Ray TracingRay Tracing

CptS 548

Advanced Computer Graphics

Why Write a Ray Tracer?Why Write a Ray Tracer?

• More elegant than polygon scan conversion

• Testbed for numerous techniques and effects– modeling– rendering– texturing

• Easiest photorealistic renderer to implement

OverviewOverview

• Vector arithmetic

• Lookat viewing coordinates

• Ray casting

• Shadows

• Reflection

• Refraction

Homogeneous CoordinatesHomogeneous Coordinates

• Point: o = (xo, yo, zo, 1)

• Direction: d= (xd, yd, zd, 0)

• Points translate, directions do not

• Homogeneous coordinate always 0 or 1

• No perspective transformation

• Ray: r = (o,d) = ((xo, yo, zo,1),(xd, yd, zd,0))

• Ray direction d always unit length

o

dr

Dot ProductDot Product

• a · b = dot(a,b) = xa xb + ya yb + za zb

• Cosine of angle between a and bif both unit length

• Used to “cast a shadow” of one unit vector onto another

a

b

(a·b) a

Cross ProductCross Product

• a b = cross(a,b)= (ya zb - za yb, za xb - xa zb, xa yb - ya xb, 0)

• Returns direction perp. to plane of a, b

• Used to set up coordinate systems

ab

c = a b

c a

Eye LUV CoordinatesEye LUV Coordinates

lookat

eye

l

up u

v

l = (lookat - eye)/||(lookat - eye)||v = (l up)/||(l up)||u = v l

Pixels in World CoordinatesPixels in World Coordinates

eye

dl

-av

-u

ll

ll = eye + dl - av - u

aspect ratio a = w/h

focal length d = 1/tan(fovy/2)

fovy

Casting Rays through PixelsCasting Rays through Pixels

for (j = 0; j < VRES; j++) {for (i = 0; i < HRES; i++) {

p = ll + 2av (double)i/HRES + 2u (double)j/VRES;d = (p - eye)/||p - eye||;r = (eye,d);color = TraceRay(r);plot(i,j,color);

}}

TraceRay AttributesTraceRay Attributes

• Invoked with ray parameter– Better if object database is global– Best if TraceRay is a member function of object

database object (yikes)

• Returns a color vector (R,G,B,) indicates transparency– color = foreground + (1- ) background

Object HierarchyObject HierarchyObject

intersects()TraceRay()

Primitive

normal()

HitObject

thitn

shade()

Composite

Sphere

Polygon

Patch

List

CSG

TraceRay DefinitionTraceRay Definition

Color TraceRay(Ray r, int depth) {HitObject ho;color c = background;if (intersects(r,ho)) {

c = ho.shade(lights); /* casts a shadow ray */c += ho.ks*TraceRay(ho.reflect(r), depth-1);c += ho.kr*TraceRay(ho.refract(r), depth-1);

}return c;

}

List DefinitionList Definitionclass List : public Composite {

Object item[100];int n;boolean intersects(Ray r, HitObject &ho) {

min_t = 1.e20;for (i = 0; i < n; i++)

if (item[i].intersects(r,o) && o.t < min_t) {min_t = o.t; ho = o;

}combine_shade(ho);return (min_t < 1.e20);

}}

shade Definitionshade DefinitionColor shade(Lights lights) {

Color c = ka;for (l = 0; l < lights.n; l++) {

ldir = normalized(lights[l].pos - hit);if (!intersects(ray(hit,ldir))) c += kd*cd*dot(n, ldir) + …

}return(c);

}

reflect Definitionreflect Definition

r.d bounce

o.n

Ray reflect(Ray r) {Ray bounce;bounce.o = hit;bounce.d = 2.0*dot(n,-r.d)*n + r.d;return(bounce);

}

Calculating RefractionCalculating RefractionSnell’s Law: i sin i = t sin t

Let =i /t = sin t / sin i

t = sin t m - cos t n

m = (cos i n - i) / sin i

= (sin t / sin i) (cos i n - i) - cos t n

cos i n - i

i

n

-n

i

t t

m

= ( cos i - cos t )n - i

itt 222 sin1sin1cos

ininint ))(1(1)( 22

refract Definitionrefract Definition

Ray refract(Ray r) {Ray t;double ni = dot(n,-r.d);eta = current_index/new_index;t.o = hit;t.d =(eta*ni - sqrt(1.0 - eta*eta*(1-ni*ni)))*n + eta*r.d;return(t);

}

ininint ))(1(1)( 22

Ray-Object IntersectionRay-Object Intersection

• intersects() member function of Object class

• Called with ray and hitobject

• Intersects ray with this object

• Returns intersection data in hitobject– t - parameter, hit - location, n - normal– new - cannot be static because of recursion– always primitive, never composite

Intersection ComputationIntersection Computation

• Parametric ray: r(t) = o + t d– t 0– If ||d|| = 1, t is distance along ray

• Implicit object: f(x) = 0

• Intersection occurs when f(r(t)) = 0– Real function of one real variable– Intersection root finding

Sphere ObjectSphere Object

class Sphere : public Primitive {Point c;double r;boolean intersects(Ray r, HitObject ho);

}

cr

Sphere IntersectionSphere Intersectionf(x)=(x - c)(x - c) - r2

f(r(t)) = (o + t d - c)(o + t d - c) - r2

= dd t2 + 2 (o-c)d t + (o-c)(o-c) - r2

A = dd = 1B = 2 (o-c)dC = (o-c)(o-c) - r2

A

ACBBt

2

42

D = B*B - 4*C;if (D < 0.0) return FALSE;rootD = sqrt(D);t0 = 0.5*(-B - rootD);t1 = 0.5*(-B + rootD);if (t0 >= 0) t = t0, return TRUE;if (t1 >= 0) t = t1, return TRUE;return FALSE;

Other HitObject ValuesOther HitObject Values

hit = r.o + t*r.d

n = hit - c

Ellipsoid IntersectionEllipsoid Intersection

f(x) = 0 f(T-1x)=0

Let T be a 4x4 transformationT distorts a sphere into an ellipsoid

f(T-1r(t))= f(T-1(o + t d))= f(T-1o + t T-1d))

Ellipsoid is implicit surface of f(T-1x)

Ellipsoid::intersects(r,&ho) {Ray Tir = (Ti * o, Ti * d);Sphere::intersects(Tir,ho);

}

o

d

T-1o

T-1d

T

xT-1x

What about the normal?What about the normal?

f(x) = 0 f(T-1x)=0T

Txx

n is tangent plane [a b c d]x is point [x y z 1]T

Such that matrix product n x = 0

n

(n Q) Tx = 0What is Q?Q = T-1

(n T-1) Tx = n (T-1 T)x = 0

New normal n’ = n T-1 = (T-1)T nT

n’

Intersecting QuadricsIntersecting Quadrics

• Just like sphere: x2 + y2 + z2 - r2

• Cylinder?– x2 + y2 - r2

• Cone?– x2 + y2 - z2

• Variations?– Use the 4x4 transformation trick

Constructive Solid GeometryConstructive Solid Geometry

• Construct shapes from primitives using boolean set operations

• Union: A+B?– A or B

• Intersection: A*B?– A and B

• Difference: A - B?– A and not B

CSG IntersectionsCSG Intersections

• List of t-values– for left object L– for right object R

• Assume ray originates outside object– t = 0 means ray originates inside object

• Traverse both lists in increasing t order

• Roth Table determine resulting t-values

Accelerating Ray TracingAccelerating Ray Tracing

• Q: Why is ray tracing so slow?

• A: It intersects every ray with every object

• Q: How can we make ray tracing faster?

• A: Coherence– Image coherence - neighboring pixel, same obj.– Spatial coherence - neighboring pt., same obj.– Temporal coherence - next frame, same image

Shadow CachingShadow Caching

• Neighboring points in a shadow are probably shadowed by the same object

• Start shadow ray intersection search with object intersected in last shadow search

Bounding VolumeBounding Volume

Cartman contains 100,000 polygons

Ray-Cartman intersection = 100,000 ray-poly intersections

Even if the ray misses Cartman

Solution:Place a sphere around Cartman

If ray misses sphere then ray misses Cartman

Bounding Volume HierarchiesBounding Volume Hierarchies

GridsGrids

• 3-D array of lists• Ray intersected with

all objects in a given cell

• Cells visited in Bresenham order

Tagging the ObjectTagging the Object

• Intersection test for entire object

• Not just portion of object in grid cell

• Only intersect object once

• Cache temporary intersection info in object

Other Partitioning StructuresOther Partitioning Structures

• Octree– Ray can parse through large empty areas– Requires less space– Subdivision takes time

• Binary Space Partition (BSP) Tree– Trees more balanced– Partitioning more flexible -> shallower trees– Added ray-plane intersections

Saving the PictureSaving the Picture

• Windows– PrtSc key

• Copies current window

• Paste into any app

• Save as any file type

– Libraries• libtiff, etc.

• Unix– xwd

• saves a window

• as an xwd file

• convert to any format

– xv• grabs a window

• save as any format

– Libraries

Submitting the PictureSubmitting the Picture

• Format– JPEG 100% uncompressed– Don’t want compression artifacts– Might be mistaken for bugs

• Web page– ~you/public_html/cs319– small compressed version– link to full-size uncompressed version

Submitting the CodeSubmitting the Code

• E-mail hart@eecs.wsu.edu

• Attach– compressed tar file– zip/gzip archive

• Contents– source code (+headers)– makefile– scene description files