ray tracing - computer graphics€¦ · classic ray tracing greeks: do light rays proceed from the...

CS348B Lecture 2 Pat Hanrahan, Spring 2005

Ray Tracing

Today

Basic algorithms

Overview of pbrt

Ray-surface intersection for single surface

Next lecture

Acceleration techniques for ray tracing large numbers of geometric primitives


Classic Ray Tracing

Greeks: Do light rays proceed from the eye to the light, or from the light to the eye?

Gauss: Rays through lenses

Three ideas about light

1. Light rays travel in straight lines

2. Light rays do not interfere with each other if they cross

3. Light rays travel from the light sources to the eye (but the physics is invariant under path reversal - reciprocity).


Ray Tracing in Computer Graphics

Appel 1968 - Ray casting

1. Generate an image by sending one ray per pixel

2. Check for shadows by sending a ray to the light


Ray Tracing in Computer Graphics

Whitted 1979

Recursive ray tracing (reflection and refraction)

Ray Tracing Video


Spheres-over-plane.pbrt (g/m 10)


Spheres-over-plane.pbrt (mirror 0)






Spheres-over-plane.pbrt (glass 0)


PBRT Architecture


PBRT Architecture

Ray-Surface Intersection


Ray-Plane Intersection

Ray:

Plane:

Solve for intersection

Substitute ray equation into plane equation

t= +P O D

( ) 00ax by cz d

− • =′+ + + =P P N

OD

t

0 t≤ < ∞ N

− • = + − • =′ ′

− •′= −

•

( ) ( ) 0( )

t

t

P P N O D P N

O P ND N

P

′P


Ray-Polyhedra

Ray-Slab

Ray-Convex PolyhedraRay-Box

1t 2t

Note: Procedural geometry


Ray-Triangle Intersection 1

1P

2P3P

P1s

2s3s

1 1 2 2 3 3s s s= + +P P P P

Barycentric coordinates

1

2

3

1 2 3

0 10 10 1

1

sss

s s s

≤ ≤

≤ ≤

≤ ≤

+ + =

Inside triangle criteria

1 2 3area( )s = ∆PP P

2 3 1area( )s = ∆PP P

3 1 2area( )s = ∆PPP


2D Homogeneous Coordinates

ax by cz• = + +L P

1 2 1 2

1 2 1 2 1 2

1 2 1 2

x b c c by c a a cz a b b a

− = = × = − −

P L L

1 2 1 2

1 2 1 2 1 2

1 2 1 2

a y z z yc z x x zc x y y x

− = = × = − −

L P P


2D Homogeneous Coordinates

1 1 1

1 2 3 2 2 2

3 3 3

area( )x y zx y zx y z

∝PP P

1 1 1

1 2 3 2 2 2

3 3 3

x y zx y zx y z

• × =P P P



= + +1 1 2 2 3 3s s sP P P P

[ ] [ ]1

1 2 3 2

3

sss

=

P P P P

[ ]1 2 3

2 3 1

3 1 2

sss

× = × ×

P PP P PP P



2 3 2 31 2 3

1 2 3

1 3 3 12 3 1

1 2 3

1 2 1 23 1 2

1 2 3

s

s

s

×= = • = • ×

∆

×= = • = • ×

∆

×= = • = • ×

∆

P P P P PP P P PP P P



[ ]1 2 3

2 3 1

3 1 2

sss

× = × ×

P PP P PP P


Ray-Polygon Intersection

1. Find intersection with plane of support

2. Test whether point is inside 3D polygon

a. Project onto xy plane

b. Test whether point is inside 2D polygon


Point in Polygon

inside(vert v[], int n, float x, float y){ int cross=0; float x0, y0, x1, y1;x0 = v[n-1].x - x;y0 = v[n-1].y - y;while( n-- ) {x1 = v->x - x;y1 = v->y - y;if( y0 > 0 ) {if( y1 <= 0 )if( x1*y0 > y1*x0 ) cross++;}else { if( y1 > 0 )if( x0*y1 > y0*x1 ) cross++;}x0 = x1; y0 = y1; v++;}return cross & 1;}


Ray-Sphere Intersection

Ray:

Sphere: ( )2 2 0R− − =P CO

C

D

( )2 2 0t R+ − − =O D C

2 0at bt c+ + =2a = D

2( )b = − •O C D2 2( )c R= − −O C

2 42

b b acta

− ± −=

t= +P O DP


Geometric Methods

Methods

Find normal and tangents

Find surface parameters

e.g. Sphere

Normal

Parameters

= −N P C

sin cossin sincos

xyz

θ φθ φθ

===

1

1

tan ( , )cos

x yz

φ

θ

−

−

=

=

N

ST

, ,u vP


Ray-Implicit Surface Intersection

1. Substitute ray equation

2. Find positive, real roots

Univariate root finding

Newton’s method

Regula-falsi

Interval methods

Heuristics

( , , ) 0f x y z =

*( ) 0f t =

0 1

0 1

0 1

x x x ty y y tz z z t

= +

= +

= +


Ray-Algebraic Surface Intersection

( , , ) 0np x y z =

*( ) 0np t =

Degree n

Linear: Plane

Quadric: Spheres, …

Quartic: Tori

Polynomial root finding

Quadratic, cubic, quartic

Bezier/Bernoulli basis

Descartes’ rule of signs

Sturm sequences

0 1

0 1

0 1

x x x ty y y tz z z t

= +

= +

= +


History

Polygons Appel ‘68Quadrics, CSG Goldstein & Nagel ‘71Tori Roth ‘82Bicubic patches Whitted ‘80, Kajiya ‘82Superquadrics Edwards & Barr ‘83Algebraic surfaces Hanrahan ‘82Swept surfaces Kajiya ‘83, van Wijk ‘84Fractals Kajiya ‘83Height fields Coquillart & Gangnet ’84, Musgrave ‘88Deformations Barr ’86Subdivision surfs. Kobbelt, Daubert, Siedel, ’98

P. Hanrahan, A survey of ray-surface intersection algorithms

ray tracing - computer graphics€¦ · classic ray tracing greeks: do light rays proceed from the...

Documents