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MIT EECS 6.837, Cutler and Durand 1

Ray Casting II

MIT EECS 6.837

Frdo Durand and Barb Cutler

Some slides courtesy of Leonard McMillan

Courtesy of James Arvo and David Kirk.Used with permission.

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Review of Ray Casting

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Ray Casting

For ever y pi xelConst r uct a r ay f r om t he eyeFor ever y obj ect i n t he scene

Fi nd i nt er sect i on wi t h t he r ay

Keep i f cl osest

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Ray Tracing

Secondary rays (shadows, reflection, refraction)

In a couple of weeks

reflection

refraction

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Ray representation

Two vectors: Origin

Direction (normalized is better) Parametric line

P(t) = R + t * D

RD

P(t)

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Explicit vs. implicit

Implicit Solution of an equation

Does not tell us how to generate a point on the plane Tells us how to check that a point is on the plane

Explicit Parametric

How to generate points

Harder to verify that a point is on the ray

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A note on shading

Normal direction, direction to light

Diffuse component: dot product

Specular component for shiny materials Depends on viewpoint

More in two weeks

Diffuse sphere shiny spheres

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Textbook

Recommended, not required

Peter Shirley

Fundamentals of Computer GraphicsAK Peters

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References for ray casting/tracing

Shirley Chapter 9

Specialized books: An Introduction to Ray Tracing, Edited by Andrew S. Glassner

Realistic Ray Tracing, Peter Shirley

Realistic Image Synthesis Using Photon Mapping, Henrik Wann Jensen

Online resourceshttp://www.irtc.org/

http://www.acm.org/tog/resources/RTNews/html/

http://www.povray.org/http://www.siggraph.org/education/materials/HyperGraph/raytrace/rtrace0.htm

http://www.siggraph.org/education/materials/HyperGraph/raytrace/rt_java/raytrace.html

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Assignment 1

Write a basic ray caster Orthographic camera

Spheres Display: constant color and distance

We provide Ray

Hit

Parsing

And linear algebra, image

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Object-oriented design

We want to be able to add primitives easily Inheritance and virtual methods

Even the scene is derived from Object3D!

Object3Dbool intersect(Ray, Hit, tmin)

Planebool intersect(Ray, Hit, tmin)

Spherebool intersect(Ray, Hit, tmin)

Trianglebool intersect(Ray, Hit, tmin)

Groupbool intersect(Ray, Hit, tmin)

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Ray

cl ass Ray {

publ i c:

// CONSTRUCTOR & DESTRUCTOR

Ray ( ) {}

Ray ( const Vec3f &di r , const Vec3f &or i g) {di r ect i on = di r ;or i gi n = or i g; }

Ray ( const Ray& r ) {*t hi s=r ; }

// ACCESSORS

const Vec3f & get Or i gi n( ) const { r et ur n or i gi n; }const Vec3f & get Di r ect i on( ) const { r et ur n di r ect i on; }Vec3f poi nt At Par amet er ( f l oat t ) const {

r et ur n or i gi n+di r ect i on*t ; }

pr i vat e:

// REPRESENTATION

Vec3f di r ect i on;Vec3f or i gi n;

};

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Hit

Store intersection point & various informationcl ass Hi t {

publ i c:

// CONSTRUCTOR & DESTRUCTOR

Hi t ( f l oat _t , Vec3f c) { t = _t ; col or = c; }~Hi t ( ) {}

/ / ACCESSORS

f l oat get T( ) const { r et ur n t ; }

Vec3f get Col or ( ) const { r et ur n col or ; }

// MODIFIER

voi d set ( f l oat _t , Vec3f c) { t = _t ; col or = c; }

pr i vat e:

// REPRESENTATION

f l oat t ;Vec3f col or ;

//Material *material;

//Vec3f normal;

};

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Tasks

Abstract Object3D

Sphere and intersection

Group class Abstract camera and derive Orthographic

Main function

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Questions?

Image removed due to copyright considerations.

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Overview of today

Ray-box intersection

Ray-polygon intersection

Ray-triangle intersection

Ray Parallelepiped

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Ray-ParallelepipedIntersection Axis-aligned

From (X1, Y1, Z1) to (X2, Y2, Z2)

Ray P(t)=R+Dt

y=Y2

y=Y1

x=X1 x=X2

R

D

N b I i

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Nave ray-boxIntersection

Use 6 plane equations

Compute all 6 intersection

Check that points are inside boxAx+by+Cz+D

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Factoringoutcomputation


Pairs of planes have the same normal

Normals have only one non-0 component

Do computations one dimension at a time

Maintain tnear and tfar (closest and farthest

so far)y=Y2

y=Y1

x=X1 x=X2

R

D

T t if ll l

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Testifparallel


If Dx=0, then ray is parallel If Rxx2 return false

y=Y2

y=Y1

x=X1 x=X2

R

D

If t ll l

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If not parallel

Calculate intersection distance t1 and t2 t1=(X1-Rx)/Dx

t2=(X2-Rx)/Dx

y=Y2

y=Y1

x=X1 x=X2

R

D

t1

t2

T t 1

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Test 1

Maintain tnear and tfar If t1>t2, swap

if t1>tnear, tnear =t1 if t2tfar, box is missed

y=Y2

y=Y1R

D

t1x t2xtnear

t1y

t2ytfar

tfar

x=X1 x=X2

T t 2

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Test 2

If tfar

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Algorithm recap

Do for all 3 axis Calculate intersection distance t1 and t2

Maintain tnear and tfar If tnear>tfar, box is missed

If tfar

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Efficiency issues

Do for all 3 axis Calculate intersection distance t1 and t2

Maintain tnear and tfar

If tnear>tfar, box is missed

If tfar

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Questions?


Overview of today

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Overview of today




Ray polygon intersection

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Ray-plane intersection Test if intersection is in the polygon

Solve in the 2D plane

D

R

Point inside/outside polygon

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Point inside/outside polygon

Ray intersection definition: Cast a ray in any direction

(axis-aligned is smarter) Count intersection

If odd number, point is inside

Works for concave and star-shaped

Precision issue

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Precision issue

What if we intersect a vertex? We might wrongly count an intersection for each

adjacent edge Decide that the vertex is always above the ray

Winding number

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Winding number

To solve problem with star pentagon Oriented edges

Signed number of intersection Outside if 0 intersection

+ -+

-

Alternative definitions

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Alternative definitions

Sum of the signed angles from point to vertices 360 if inside, 0 if outside

Sum of the signed areas of point-edge triangles Area of polygon if inside, 0 if outside

How do we project into 2D?

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How do we project into 2D?

Along normal Costly

Along axis Smarter (just drop 1 coordinate)

Beware of parallel planeD

R

D

R

Questions?

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Questions?


Overview of today

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Overview of today




Ray triangle intersection

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Ray triangle intersection

Use ray-polygon Or try to be smarter

Use barycentric coordinates

c

ab

PRD

Barycentric definition of a plane

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Barycentric definition of a plane

[Mbius, 1827] P( , , )=a+b+cwith + + =1

Is it explicit or implicit?

c

P

a b

Barycentric definition of a triangle

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Barycentric definition of a triangle

P( , , )=a+b+cwith + + =1

0<

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G ve , ow ca we co pute , , ?


Compute the areas of the opposite subtriangle Ratio with complete area

=Aa/A, =Ab/A =Ac/AUse signed areas for points outside the triangle

c

a b

P TaT

Intuition behind area formula

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u o be d a ea o u a

P is barycenter of a and Q A is the interpolation coefficient on aQ

All points on line parallel to bc have the same All such Ta triangles have same height/area

c

a b

P Q

Simplify

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p y

Since + + =1we can write =1

P(, )=(1) a + b +c

c

a b

P

Simplify

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p y

P(, )=(1) a + b +c P(, )=a + (b-a) +(c-a)

Non-orthogonal coordinate system of the plane

c

a b

P

How do we use it for intersection?

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Insert ray equation into barycentric expression oftriangle

P(t)= a+ (b-a)+ (c-a) Intersection if +

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Rx+tDx= ax+ (bx-ax)+ (cx-ax) Ry+tDy= ay+ (by-ay)+ (cy-ay)

Rz+tDz= az+ (bz-az)+ (cz-az)

c

ab

PRD

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Cramers rule

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| | denotes the determinant

Can be copied mechanically in the code

ab

PRD

A

DcaRa

DcaRa

DcaRa

zzzzz

yyyyy

xxxxx

= A

DRaba

DRaba

DRaba

zzzzz

yyyyy

xxxxx

= A

Racaba

Racaba

Racaba

t zzzzzz

yyyyyy

xxxxxx

=

c

Advantage

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Efficient Store no plane equation

Get the barycentric coordinates for free Useful for interpolation, texture mapping

ab

PRD

c

Plucker computation

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Plucker space:6 or 5 dimensional space describing 3D lines

A line is a point in Plucker space

Plucker computation

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The rays intersecting a line are a hyperplane A triangle defines 3 hyperplanes

The polytope defined by the hyperplanes is theset of rays that intersect the triangle

Plucker computation

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The rays intersecting a line are a hyperplane A triangle defines 3 hyperplanes

The polytope defined by the hyperplanes is theset of rays that intersect the triangle

Ray-triangleintersectionbecomes a polytope

inclusion Couple of additional

issues

Next week: Transformations

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Permits 3D IFS ;-)


Documents

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