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1GR2-00
GR2Advanced Computer
GraphicsAGR
GR2Advanced Computer
GraphicsAGR
Lecture 13
An Introduction to Ray Tracing
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Ray TracingRay 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
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Ray Tracing ExampleRay Tracing Example
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Firing RaysFiring Rays
pixel positionson view plane
camera
The view planeis marked with agrid correspondingto pixel positionson screen.
A ray is traced fromthe camera througheach pixel in turn.
light
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Finding IntersectionsFinding Intersections
pixel positionson view plane
camera
We calculate theintersection of the ray with all objectsin the scene.
The nearest inter-section will be thevisible surface forthat ray
light
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Intensity CalculationIntensity Calculation
pixel positionson view plane
camera
The intensity at this nearest intersectionis found from the usual Phongreflection model.
Stopping at thispoint gives us a verysimple renderingmethod.
Often called:ray casting
light
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Phong Reflection ModelPhong Reflection Model
lightsourceN
LR
Veye
surface
I() = Ka()Ia() + ( Kd()( L . N ) + Ks( R . V )n ) I*() / dist
Note: R.V calculation replaced by H.N for speed - H = (L+V)/2
dist = distance attenuation factor
Here V is direction of incoming ray, N is normal, L is direction tolight source, and R is direction of perfect spec. reflection.
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Intensity CalculationIntensity Calculation
pixel positionson view plane
camera
Thus Phong reflectionmodel gives us intensity at point based on alocal model:
I = I local
whereI local = I ambient +
I diffuse + I specular
Intensity calculatedseparately for red, green, blue
light
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ShadowsShadows
pixel positionson view plane
camera
To determine ifthe point is in shadow,we can fire a ray atthe light source(s) -in direction L.
If the ray intersectsanother object, then point is in shadowfrom that light.In this case, we just useambient component:I local = I ambient
Otherwise the point isfully lit.
light
shadowray
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Reflected RayReflected Ray
pixel positionson view plane
camera
Ray tracing modelsreflections by lookingfor light coming inalong a secondaryray, making a perfectspecular reflection.
R1 = V - 2 (V.N) N
R1
light
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Reflected Ray- Intersection and Recursion
Reflected Ray- Intersection and Recursion
We calculate theintersection of thisreflected ray, with allobjects in the scene.
The intensity at thenearest intersection point is calculated, and added as a contributionattenuated by distance.
This is done recursively.pixel positionson view plane
camera
light
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Reflected Ray - Intensity Calculation
Reflected Ray - Intensity Calculation
pixel positionson view plane
camera
light
The intensity calculationis now:I = I local + k r * I reflected
Here I reflected
is calculated recursively
k r is a reflection coefficient (similar to k s )
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Ray TerminationRay Termination
pixel positionson view plane
camera
Rays terminate:- on hitting diffusesurface- on going to infinity- after severalreflections - why?
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Transmitted RayTransmitted Ray
If the object is semi-transparent, we alsoneed to take into account refraction
pixel positionson view plane
camera
light
Thus we follow alsotransmitted rays,eg T1.
T1
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RefractionRefraction
N
V
T
ir
r
i Change in directionis determined by therefractive indices ofthe materials, i and r
Snell’s Law:sin r = (i / r ) * sin i
T = (i / r ) V - ( cos r - (i / r ) cos i ) N
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Refraction ContributionRefraction 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
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Intensity CalculationIntensity Calculation
pixel positionson view plane
camera
The intensity calculationis now:I = I local + k r * I reflected
+ k t * I transmitted
I reflected and I transmitted
are calculated recursively
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Binary Ray Tracing TreeBinary Ray Tracing Tree
S1
S2 S3
S4
pixel positionson view plane
camera
S1
S2 S3
S4
R1T1
T1R1
The intensity iscalculated bytraversing the treeand accumulatingintensities
light
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Calculating Ray Intersections
Calculating Ray Intersections
A major part of ray tracing calculation is the intersection of ray with objects
Two important cases are:– sphere– polygon
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Ray - Sphere IntersectionRay - 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 * iy(t) = y1 + t * (y2 - y1) = y1 + t * jz(t) = z1 + t * (z2 - z1) = z1 + t * k
As t increases from zero, x(t), y(t), z(t) traces outthe 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
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Ray - Sphere IntersectionRay - Sphere Intersection
Putting the parametric equations for x(t), y(t), z(t) inthe 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 intersectionsb2 - 4ac = 0 ray is tangentb2 - 4ac > 0 two intersections, we want smallest positive
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Ray - Polygon IntersectionRay - Polygon Intersection
Equation of ray:x(t) = x1 + t * iy(t) = y1 + t * jz(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.
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Ray Tracing - LimitationsRay 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.
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Ray Tracing ExampleRay Tracing Example
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Depth 0Depth 0
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Depth 1Depth 1
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Depth 2Depth 2
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Depth 3Depth 3
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Depth 4Depth 4
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Depth 7Depth 7
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AcknowledgementsAcknowledgements
As ever, thanks to Alan Watt for the excellent images
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