ray tracing i: basics - computer graphics · cs348b lecture 2 pat hanrahan, spring 2016 ray tracing...

CS348b Lecture 2 Pat Hanrahan, Spring 2016

Ray Tracing I: Basics

Today

￭ Basic algorithms

￭Overview of pbrt

￭ Ray-surface intersection

Next lecture

￭ Techniques to accelerate ray tracing of large numbers of geometric primitives


Light Rays

Three ideas about light rays

1. Light travels in straight lines (mostly)

2. Light rays do not interfere with each other if they cross (light is invisible!)

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 casting one ray per pixel

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

Ray Tracing in Computer Graphics


Spheres and Checkerboard, T. Whitted, 1979

“An improved Illumination model for shaded display” T. Whitted, CACM 1980

Time: - VAX 11/780 (1979) 74m - PC (2006) 6s - GPU (2012) 1/30s

512 x 512

Ray Tracing Video

Spheres-over-plane.pbrt (depth=1)


Spheres-over-plane.pbrt (mirror depth=2)


Spheres-over-plane.pbrt (glass depth=1)


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


PBRT Render Loop


PBRT Intersection Methods


PBRT Intergrater


Ray-Surface Intersection


Ray-Plane Intersection

Ray:

t= +P O D!! !

O!

D!0 t≤ < ∞

P!

t

N!



Ray:

Plane:

t= +P O D!! !

O!

D!0 t≤ < ∞

t

N!

!P!( ) 0

0ax by cz d− • ="

+ + + =

P P N! ! !

P!



Ray:

Plane:

Solve for intersection

t= +P O D!! !

O!

D!0 t≤ < ∞

t

N!

!P!( ) 0

0ax by cz d− • ="

+ + + =

P P N! ! !

− • = + − • =" "

− •"= −

•

!! ! ! ! ! !

! ! !! !

( ) ( ) 0( )

t

t

P P N O D P NO P ND N

P!

Optimize Ray-Convex Polyhedra?


Polyhedra defined as the intersection of N half-planes

Triangles


1P!

2P!3P

! P!

1s

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

s1 = area(4PP2P3)/area(4P1P2P3)

s2 = area(4P1PP3)/area(4P1P2P3)

s3 = area(4P1P2P)/area(4P1P2P3)


Triangle Inside Test

Test whether a point is inside the triangle

= + +! ! ! !

1 1 2 2 3 3s s sP P P P

[ ] [ ]! "# $ =# $# $% &

1

1 2 3 2

3

sss

P P P P

[ ] [ ]−

" #$ % =$ %$ %& '

11

2 1 2 3

3

sss

P P P P

Moller-Trumbore Algorithm


+ = − − + +! ! ! ! !

1 2 0 1 1 2 2(1 )t b b b bO D P P P

= −! ! !1 1 0E P P= −

! ! !2 2 0E P P

= −! ! !

0S O P

! "•! "# $# $ = •# $# $ • # $# $ •% & % &

! !! !

! !! !

2 2

1 11 1

2 2

1tbb

S ES S

S ES D

= ×! ! !1 2S D E= ×

! ! !2 1S S E

Where:

Cost = (1 div, 27 mul, 17 add)

Ray-Sphere Intersection


Ray:

Sphere: ( )2 2 0R− − =P C!!

O!

C!

D! t= +P O D

!! !

P!

Ray-Sphere Intersection


Ray:

Sphere:O!

C!

D!

( )2 2 0t R+ − − =O D C! !!

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

2( )b = − •O C D! ! !

2 2( )c R= − −O C! !

2 42

b b acta

− ± −=

P!

( )2 2 0R− − =P C!!

t= +P O D!! !

Shapes

Shapes in pbrt


class Shape {

public:

// Shape interface

virtual BBox ObjectBound() const = 0;

virtual BBox WorldBound() const;

virtual bool CanIntersect() const;

virtual void Refine(vector<Reference<Shape>> &refined) const;

virtual bool Intersect(const Ray &ray, float *tHit,

float *rayEpsilon, DifferentialGeometry *dg) const;

virtual bool IntersectP(const Ray &ray) const;

virtual void GetShadingGeometry(const Transform &obj2world,

const DifferentialGeometry &dg,

DifferentialGeometry *dgShading) const {

*dgShading = dg;

}

virtual float Area() const;

// Shape Public Data

const Transform *ObjectToWorld, *WorldToObject;

};

Quadrics



Geometric Methods: Normals

e.g. Sphere

= −N P C!! !

θ φ

θ φ

θ

=

=

=

=!

sin cossin sincos( , , )

xyz

x y zP

N!

S! T

!, ,u vP!

θ φ θ φ θθ

∂= −

∂

!(cos cos ,cos sin , sin )P

θ φ θ φφ

∂= −

∂

!( sin sin ,sin cos ,0)P

θ φ

∂ ∂= ×∂ ∂

! !! P PN


Geometric Methods: Parameters

e.g. Sphere

sin cossin sincos

xyz

θ φ

θ φ

θ

=

=

=

1

1

tan ( , )cos

x yz

φ

θ

−

−

=

=

θ θ θ θ

φ φ

= + −

=min max min

max

( )vu

θ θ θ θ

φ φ

= − −

=min max min

max

( ) /( )/

vu

Differential Geometry Representation


struct DifferentialGeometry { // Filled in by Shape::Intersect() Point p; Normal nn; float u, v; const Shape *shape; Vector dpdu, dpdv; Normal dndu, dndv; };


Ray-Implicit Surface Intersection

1. Substitute ray equation

2. Find positive, real roots

Univariate root finding

￭Newton’s method

￭ Regula-falsi

￭ Interval methods

￭…

( , , ) 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

￭…

0 1

0 1

0 1

x x x ty y y tz z z t

= +

= +

= +

ray tracing i: basics - computer graphics · cs348b lecture 2 pat hanrahan, spring 2016 ray tracing...

Documents