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Dr. Scott Schaefer

Geometric Modeling CSCE 645/VIZA 675

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Course Information

Instructor Dr. Scott Schaefer HRBB 527B Office Hours: MW 9:00am – 10:00am

(or by appointment)

Website: http://courses.cs.tamu.edu/schaefer/645_Spring2013

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Geometric Modeling

Surface representations Industrial design

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Geometric Modeling

Surface representations Industrial design Movies and animation

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Geometric Modeling

Surface representations Industrial design Movies and animation

Surface reconstruction/Visualization

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Topics Covered Polynomial curves and surfaces

Lagrange interpolation Bezier/B-spline/Catmull-Rom curves Tensor Product Surfaces Triangular Patches Coons/Gregory Patches

Differential Geometry Subdivision curves and surfaces Boundary representations Surface Simplification Solid Modeling Free-Form Deformations Barycentric Coordinates

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What you’re expected to know

Programming Experience Assignments in C/C++

Simple Mathematics

Graphics is mathematics made visible

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How much math?

General geometry/linear algebra Matrices

Multiplication, inversion, determinant, eigenvalues/vectors

Vectors Dot product, cross product, linear independence

Proofs Induction

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Required Textbook

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Grading

50% Homework 50% Class Project

No exams!

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Class Project

Topic: your choice Integrate with research Originality

Reports Proposal: 2/7 Update #1: 3/7 Update #2: 4/9 Final report/presentation: 4/25

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Class Project Grading

10% Originality 20% Reports (5% each) 5% Final Oral Presentation 65% Quality of Work

http://courses.cs.tamu.edu/schaefer/645_Spring2013/assignments/project.html

Honor Code

Your work is your own You may discuss concepts with others Do not look at other code.

You may use libraries not related to the main part of the assignment, but clear it with me first just to be safe.

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Questions?

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Vectors

v

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Vectors

v

u

?uv

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Vectors

v

uuv

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Vectors

v

uuv

v

u

? uv

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Vectors

v

uuv

vu

uv

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Vectors

v

uuv

v

?vcvu

uv

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Vectors

v

uuv

v

vc

vu

uv

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Points

p

q

? qp

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Points

p

q

undefinedqp

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Points

p

q

?2

qp

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Points

p

q

2

qp

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Points

1 p=p 0 p=0 (vector) c p=undefined where c 0,1 p – q = v (vector)

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Points

n

kkk pc

0

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Points

n

kkk

n

kk

n

kkk ppcpcpc

000

00

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Points

n

kkk

n

kk

n

kkk ppcpcpc

000

00

00

n

kkc

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Points

n

kkk

n

kk

n

kkk ppcpcpc

000

00

n

kkk

n

kk

n

kkk

n

kk ppcpcpcc

000

000

0

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Points

n

kkk

n

kk

n

kkk ppcpcpc

000

00

vectorppcpccn

kkk

n

kkk

n

kk

00

00

0

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Points

n

kkk

n

kk

n

kkk ppcpcpc

000

00

vectorppcpccn

kkk

n

kkk

n

kk

00

00

0

10

n

kkc

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Points

n

kkk

n

kk

n

kkk ppcpcpc

000

00

vectorppcpccn

kkk

n

kkk

n

kk

00

00

0

n

kkk

n

kk

n

kkk

n

kk ppcpcpcc

000

000

1

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Points

n

kkk

n

kk

n

kkk ppcpcpc

000

00

vectorppcpccn

kkk

n

kkk

n

kk

00

00

0

pointppcppccn

kkk

n

kkk

n

kk

000

00

1

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Barycentric Coordinates

1p

3p2p

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Barycentric Coordinates

v

1p

3p2p

k

kk pv

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Barycentric Coordinates

v

1p

3p2p

k

kk pv

1k

k

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Barycentric Coordinates

v

1p

3p2p

k

kk pv

1k

k

0321 kpppv

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Barycentric Coordinates

v

1p

3p2p

k

kk pv

1k

k

0321 kpppv 01,1 kkjkk ppv

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Barycentric Coordinates

v

1p

3p2p

k

kk pv

1k

k

1111

3

2

1321 vppp

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Barycentric Coordinates

v

1p

3p2p

k

kk pv

1k

k

1111

1

321

3

2

1 vppp

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Convex Sets

If , then the form a convex combination

k

kk pv

0k k

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Convex Hulls

Smallest convex set containing all the kp

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Convex Hulls

Smallest convex set containing all the kp

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Convex Hulls

If pi and pj lie within the convex hull,

then the line pipj is also contained within the convex hull

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Convex Hulls

If pi and pj lie within the convex hull,

then the line pipj is also contained within the convex hull

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Convex Hulls

If pi and pj lie within the convex hull,

then the line pipj is also contained within the convex hull

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Convex Hulls

If pi and pj lie within the convex hull,

then the line pipj is also contained within the convex hull

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Convex Hulls

If pi and pj lie within the convex hull,

then the line pipj is also contained within the convex hull

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Convex Hulls

If pi and pj lie within the convex hull,

then the line pipj is also contained within the convex hull

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Convex Hulls

If pi and pj lie within the convex hull,

then the line pipj is also contained within the convex hull

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Convex Hulls

If pi and pj lie within the convex hull,

then the line pipj is also contained within the convex hull

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Convex Hulls

If pi and pj lie within the convex hull,

then the line pipj is also contained within the convex hull

kk

kkk ppv ConvexHull0 v

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Affine Transformations

Preserve barycentric combinations

Examples: translation, rotation, uniform scaling, non-uniform scaling, shear

k k

kkkk pvpv )()(

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Other Transformations

Conformal Preserve angles under transformation Examples: translation, rotation, uniform

scaling Rigid

Preserve angles and length under transformation

Examples: translation, rotation

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Vector Spaces

A set of vectors vk are independent if

The span of a set of vectors vk is

A basis of a vector space is a set of

independent vectors vk such that

kj

jjkj vvtsk ..existnotdoesthere,

}|{ k

kkvvv

)(span, kvvv