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Ray Tracing

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What is ray tracing

• Ray tracing is a technique for rendering three-dimensional graphics with very com-plex light interactions. This means you can create pictures full of mirrors, transparent surfaces, and shadows, with stunning results.

• A very simple method to both understand and implement.

• It is based on the idea that you can model re-flection and refraction by recursively follow-ing the path that light takes as it bounces through an environment

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PCKTWTCH by Kevin Odhner, POV-Ray

Raytraced Images

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Kettle, Mike Miller, POV-Ray

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Ray Tracing Model

Source: wikipedia

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Ray tracing

Shadow rays

Reflection ray

refracted ray

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Ray tracing algorithm

• Builds the image pixel by pixel• Cast additional rays from the hit point to de-

termine the pixel color– Shoot rays toward each light. If they hit some-

thing, the object is shadowed from that light, otherwise use “standard model” for the light

– Reflection rays for mirror surfaces, to see what should be reflected in the mirror

– Refraction rays to see what can be seen through transparent objects

– Sum all the contributions to get the pixel color

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Recursive ray tracing

• When a reflected or refraction ray hits a sur-face, repeat the whole process from that point– Send more out shadow rays– Send out new reflected rays (if required)– Send out a new refracted ray (if required)– Generally, reduce the weight of each additional ray

when computing the contributions to the surface color

– Stop when the contribution from a ray is too small to notice or maximum recursion level has been reached

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Ray tracing implementation

• Ray tracing breakdown into two tasks– Constructing the ray to cast– Intersection rays with geometry

• The former problem is simple vector arithmetic

• Intersection calculation can be done in world coordinates or model coor-dinates

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Constructing Rays• Define rays by an initial point and a direction:

x(t)=x0+td• Eye rays: Rays from the eye through a pixel

– Construct using the eye location and the pixel’s location on the image plane. X0 = eye

• Shadow rays: Rays from a point on a surface to the light.– X0 = point on surface

• Reflection rays: Rays from a point on a surface in the re-flection direction– Construct using laws of reflection. X0 = surface point

• Transmitted rays: Rays from a point on a transparent sur-face through the surface– Construct using laws of refraction. X0 = surface point

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Ray-Object Intersections

• Aim: Find the parameter value, ti , at which the ray first meets object i

• Write the surface of the object implicitly: f(x)=0– Unit sphere at the origin is x•x-1=0– Plane with normal n passing through origin is: n•x=0

• Put the ray equation in for x– Result is an equation of the form f(t)=0 where we

want t– Now it’s just root finding

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Ray-Sphere Intersection

• Quadratic in t– 2 solutions: Ray passes through sphere - take mini-

mum value that is > 0– 1 solution: Ray is tangent - use it if >0– 0 solutions: Ray does not hit sphere

• Numerical stability is very important. For example, can a reflection ray hit the same sphere?

012:

01 :Substitute

01 :Sphere

)( :Ray

0002

00

0

xxdxdd

dxdx

xx

dxx

tt

tt

tt

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Sphere Intersection

A sphere is defined by its center, s, and its radius r. The intersec-tion of a ray with a sphere can be computed as follows:

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Ray-Plane Intersections

• This is plane going through the origin– What about an arbitrary plane?

• To do polygons, intersect with plane then do point-in-polygon test…

dn

xn

xndn

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xn

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Intersections

• It is necessary to determine the in-tersection between a ray with objects to decide which pixel is

• Intersection with a sphere• Intersection with a plane• …

Ray Tracing IlluminationRecursive

dtransmittereflecteddirect IIIVEI ),(

),( dtransmittetdtransmitte VPIkI

shinyn

sdlightambientadirect RVkLNkIIkI ˆˆˆˆ

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directIVEI ),( reflecteddirect IIVEI ),(

),( reflectedrreflected VPIkI

Check for shadowing (intersection with object along ray (P,L))

The Ray Tree

R2

R1

R3

L2

L1 L3 N1

N2

N3

T1

T3

Ni surface normal

Ri reflected ray

Li shadow ray

Ti transmitted (refracted) ray

View-point

L1

T3 R3

L3 L2

T1 R1

R2

Eye

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Reflection Rays

• The laws of reflection

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Reflection

• Reflection angle = view angle

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Reflection

• The maximum depth of the tree affects the handling of re-fraction

• If we send another reflected ray from here, when do we stop? 2 solutions (complementary)– Answer 1: Stop at a fixed depth. – Answer 2: Accumulate product of reflection coefficients and

stop when this product is too small.

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Reflection

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Refraction

Snell’s Lawsin

sint i

ri t

NMT ttˆcosˆsinˆ

i

i INM

sin

)ˆcosˆ(ˆ

NINT tii

t ˆcos)ˆcosˆ(sin

sinˆ

INT rtirˆˆ)coscos(ˆ

))ˆˆ(1(1sin1sin1cos

ˆˆcos

22222 IN

IN

rirtt

i

INININT rrrˆˆ))ˆˆ(1(1)ˆˆ(ˆ 22

N

N

M

T

Ii

t

IN iˆcosˆ

iN cosˆ

Note that I is the nega-tive of the incoming ray

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Refraction

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Shadow Rays

• Shadows are important lighting effect that can be easily with ray tracing

• If we wish to compute the illinumination with shadows for a point, we shoot an additional ray from the point to every light source

• A light is only allowed to contribute to the final color if the ray doesn’t hit any-thing between the point and a light source

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Pseudo Code for Ray Tracingrgb lsou; // intensity of light source rgb back; // background intensity rgb ambi; // ambient light intensity

Vector L // vector pointing to light sourceVector N // surface normalObject objects [n] //list of n objects in scene float Ks [n] // specular reflectivity factor for each objectfloat Kr [n] // refractivity index for each object float Kd [n] // diffuse reflectivity factor for each object Ray r;

void raytrace() {for (each pixel P of projection viewport in raster order) { r = ray emanating from viewer through P

int depth = 1; // depth of ray tree consisting of multiple paths the pixel color at P = intensity(r, depth)}

}

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rgb intensity (Ray r, int depth) {Ray flec, frac;rgb spec, refr, dull, intensity;

if (depth >= 5) intensity = back; else {

find the closest intersection of r with all objects in scene

if (no intersection) { intensity =back;} else { Take closest intersection which is object[j] compute normal N at the intersection point if (Ks[j] >0) { // non-zero specular reflectivity compute reflection ray flec; refl = Ks[j]*intensity(flec, depth+1); } else refl =0; if (Kr[j]>0) { // non-zero refractivity compute refraction ray frac; refr = Kr[j]*intensity(frac, depth+1); } else refr =0; check for shadow; if (shadow) direct = Kd[j]*ambi else direct = Phong illumination computation; intensity = direct + refl +refr;

} } return intensity; }

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Thank for attention!