
originWS
directionWS
qWS = originWS + tWS * directionWS
originOS
directionOS
qOS = originOS + tOS * directionOS
World Space
Object Space

31. What to do about the depth, t
If M includes scaling, directionOS ends up NOT being normalized after transformation
Two solutions:
Normalize the direction
Don't normalize the direction

32. Don't normalize direction
Highly recommended
Don't normalize direction
tOS = tWS  convenient!
But you should not rely on tOS being true distance during intersection routines
tWS
tOS
World Space
Object Space

33. View-reference to World transform
Given the screen point in terms of viewing- space coordinates (u, v, w), transform to world-space (x, y, z):
The viewing transform takes a point from world space to view space (change of basis, CoB): y v s w
Window u x z

34. View-reference to World transform
Given the screen point in terms of viewing-space coordinates (u, v, w), transform to world-space (x, y, z):
We want to reverse this: y v s w
Window u x z
or sWorld = LookAt + usu + vsv + wsw

35. Ray/Object intersection
Can intersect ray with many objects
Planes, Spheres, Cylinders, Cones
Boxes (axis-aligned or otherwise)
Even procedural/implicit objects!
Constructive solid geometry (CSG)
Implicit surfaces / equations

36. Remember: Ray Representation
A ray can be represented explicitly (in parametric form) as an origin (point) and a direction (vector):
Origin:
Direction
The ray consists of all points:
r(t) = ro + rdt

37. Ray-Object Intersections
Many objects can be represented as implicit surfaces:
Sphere (with center at c and radius R):
fs (p) = ||p – c||2 - R2 = 0
Plane (with normal n and distance to origin d):
fp (p) = p · n + D = 0
To determine where a ray intersects an object:
Plug the ray equation into the surface equation and solve for t:
f(ro + rdt) = 0
Substitute t back into ray equation to find intersection point p:
p = r(t) = ro + rdt

38. Ray-Sphere Intersections
To find the intersection points of a ray with a sphere:
Sphere is represented as:
center: c = (xc, yc, zc)
radius: R
The sphere is the set of all points (x, y, z) such that:
(x-xc)2 + (y - yc)2 + (z - zc)2 = R2

39. Ray-Sphere Intersections
First, split the ray into its component equations:
x = xo + xdt
y = yo + ydt
z = zo + zdt
Then substitute the ray equation into the sphere equation:
(x-xc)2 + (y - yc)2 + (z - zc)2 = R2
Giving:
(xo + xdt - xc)2 + (yo + ydt - yc)2 + (zo + zdt - zc)2 = R2

40. Ray-Sphere Intersections
Next multiply out the squared terms:
(x0 + xdt - xc)2 + (y0 + ydt - yc)2 + (z0 + zdt - zc)2 = R2
Þ (xd2 + yd2 + zd2)t2 +
2(xdxo - xdxc+ ydyo - ydyc+ zdzo - zdzc)t +
(xo2- 2xoxc+ xc2+ yo2- 2yoyc+ yc2+ zo2-2zozc+ zc2)
= R2
How do we solve for t?
Use the Quadratic Equation

41. Ray-Sphere Intersections
Let
A = xd2 + yd2 + zd2 = 1
B = 2(xdxo - xdxc+ ydyo - ydyc+ zdzo - zdzc)
C = xo2- 2xoxc+ xc2+ yo2- 2yoyc+ yc2+ zo2-2zozc+ zc2 - r2 So
At2 + Bt + C = 0
and we can solve this using the quadratic equation:

42. Ray-Sphere Intersections
3 possibilities:
Case 1: B2 – 4C < 0
Zero real roots. No intersection.
Case 2: B2 – 4C = 0
One real root
t0 = t1 = -B/2
Case 3: B2 – 4C > 0
Two real roots
Subcase a: t0 < 0, t1 > 0
t1 is the correct answer
Subcase b: 0 < t0 < t1
t0 is the correct answer
case 1
case 2 t1
case 3a t0 t0 t1
case 3b

43. Ray-Sphere Intersections
If the discriminant B2 – 4C < 0, the ray misses the sphere.
The smaller positive root (if one exists) is the closest intersection point.
We can save time by computing the small root first and only computing the large root if necessary.

44. Ray-Sphere Intersections: Algorithm
Algorithm for ray-sphere intersection:
Calculate B and C of the quadratic
Calculate the discriminant: D = B2 – 4C
If D < 0 return false (no intersection point)
Calculate smaller t-value t0:
If t0  0 then
calculate larger t-value t1:
If t1  0 return false (intersection is behind ray)
else set t = t1
else
set t = t0
Return intersection point: p = r(t) = ro + rdt

45. Ray-Sphere Intersections: Normal
The normal n at an intersection point p on a sphere is: n p R c

46. Ray-Sphere Intersections
Computation time per ray-sphere test:
17 additions / subtractions
17 multiplies
1 square root
Can we reduce the number of intersection calculations?
Yes, use a geometric approach (not here).

47. Ray-Plane Intersections
To find the intersection point of a ray with a plane:
Plane equation: ax +by +cz +d = 0
with a2 + b2 +c2 = 1
normal n = (a, b, c)
distance from (0, 0, 0) to plane is d
Ray equation:
Origin: ro = (xo, yo, zo)
Direction: rd = (xd, yd, zd)
Substitute the ray equation into the plane equation:
a(x0 + xdt) + b(y0 + ydt) + c(z0 + zdt) + d = 0
Þ ax0 + axdt + by0 + bydt + cz0 + czdt + d = 0
Solving for t we get:
t = -(ax0 + by0 + cz0 + d) / (axd + byd + czd)

48. Ray-Plane Intersections
In vector form:
If the ray is parallel to the plane and does not intersect
If the plane’s normal points away from the ray and thus the plane is culled.
If t  0 then intersection point is behind the ray, so no real intersection occurs
Otherwise, compute intersection: p = ro + rdt

49. Ray/Plane Intersection
Basic Algorithm:
vd = n·rd
if vd ³ 0 and plane has one side then return
if vd = 0 then return (ray is parallel to plane)
t = -(n·ro + d) / vd
if t < 0 then return
return r = (xo + xdt, yo + ydt, zo + zdt)

50. Ray/Polygon Intersection
Assume planar polygons
First, intersect the plane the polygon is on with the ray.
Next, determine if the intersection is within the polygon.
How do we quickly tell if the intersect is within the polygon?

51. Ray/Polygon Intersection
Idea: shoot a ray from the intersection point in an arbitrary direction
if the ray crosses an even number of polygon edges, the point is outside the polygon
if the ray crosses an odd number of polygon edges, the point is inside the polygon
11 crossings -
point inside
polygon
4 crossings -
point outside
polygon

52. Ray/Polygon Intersection
a set of N points Pj = (xj, yj, zj), j = 0, 1, … N-1
Plane:
ax + by + cz + d = 0, with normal n = (a, b, c)
Intersection point:
pi = (xi, yi, zi), pi on the plane

53. Ray/Polygon Intersection Algorithm
Step 1:
Project the polygon onto a coordinate plane
Simply discard one of the coordinates.
This will project the polygon onto the plane defined by the other two coordinates.
This doesn’t preserve area, but does preserve topology.
Discard the coordinate whose magnitude in the plane normal is the largest.
If n = (3, -1, -5) we would throw away the z coordinates.

54. Ray/Polygon Intersection Algorithm
Step 2:
Translate the polygon so that the intersection point is at the origin.
Translate all its vertices.
Step 3:
Send a ray down a coordinate axis and count the number of times it intersects the polygon.
This can be done simply by counting the number of edges that cross the coordinate axis.

55. Ray/Polygon Intersection Algorithm
Problems:
Vertices that lie exactly on the ray
Edges that lie exactly on the ray

56. Ray/Polygon Intersection Algorithm
How do we handle these cases?
Essentially, shift the vertex or edge up by e so that it will be just above the axis.

57. Ray/Polygon Intersection Algorithm
The effect of doing this e shift is:
These cases all work correctly now.

58. The Complete Algorithm - part 1
Find the coordinate of n with the largest magnitude.
For each vertex (xj, yj, zj), j = 0, 1, …, N-1 of the polygon, project the vertex onto the coordinate plane of the other two coordinates. This gives us a new coordinates system (uj, vj).
Project the intersection point (xi, yi, zi) onto the same coordinate plane as the vertices.
Translate all polygon vertices by (-ui, -vi).
(uj’, vj’) = (uj, vj) - (ui, vi).
5. Set numCrossings = 0

59. The Complete Algorithm - part 2
If v0’ < 0, set sign = -1, otherwise set sign = 1
For i = 0 to N-1 (let uN = u0, vN = v0 )
if vi+1’ < 0 set nextSign = -1
else set nextSign = 1
if (sign ≠ nextSign)
if (ui’ > 0 and ui+1’ > 0) numCrossings++
else
if (ui’ > 0 or ui+1’>0) \\ the edge might cross +u’
\\ compute the intersection with the u’ axis
uc = ui’ - vi’ * (ui+1’ -ui’)/(vi+1’ -vi’)
if uc > 0 numCrossings++
sign = nextSign
If numCrossings is odd, the intersection is inside the polygon.

60. Example Given a polygon: and intersection point
Ri = (-2, -2, 4)
Does the intersection point lie within the polygon? P0 P1 P2

61. Example Step 1: Get the plane normal, determine dominant coordinate
n can be computed from the cross product of two vectors in the plane
The vertices of the polygon can be used to compute vectors in the plane P0 P1 P2
P0 = (-3, -3, 7)
P1 = (3, -4, 3)
P2 = (4, -5, 4)
Ri = (-2, -2, 4)

62. Example Compute the Normal: v1 = P0 - P1 = (-3, -3, 7) - (3, -4, 3)
First, compute edge vectors from the vertices P0 P1
v1 = P0 - P1 = (-3, -3, 7) - (3, -4, 3)
= (-6, 1, 4) P2
P0 = (-3, -3, 7)
P1 = (3, -4, 3)
P2 = (4, -5, 4)
Ri = (-2, -2, 4)
v2 = P2 - P1 = (4, -5, 4) - (3, -4, 3)
= (1, -1, 1)

63. Example The plane normal is then v2 x v1 n = (-6, 1, 4) x (1, -1, 1)
= (5, 10, 5) P0 n P1
So the dominant coordinate is y P2
P0 = (-3, -3, 7)
P1 = (3, -4, 3)
P2 = (4, -5, 4)
Ri = (-2, -2, 4)

64. Example Step 2: Project the vertices
Step 3: Project the intersection point
project P0 = (-3, -3, 7) Þ (-3, 7)
project P1 = (3, -4, 3) Þ (3, 3)
project P2 = (4, -5, 4) Þ (4, 4) u’
P1’ Ri
P2’
P0’ v’
project Ri = (-2, -2, 4) Þ (-2, 4)

65. Example Step 4: Translate the vertices
Ri’ = (-2, 4) - (-2, 4) Þ (0, 0)
P1’ Ri u’
P2’
P0’ v’

66. Example Step 5: Set numCrossings = 0 Step 6: v0’= 3, so sign = 1 P1’Ri v’
P2’
P0’ u’

67. Example Since numCrossings is even, the point is outside the polygon
(Figure is drawn upside-down. Sorry.
The positive v’ axis points down.
Step 7:
i sign nextSign numCrossings intersection point
For i = 0 to N \\ (let uN = u0, vN = v0 )
if vi+1’ < 0 set nextSign = -1
else set nextSign = 1
if (sign ≠ nextSign)
if (ui’ > 0 and ui+1’ > 0) numCrossings++
else
if (ui’ > 0 or ui+1’>0) \\ the edge might cross +u’
\\ compute the intersection with the u’ axis
uc = ui’ - vi’ * (ui+1’ -ui’)/(vi+1’ -vi’)
if uc > 0 numCrossings++
sign = nextSign
P1’=(5,-1) Ri u’
P2’ = (6,0)
P0’=(-1,3) v’ +1 -1 1
-1-3*(5-(-1))/(-1-3) = 3.5 1 -1 +1 2 2 +1 +1
Since numCrossings is even, the point is outside the polygon

68. Shadows Send a shadow ray from intersection point to the light:
Compute the following shadow ray properties:
Shadow ray direction: sd = (l – p) / ||l – p||
Distance to the light from the intersection point:
tl = ||l – p||
Test if shadow ray intersects an object before reaching the light:
In theory, the possible range of t-values is t  [0, tl]
Due to numerical (floating-point) error, test in the range t  [e, t1] where e is some small value (such as 2–15)

69. Recursive Ray Tracing
Basic ray tracing results in basic Phong illumination plus hidden surfaces.
Shadows require only one extra ray per light source
Shadow rays do not reflect or refract.
No need to find the closest object, only need to hit once before reaching the light.
Reflection and refraction rays add a lot to the images.
Reflection and refraction can spawn many new rays since light can keep bouncing.

70. Computing ray colours How to get the final pixel colour? End result:
Every material has local colour c
Define reflexivity, refraction factors ρ (i.e. between 1.0 (mirror) and 0.0 (no reflection)
Each hit returns its colour down the chain
End result:
colour = (1 –ρ) c1 + ρ1 ( (1- ρ2)c2 + ρ2 (c3 …. )))
Local colour also influenced by shadow rays!

71. Computing reflection rays
Reflection is pretty simple
Incident angle = Exit angle
Incident ray: x + t*d
Reflection ray: a + t*(d – 2*n*dot(n,d)) (careful, assumes n and d normalized!) n a

72. Specular Reflection
Perfect reflection: angle of reflection = angle of incidence
Light continues to bounce:
Typically, some energy is lost on each bounce
n and l are unit vectors n r l

73. Specular Reflection
Implement specular reflection with a recursive call:
colour = ambient + diffuseobj + specularobj +
ks reflectedcolour
where ks represents how “perfect” the mirror is and reflected colour is the recursive call
Limit recursion: max depth or when the contribution of a ray is negligible

74. Refraction (transparency)
Light transmits through transparent objects.
Light bends when moving from one medium to another according to Snell’s law: n1
air
glass n2

75. Refraction Indices Material Index Vacuum 1.0 Air 1.0003 Water 1.33
Alcohol
Fused quartz 1.46
Crown glass 1.52
Flint glass 1.65
Sapphire 1.77
Heavy flint glass 1.89
Diamond 2.42

76. Refraction Total internal reflection
When going from a dense to less dense medium, the angle of refraction becomes more shallow
If the angle of incidence is too shallow, it can get trapped in the dense material
Optical cable
Diamonds
Calculating the transmitted vector requires:
Incident ray direction
Normal
Indices of refraction
Some trigonometry
air d
glass n

77. Ray Tracing
Ambient Light only

78. Ray Tracing
With shadows

79. Ray Tracing
Primary rays only

80. Ray Tracing
With Phong shading

81. Ray Tracing
With reflections

82. Ray Tracing
With transmissions

83. Other Ray Traced Images