Ray Tracing I: Basics

CS348b Lecture 2

Ray Tracing I: Basics

Today

￭ pbrt overview

￭ Basic algorithms

￭ Ray-surface intersection

￭ Efficiency

￭ Floating-point error

Next lecture

￭ Accelerating ray tracing of large numbers of geometric primitives

Pat Hanrahan / Matt Pharr, Spring 2021

CS348b Lecture 2

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).


CS348b Lecture 2

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

CS348b Lecture 2 Pat Hanrahan / Matt Pharr, Spring 2021

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

1. Always send ray to the light source (unless glass or mirror)

2. Recursively generate reflected rays (mirror) and transmitted rays (glass)

PBRT Overview

http://www.pbr-book.org/

Mirror, depth 1

Mirror, depth 2

Mirror, depth 3

Mirror, depth 10

Glass, depth 1

Glass, depth 2

Glass, depth 3

Glass, depth 10

Bidirectional paths

Shape Interface (Simplified)


classShape{public:Bounds3fObjectBound()const;Bounds3fWorldBound()const;boolIntersect(constRay&ray,Float*tHit,SurfaceInteraction*isect,booltestAlphaTexture)const;boolIntersectP(constRay&ray,booltestAlphaTexture);FloatArea()const;//…};

Surface Interaction (Simplified)


classSurfaceInteraction{Point3fp;Normal3fn;

Point2fuv;Vector3fdpdu,dpdv;Normal3fdndu,dndv;

struct{Normal3fn;Vector3fdpdu,dpdv;Normal3fdndu,dndv;}shading;

//…};

Ray-Surface Intersection

CS348b Lecture 2

Ray-Plane Intersection


Rayr(t) = o+ t

�!d

o

�!d

r(t)Want t where the ray intersects the plane

Plane

(p� p0) ·�!n = 0�!np

p0

CS348b Lecture 2

Ray-Plane Intersection


Ray: r(t) = o+ t�!d

Plane: (p� p0) ·�!n = 0

Substitute ray equation into plane equation:

t = � (o� p) ·�!n�!d ·�!n

(p� r(t)) ·�!n = (p� (o+ t�!d )) ·�!n = 0

CS348b Lecture 2

Finding The Closest Intersection


o

�!d

r(t) r(t0)

r(t1)

r(0)

CS348b Lecture 2

Optimizing Ray-Box


t = � (o� p) ·�!n�!d ·�!n

o

�!d

r(t)

t = �ox � px�!dx

CS348b Lecture 2

What About Rays Parallel to a Plane?


o

�!d

t = �ox � px�!dx

xy

IEEE Floating Point Standard says:

1. positive num / 0 = +Inf

negative num / 0 = -Inf

2. +Inf > all other floats

-Inf < all other floats

Math says: t = �ox � px

0= ±1

ray

plane

CS348b Lecture 2

Ray-Sphere Intersection


o

�!d

r(t)

Ray: r(t) = o+ t�!d

Sphere:

c

p

||p� c||2 � r2 = 0

r

(o+ t�!d � c)2 � r2 = 0

at2 + bt+ c = 0

a =�!d ·

�!d

b = 2(o� c) ·�!d

c = ((o� c) · (o� c))� r2

Ray-Triangle Intersection

CS348b Lecture 2


1. Find ray-plane intersection point using the methods developed previously

2. Test whether the intersection point is inside the triangle


The signed area of the parallelogram given by the vectors and is given by

CS348b Lecture 2

Review: Geometric Building-Blocks


Half of this area is the area of the triangle determined by the 3 points

��x1 x2

y1 y2

�� = (x1y2)� (x2y1)

v1 = (x1, y1) v2 = (x2, y2)

v1

v1

v2

v2

Area of a triangle with vertices etc.

CS348b Lecture 2

Review: Geometric Building-Blocks


p0 = (x0, y0)

1

2

��x1 � x0 x2 � x0

y1 � y0 y2 � y0

�� =1

2((x1 � x0)(y2 � y0)� (x2 � x0)(y1 � y0))

p0

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p1

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p2

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CS348b Lecture 2

Barycentric Coordinates


a0(p) = Area(p1, p2, p)

a1(p) = Area(p2, p0, p)

a2(p) = Area(p0, p1, p)

bi =ai

Area(p0, p1, p2), 0 i 2

Define barycentric coordinates:

is inside the triangle ifp

pp0

p1

p2

b0 > 0, b1 > 0, b2 > 0

CS348b Lecture 2


Points on a plane:

1. Find ray-plane intersection point

2. Test whether that point is inside the triangle

Inside iff


p = b0p0 + b1p1 + b2p2

⇥p0 p1 p2

⇤2

4b0b1b2

3

5 =⇥p

⇤

2

4b0b1b2

3

5 =⇥p0 p1 p2

⇤�1 ⇥p

⇤

b0 > 0, b1 > 0, b2 > 0

CS348b Lecture 2

Ray Triangle Intersection


o+ t�!d = (1� b1 � b2)p0 + b1p1 + b2p2

[Möller and Trumbore 1997]

2

4tb1b2

3

5 =1

�!s1 ·�!e1

2

4�!s2 ·�!e2�!s1 ·�!s�!s2 ·

�!d

3

5�!e1 = �!p1 ��!p0�!e2 = �!p2 ��!p0�!s = �!o ��!p0

�!s1 =�!d ⇥�!e2

�!s2 = �!s ⇥�!e1

Where

Contemporary Bathroom Scene


525M triangles

12.9B ray-triangle tests

~26 tests per triangle

3.2B hits (~25% of tests)

~30 minutes with 8 threads

5m 6s in ray/triangle (17%)

Conserve Memory vs. Coherence


classTriangle{//…int*v;TriangleMesh*mesh;};

classTriangleMesh{Point3f*P;//…};

pbrt Triangle

classTriangle{//…Point3fP[3];};

Alternative Triangle

CS348b Lecture 2

Ray-Implicit Surface Intersection


f(x, y, z) = 0

Implicit surface

x = ox + tdx

y = oy + tdy

z = oz + tdz

f⇤(t) = 0

Substitute ray equation

Univariate root finding

Floating-Point Error

CS348b Lecture 2

Floating-point Representation

Scientific notation

￭with a fixed sized mantissa (23-bits),

￭ a limited exponent range (8-bits, e-127),

￭ signed bit


±1.m⇥ 2e

2.5 = 1.25⇥ 21 = 1.01b ⇥ 21

1/3 ⇡ 1.01010101010101010101011b ⇥ 2�2

0 = ?

Floating-point Representation


Roundoff Error


Catastrophic Cancellation


Catastrophic Cancellation


A

B

A’

B’

A - B vs A’ - B’ …

CS348b Lecture 2

Effect of Roundoff Error


CS348b Lecture 2

Problems This Causes

Pat Hanrahan / Matt Pharr, Spring 2021Effect of Floating-Point Roundoff Error

CS348b Lecture 2

Round-off Error Remedies

Use double (fp64) rather than float (fp32)

￭ Can help, but also avoids the root problem

￭ Increases memory use, reduces performance

Ignore reintersection with the last object hit

￭Only works for flat objects (e.g. triangles)

￭No help if model has coincident triangles

Have a tmin along ray to ignore close intersections

￭What value should tmin take?


CS348b Lecture 2

Refine Intersection Point, Bound Error


See pbrt 3.9 for details

Ray Tracing I: Basics

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