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CS 586/480Computer Graphics II

Dr. David BreenMatheson 408Thursday 6PM Æ 8:50PM

Presentation 210/7/04

Start Upn Any questions from last time?n Presentation assignmentsn Shailajan Ilyan Dmitriy

n Programming assignment submissionsn Go over sampling image plane?

Slide Creditsn Leonard McMillan, Seth Teller,

Fredo Durand, Barb Cutler - MITn G. Drew Kessler, Larry Hodges - Georgia

Institute of Technologyn John Hart - University of Illinois

More Geometry & Intersections

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Ray/Plane IntersectionRay is defined by R(t) = Ro + Rd*t where t ≥ 0

Ro = Origin of ray at (xo, yo, zo)Rd = Direction of ray [xd, yd, zd] (unit vector)

Plane is defined by [A, B, C, D]Ax + By + Cz + D = 0 for a point in the planeNormal Vector, N = [A, B, C] (unit vector)A2 + B2 + C2 = 1D = - N • P0 (P0 - point in plane)

Ray/Plane (cont.)Substitute the ray equation into the planeequation:

A(xo + xdt) + B(yo + ydt) + C(zo +zdt) + D = 0

Solve for t:t = -(Axo + Byo + Czo + D) / (Axd + Byd + Czd)t = -(N • Ro + D) / (N • Rd)

What Can Happen? N • Rd = 0 N • Rd > 0

t < 0

Ro

t > 0

Ro

Ray/Plane SummaryIntersection point:

(xi, yi, zi) = (xo + xdti, yo + ydti, zo + zdti)

1. Calculate N • Rd and compare it to zero.2. Calculate ti and compare it to zero.3. Compute intersection point.4. Flip normal if N • Rd is positive

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Ray-Parallelepiped Intersectionn Axis-alignedn From (X1, Y1, Z1) to (X2, Y2, Z2)n Ray P(t)=Ro+Rdt

y=Y2

y=Y1

x=X1 x=X2

Ro

Rd

Naïve ray-box Intersectionn Use 6 plane equationsn Compute all 6 intersectionn Check that points are inside box

Ax+By+Cz+D ≤ 0y=Y2

y=Y1

x=X1 x=X2

Ro

Rd

Factoring out computationn Pairs of planes have the same normaln Normals have only one non-0 componentn Do computations one dimension at a timen Maintain tnear and tfar (closest and

farthest so far)y=Y2

y=Y1

x=X1 x=X2

Ro

Rd

Test if paralleln If Rdx = 0, then ray is paralleln If Rox < X1 or Rox > x2 return false

y=Y2

y=Y1

x=X1 x=X2

Ro

Rd

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If not paralleln Calculate intersection distance t1 and t2n t1 = (X1-Rox)/Rdxn t2 = (X2-Rox)/Rdx

y=Y2

y=Y1

x=X1 x=X2

Ro

Rd

t1

t2

Test 1n Maintain tnear and tfar

n If t1 > t2, swapn if t1 > tnear, tnear = t1n if t2 < tfar, tfar = t2

n If tnear > tfar, box is missed

y=Y2

y=Y1

x=X1 x=X2

Ro

Rd

t1x t2xtnear

t1y

t2ytfar

tfar

Test 2n If tfar < 0, box is behind

y=Y2

y=Y1

x=X1 x=X2

Ro

Rd

t1xt2xtfar

Algorithm recapn Do for all 3 axes

n Calculate intersection distance t1 and t2n Maintain tnear and tfarn If tnear > tfar, box is missedn If tfar < 0, box is behind

n If box survived tests, report intersection at tneary=Y2

y=Y1

x=X1 x=X2Ro

Rd

tnear

t1y

tfar

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Ray/Ellipsoid IntersectionRay/Cylinder Intersection

Ellipsoid's surface is defined by the set of points{(xs, ys, zs)} satisfying the equation:

(xs/ a)2 + (ys/ b)2 + (zs/ c)2 - 1 = 0

Cylinder's surface is defined by the set of points{(xs, ys, zs)} satisfying the equation:

(xs)2 + (ys)2 - r2 = 0 -z0 ≤ zs ≤ z0

Centers at origin

ß Substitute ray equation into surfaceequationsß This is a quadratic equation in t:ß At2 + Bt + C = 0

ß Analyze as beforeß Solve for t with quadratic equationß Plug t back into ray equation - Doneß Well,… not exactly

Ray/Ellipsoid IntersectionRay/Cylinder Intersection

Ray/Cylinder Intersectionß Is intersection point Pi between -Z0 and Z0?ß If not, Pi is not validß Also need to do intersection test with

z = -Z0 | Z0 planeß If (Pix)2 + (Piy)2 ≤ r2, you’ve intersected a

“cap”ß Which valid intersection is closer?

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Superquadrics

†

xa1

Ê

Ë Á

ˆ

¯ ˜

2e 2

+ya2

Ê

Ë Á

ˆ

¯ ˜

2e 2

È

Î

Í Í

˘

˚

˙ ˙

e 2e1

+za3

Ê

Ë Á

ˆ

¯ ˜

2e1

=1

SMF Triangle Meshesv -1 -1 -1v 1 -1 -1v -1 1 -1v 1 1 -1v -1 -1 1v 1 -1 1v -1 1 1v 1 1 1f 1 3 4f 1 4 2f 5 6 8f 5 8 7f 1 2 6f 1 6 5f 3 7 8f 3 8 4f 1 5 7f 1 7 3f 2 4 8f 2 8 6

vertices

triangles

Draw data structure

Triangle Meshes (.iv) Transformations &Hierarchical Models

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n Add an extra dimension• in 2D, we use 3 x 3 matrices• In 3D, we use 4 x 4 matrices

n Each point has an extra value, w

Homogeneous Coordinates

x'

y'

z'

w'

=

x

y

z

w

a

e

i

m

b

f

j

n

c

g

k

o

d

h

l

p

p' = M p

n Most of the time w = 1, and we canignore it

Homogeneous Coordinates

x'

y'

z'

1

=

x

y

z

1

a

e

i

0

b

f

j

0

c

g

k

0

d

h

l

1

Translate (tx, ty, tz)n Why bother with the

extra dimension?Because now translationscan be encoded in the matrix!

Translate(c,0,0)

x

y

p p'

c

x'

y'

z'

1

=

x

y

z

1

1

0

0

0

0

1

0

0

0

0

1

0

tx

ty

tz

1

x'

y'

z'

Scale (sx, sy, sz)n Isotropic (uniform)

scaling: sx = sy = sz

n You only have to implement uniform scaling

x'

y'

z'

1

=

x

y

z

1

sx

0

0

0

0

sy

0

0

0

0

sz

0

0

0

0

1

Scale(s,s,s)

x

p

p'

qq'

y

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Rotationn About z axis

x'

y'

z'

1

=

x

y

z

1

cos f

sin f

0

0

-sin f

cos f

0

0

0

0

1

0

0

0

0

1

ZRotate(f)

x

y

z

p

p'

_

Rotation

n Aboutx axis:

n Abouty axis:

x'

y'

z'

1

=

x

y

z

1

0cos a

sin a

0

0-sin a

cos a

0

1

0

0

0

0

0

0

1

x'

y'

z'

1

=

x

y

z

1

cos q

0

-sin q

0

sin q

0cos q

0

0

1

1

0

0

0

0

1

Rotationn About (kx, ky, kz), an arbitrary unit vector (Rodrigues Formula)

x'

y'

z'

1

=

x

y

z

1

kxkx(1-c)+c

kykx(1-c)+kzs

kzkx(1-c)-kys

0

0

0

0

1

kykx(1-c)-kzs

kyky(1-c)+c

kzky(1-c)+kxs

0

kxkz(1-c)+kys

kykz(1-c)-kxs

kzkz(1-c)+c

0

where c = cos q & s = sin q

Rotate(k, q)

x

y

z

_

k

How are transforms combined?

(0,0)

(1,1)(2,2)

(0,0)

(5,3)

(3,1)Scale(2,2) Translate(3,1)

TS =2

0

0

2

0

0

1

0

0

1

3

1

2

0

0

2

3

1=

Scale then Translate

Use matrix multiplication: p' = T ( S p ) = ((TS) p)

Caution: matrix multiplication is NOT commutative!

0 0 1 0 0 1 0 0 1

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Non-commutative CompositionScale then Translate: p' = T ( S p ) = TS p

Translate then Scale: p' = S ( T p ) = ST p

(0,0)

(1,1)(4,2)

(3,1)

(8,4)

(6,2)

(0,0)

(1,1)(2,2)

(0,0)

(5,3)

(3,1)Scale(2,2) Translate(3,1)

Translate(3,1) Scale(2,2)

TS =2

0

0

0

2

0

0

0

1

1

0

0

0

1

0

3

1

1

ST =2

0

0

2

0

0

1

0

0

1

3

1

Non-commutative CompositionScale then Translate: p' = T ( S p ) = TS p

2

0

0

0

2

0

3

1

1

2

0

0

2

6

2

=

=

Translate then Scale: p' = S ( T p ) = ST p

0 0 1 0 0 1 0 0 1

Transformations in Ray Tracing

Transformations in Modelingn Position objects in a scenen Change the shape of objectsn Create multiple copies of objectsn Projection for virtual camerasn Animations

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Scene Description

Scene

LightsCamera ObjectsMaterialsBackground

Simple Scene Description File

Camera { center 0 0 10 direction 0 0 -1 up 0 1 0 }

Lights { numLights 1 DirectionalLight { direction -0.5 -0.5 -1 color 1 1 1 } }

Background { color 0.2 0 0.6 }

Materials { numMaterials <n>

<MATERIALS> }

Group { numObjects <n> <OBJECTS> }

Hierarchical Models

n Logical organization of sceneGroup { numObjects 3 Group { numObjects 3 Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { numObjects 2 Group { Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { Box { <BOX PARAMS> } Sphere { <SPHERE PARAMS> } Sphere { <SPHERE PARAMS> } } } Plane { <PLANE PARAMS> } }

Simple Example with Groups

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Group { numObjects 3 Group { numObjects 3 Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { numObjects 2 Group { Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { Box { <BOX PARAMS> } Sphere { <SPHERE PARAMS> } Sphere { <SPHERE PARAMS> } } } Plane { <PLANE PARAMS> } }

Adding Materials Adding Transformations

Using Transformations

n Position the logicalgroupings of objectswithin the scene

n Transformation in group

Directed Acyclic Graphis more efficient and useful

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Processing ModelTransformattionsn Goaln Get everything into world coordinates

n Traverse graph/tree in depth-first ordern Concatenate transformationsn Can store intermediate transformationsn Apply/associate final transformation to

primitive at leaf noden What about cylinders, superquadrics, etc.?n Transform ray!

Transform the Ray

n Map the ray from World Spaceto Object Space

World Space

r majorr minor

(x,y)

(0,0) Object Space

r = 1

pWS = M pOS

pOS = M-1 pWS

Transforming Points &Directions

n Transform point

n Transform direction

n Map intersection point and normal back toworld coordinates

Homogeneous Coordinates: (x,y,z,w)

W = 0 is a point at infinity (direction)

Why is the Normal important?n It's used for shadingn makes things look 3D!

object color only(Assignment 1)

Diffuse Shading(Assignment 3)

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Given overlapping shapes A and B:

Union Intersection Subtraction

Constructive Solid Geometry(CSG) For example:

How can we implement CSG?

Points on A,Outside of B

Points on B,Outside of A

Points on B,Inside of A

Points on A,Inside of B

Union Intersection Subtraction

Collect all the intersections

Union Intersection Subtraction

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Implementing CSG

1. Test "inside" intersections:• Find intersections with A,

test if they are inside/outside B• Find intersections with B,

test if they are inside/outside A

2. Overlapping intervals:• Find the intervals of "inside"

along the ray for A and B• Compute

union/intersection/subtractionof the intervals

Wrap Upn Discuss next programming assignmentn Due 10/24/04

n I am away next week. No class!n Discuss status/problems/issues with

this week’s programming assignment