1GR2-00 GR2 Advanced Computer Graphics AGR Lecture 13 An Introduction to Ray Tracing.

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


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


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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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


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


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


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


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


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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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


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

1GR2-00 GR2 Advanced Computer Graphics AGR Lecture 13 An Introduction to Ray Tracing.

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Transcript of 1GR2-00 GR2 Advanced Computer Graphics AGR Lecture 13 An Introduction to Ray Tracing.