09 - Designing Surfaces - New York Universitypanozzo/cg/09 - Designing Surfaces.pdf · 09 -...

CSCI-GA.2270-001 - Computer Graphics - Fall 16 - Daniele Panozzo

09 - Designing Surfaces


Triangular surfaces• A surface can be discretized by a collection of points and triangles

• Each triangle is a subset of a plane

• Every point on the surface can be expressed as an affine combination of three vertices

• The surface is C0


Barycentric coordinates• The points inside a triangle, are usually parametrized using barycentric

coordinates w:

• The barycentric coordinates naturally defines a parametrization

v1

v2

v3

p

w1

w2

w3

px

py=

11 1 1

v1y v2y v3y

v1x v2x v3x


Tensor Surfaces


Bilinear Interpolation• Linear interpolation

• “Simplest” curve between two points

• Bilinear interpolation

• “Simplest” surface between four points

=�1� u u

�✓b00 b01b10 b11

◆✓1� vv

◆Domain

Isoparametric curve

x(u, v) =1X

i=0

1X

j=0

bi,jB1i (u)B

1j (v)


Bilinear Interpolation• Linear interpolation

• “Simplest” curve between two points

• Bilinear interpolation

• “Simplest” surface between four points

b0100 = (1� v)b00 + vb01

b0110 = (1� v)b10 + vb11

b0011 = (1� u)b0100 + ub0110


b00

b10

b11

b20 b21

b23

b33

b13

b03

Bézier Patches• Build on Bézier curves

• Control points as curves

• Keep one parameter fixed: iso-parameter curves

bm(u) =mX

i=0

biBmi (u)

bi = bi(v) =nX

j=0

bijBnj (v)

bmn(u, v) =mX

i=0

nX

j=0

bijBmi (u)Bn

j (v)


Properties• Affine invariance



• Repeated (bi-)linear combinations

• Convex hull property





• Partition of unity and non-negativity

• Polynomial boundary curves and corner interpolation





• Partition of unity and non-negativity

• Polynomial boundary curves and corner interpolation

• Variation diminishing


De Casteljau Algorithm• Bézier curves created by repeated linear interpolation

• Surfaces: repeated bilinear interpolation


Demo

http://cagd-applets.webarchiv.kit.edu/mocca/html/noplugin/inhalt.html


NURBS Surfaces

• Standard in most advanced modeling systems

x(u, v) =

!i

!j wi,jdi,jN

mi (u)Nn

j (v)!

i

!j wi,jN

mi (u)Nn

j (v)

projection of tensor product patches ≠ tensor product surface! (basis is not separable)


NURBS Surfaces• Standard in most advanced modeling systems


NURBS Surfaces• Influence of weights


Subdivision Surfaces


Why?• Tensor product surfaces are defined on “regular grids”

• They cannot be defined on arbitrary meshes


A different way of constructing curves!


Interpolating: 4 point scheme


Catmull-Clark scheme

evenodd

Converges to cubic B-spline


Subdivision Methods• Principal characteristics:

• triangular or quadrangular meshes

• approximating or interpolating

• smoothness of the limit surface

• We will study 2 of them:

• Loop subdivision for triangle meshes (Approximating, C2)

• Catmull-Clark subdivision for quadrilateral meshes (Approximating, C2)

• Other famous schemes (see the references for details)

• Butterfly, Kobbelt, Doo-Sabin, Midedge, Biquartic


Loop subdivision

w(n) =5

8� (

3

8+

1

4cos

2⇡

n

)2


Special stencils• Special stencils exist that allow to evaluate:

• The limit position of a vertex: that is, in a single step you can compute the position of one vertex after an infinite number of subdivisions

• The tangent plane at a vertex: in this case, two stencils are used and each stencil generates a vector that lies in the tangent plane

• It is also possible to evaluate the surface analytically, in other words it is possible to find the mapping between every point of the control mesh and the limit surface. For details see http://www.autodeskresearch.com/publications/loopsubdiv

http://www.autodeskresearch.com/publications/loopsubdiv


Catmull Clark subdivision

� =3

2N

� =1

4N

On a regular grid, the Catmull clark subdivision is a collection of bicubic

Bézier patches


Catmull-Clark• It is the standard in the animation industry

• All the major 3D modeling softwares support it

• Similarly to Loop:

• stencils for the limit surface

• stencils for tangent plane

• it can be evaluated analytically


How are they used in practice?


Efficient implementation• At every subdivision step, we add new vertices and move the existing ones

in new positions

• Every position is computed as a weighted average of existing vertices

• This means that the process is linear!

• For a fixed number of levels of subdivision, the vertices of subdivided surface can be computed as:

• where S is a sparse and fixed matrix

p = SqSubdivided verticesControl vertices


T-spline

• Extension of splines for non-rectangular grids

• http://www.youtube.com/watch?v=k1ro9S-cAwI

http://www.youtube.com/watch?v=k1ro9S-cAwI


References

Fundamentals of Computer Graphics, Fourth Edition 4th Edition by Steve Marschner, Peter Shirley

Chapter 15

Curves and Surfaces for CAGD - Gerald Farin

Subdivision Zoo - Denis Zorin http://www.cmlab.csie.ntu.edu.tw/~robin/courses/gm/note/subdivision-prn.pdf

https://www.amazon.com/s/ref=dp_byline_sr_book_1?ie=UTF8&text=Steve+Marschner&search-alias=books&field-author=Steve+Marschner&sort=relevancerank

https://www.amazon.com/Peter-Shirley/e/B00IZTGJ9O/ref=dp_byline_cont_book_2


Projective Transformations


Viewing Transformationobject space

model camera projection viewport

canonical view volumeworld space

camera space screen space


Orthographic Projection

projection

canonical view volume

camera space

xz

y(l,b,n)

(r,t,f)

Morth

=

2

664

2r�l

0 0 � r+l

r�l

0 2t�b

0 � t+b

t�b

0 0 2n�f

�n+f

n�f

0 0 0 1

3

775


Perspective Projection• In Orthographic projection, the size of the objects does not change

with distance

• In Perspective projection, the objects that are far away look smallerImage Plane

ysy

dzys =

dy

z

Image Plane


Divisions in Matrix Form• We would like to reuse the matrix machinery that we built in the

previous lectures

• How do we encode divisions?

• We extend homogeneous coordinatesys

y

dz

ys =dy

zImage Plane


• What do we have left?

• We can use the last row of the transformation:

Until now…

0

@a1 b1 c1

a2 b2 c2

0 0 1

1

A ·

0

@x

y

1

1

A =

0

@a1x+ b1y + c1

a2x+ b2y + c2

1

1

A

0

@a1 b1 c1

a2 b2 c2

e f g

1

A ·

0

@x

y

1

1

A =

0

@a1x+ b1y + c1

a2x+ b2y + c2

ex+ fy + g

1

A⇠

0

B@

a1x+b1y+c1

ex+fy+g

a2x+b2y+c2

ex+fy+g

1

1

CA


Intuition• Purely algebraic:

• Or as a projection, where each line is identified by a point on the plane z=1

• Note that in this case, you can think of it as a transformation in a space with one more dimension

0

@x

y

w

1

A ⇠

0

@x/w

y/w

1

1

A


Projective Transformation• A transformation of this form is called a projective transformation (or a homography)

• The points are represented in homogeneous coordinates

0

@a1 b1 c1

a2 b2 c2

e f g

1

A ·

0

@x

y

1

1

A =

0

@a1x+ b1y + c1

a2x+ b2y + c2

ex+ fy + g

1

A ⇠

0

B@

a1x+b1y+c1

ex+fy+g

a2x+b2y+c2

ex+fy+g

1

1

CA


Example

• It transforms a square into a quadrilateral — note that straight lines are preserved, but parallel lines are not!

• Note that you can use homogeneous coordinates for as many transformations as you want, only when you need the cartesian representation you have to normalize

0,0

1,1

0,0 3,0

1,3

M =

0

@2 0 �10 3 00 2/3 1/3

1

A


Perspective Projection• Perspective projection is easily implementable using this machinery

ysy

dz

ys =d

zImage Plane

✓ys1

◆⇠

✓d 0 00 1 0

◆0

@yz1

1

A


• We will use the same conventions that we used for orthographic:

• Camera at the origin, pointing negative z

• We scale x, y and “bring along” the z

`

xz

y(l,b,n)

(r,t,f)P =

0

BB@

n 0 0 00 n 0 00 0 n+ f �fn0 0 1 0

1

CCA


Effect on the points

xz

y(l,b,n)

(r,t,f)P =

0

BB@

n 0 0 00 n 0 00 0 n+ f �fn0 0 1 0

1

CCA

P

0

BB@

x

y

z

1

1

CCA =

0

BB@

nx

ny

(n+ f)z � fn

z

1

CCA ⇠

0

BB@

nx

z

ny

z

n+ f � fn

z

1

1

CCA


Orthographic Projection

projection

canonical view volume

camera space

xz

y(l,b,n)

(r,t,f)

Morth

=

2

664

2r�l

0 0 � r+l

r�l

0 2t�b

0 � t+b

t�b

0 0 2n�f

�n+f

n�f

0 0 0 1

3

775


Complete Perspective Transformation

P =

0

BB@

n 0 0 00 n 0 00 0 n+ f �fn0 0 1 0

1

CCA

canonical view volumecamera space

P Morth

Morth

=

2

664

2r�l

0 0 � r+l

r�l

0 2t�b

0 � t+b

t�b

0 0 2n�f

�n+f

n�f

0 0 0 1

3

775


Parameters?• How to set the parameters of the transformation?

• If we look at the center of the center of the window then the barycenter of the front back should be at (0,0,f)

• If we want no distortion on the image we need to keep a fixed aspect ratio:

• width/height = r/t (width and height are the size in pixels of the final image)

• There is only one degree of freedom left, the field of view angle :

•

• The parameters can thus by found by fixing n and . You can then compute t an consequently all the other parameters needed to construct the transformation

xz

y

(l,b,n)

(r,t,f)

✓

tan✓

2=

t

|n|

✓

✓


References

Fundamentals of Computer Graphics, Fourth Edition 4th Edition by Steve Marschner, Peter Shirley

Chapter 7

https://www.amazon.com/s/ref=dp_byline_sr_book_1?ie=UTF8&text=Steve+Marschner&search-alias=books&field-author=Steve+Marschner&sort=relevancerank

https://www.amazon.com/Peter-Shirley/e/B00IZTGJ9O/ref=dp_byline_cont_book_2

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