1. GR2 Advanced Computer Graphics AGR
Lecture 13
An Introduction to Ray Tracing

2. Ray Tracing
Ray tracing is an alternative rendering approach to the polygon projection, shading and z-buffer way of working
It is capable of high quality images through taking account of global illumination

3. Ray Tracing Example

4. Firing Rays The view plane is marked with a grid corresponding
pixel positions
on view plane
camera
light
The view plane
is marked with a
grid corresponding
to pixel positions
on screen.
A ray is traced from
the camera through
each pixel in turn.

5. Finding Intersections
pixel positions
on view plane
camera
light
We calculate the
intersection of the
ray with all objects
in the scene.
The nearest inter-
section will be the
visible surface for
that ray

6. Intensity Calculation
pixel positions
on view plane
camera
light
The intensity at this
nearest intersection
is found from the
usual Phong
reflection model.
Stopping at this
point gives us a very
simple rendering
method.
Often called:
ray casting

7. Phong Reflection Model
light
source N R
eye L  V 
surface
I() = Ka()Ia() + ( Kd()( L . N ) + Ks( R . V )n ) I*() / dist
dist = distance attenuation factor
Here V is direction of incoming ray, N is normal, L is direction to
light source, and R is direction of perfect spec. reflection.
Note: R.V calculation replaced by H.N for speed - H = (L+V)/2

8. Intensity Calculation
pixel positions
on view plane
camera
light
Thus Phong reflection
model gives us intensity
at point based on a
local model:
I = I local
where
I local = I ambient +
I diffuse + I specular
Intensity calculated
separately for red,
green, blue

9. Shadows To determine if the point is in shadow, we can fire a ray at
pixel positions
on view plane
camera
light
To determine if
the point is in shadow,
we can fire a ray at
the light source(s) -
in direction L.
If the ray intersects
another object, then
point is in shadow
from that light.
In this case, we just use
ambient component:
I local = I ambient
Otherwise the point is
fully lit.
shadow
ray

10. Reflected Ray R1 R1 = V - 2 (V.N) N Ray tracing models
pixel positions
on view plane
camera
light
Ray tracing models
reflections by looking
for light coming in
along a secondary
ray, making a perfect
specular reflection.
R1 = V - 2 (V.N) N R1

11. Reflected Ray- Intersection and Recursion
pixel positions
on view plane
camera
light
We calculate the
intersection of this
reflected ray, with all
objects in the scene.
The intensity at the
nearest intersection
point is calculated, and
added as a contribution
attenuated by distance.
This is done recursively.

12. Reflected Ray - Intensity Calculation
pixel positions
on view plane
camera
light
The intensity calculation
is now:
I = I local + k r * I reflected
Here I reflected
is calculated recursively
k r is a reflection
coefficient (similar to k s )

13. Ray Termination Rays terminate: - on hitting diffuse surface
pixel positions
on view plane
camera
Rays terminate:
- on hitting diffuse
surface
- on going to infinity
- after several
reflections - why?

14. Transmitted Ray If the object is semi- transparent, we also
pixel positions
on view plane
camera
If the object is semi-
transparent, we also
need to take into
account refraction
light T1
Thus we follow also
transmitted rays,
eg T1.

15. Refraction r i r i Snell’s Law: sin  r = ( i /  r ) * sin  i
Change in direction
is determined by the
refractive indices of
the materials, i and r T r N i V
Snell’s Law:
sin  r = ( i /  r ) * sin  i
T = ( i /  r ) V - ( cos  r - ( i /  r ) cos  i ) N

16. Refraction Contribution
The contribution due to transmitted light is taken as:
kt * It(  )
where
kt is the transmission coefficient
It(  ) is the intensity of transmitted light, again calculated separately for red, green, blue

17. Intensity Calculation
pixel positions
on view plane
camera
The intensity calculation
is now:
I = I local + k r * I reflected
+ k t * I transmitted
I reflected and I transmitted
are calculated recursively

18. Binary Ray Tracing TreeS4
pixel positions
on view plane
camera
light S1 T1 R1 S2 S3 R1 S2 S3 T1 S1 S4
The intensity is
calculated by
traversing the tree
and accumulating
intensities

19. Calculating Ray Intersections
A major part of ray tracing calculation is the intersection of ray with objects
Two important cases are:
sphere
polygon

20. Ray - Sphere Intersection
Let camera position be (x1, y1, z1)
Let pixel position be (x2, y2, z2)
Any point on ray is:
x(t) = x1 + t * (x2 - x1) = x1 + t * i
y(t) = y1 + t * (y2 - y1) = y1 + t * j
z(t) = z1 + t * (z2 - z1) = z1 + t * k
As t increases from zero, x(t), y(t), z(t) traces out
the line from (x1, y1, z1) through (x2, y2, z2).
Equation of sphere, centre (l, m, n) and radius r:
(x - l)2 + (y - m)2 + (z - n)2 = r2

21. Ray - Sphere Intersection
Putting the parametric equations for x(t), y(t), z(t) in
the sphere equation gives a quadratic in t:
at2 + bt + c = 0
Exercise: write down a, b, c in terms of i, j, k, l, m, n, x1, y1, z1
Solving for t gives the intersection points:
b2 - 4ac < 0 no intersections
b2 - 4ac = 0 ray is tangent
b2 - 4ac > 0 two intersections, we want smallest positive

22. Ray - Polygon Intersection
Equation of ray:
x(t) = x1 + t * i
y(t) = y1 + t * j
z(t) = z1 + t * k
Equation of plane:
ax + by + cz + d = 0
Intersection:
t = - (ax1 + by1 + cz1 + d) / (ai + bj + ck)
Need to check intersection within the extent of the polygon.

23. Ray Tracing - Limitations
Ray tracing in its basic form is computationally intensive
Much research has gone into increasing the efficiency and this will be discussed in next lecture.

24. Ray Tracing Example

25. Depth 0

26. Depth 1

27. Depth 2

28. Depth 3

29. Depth 4

30. Depth 7

31. Acknowledgements
As ever, thanks to Alan Watt for the excellent images