Curves & Surfaces in CG - Telecom Paris

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Curves & Surfaces in CG

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Position wrt previous lectures

● Modeling-oriented definition(s) and example(s) of :– Curves– Surfaces

● Previously :– Differential Geometry (curves and surfaces in

math)– Meshes (raw representation of surfaces in CS)

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Curves : reminder

c : I c (I )∈ℝ3

TT

T

N

N

N

BB

B

unit speed :‖c '(s)‖=1

torque

N ' .T=κ

−N .T '=κ

−N ' .B=τ

curvature

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A gentle intro : Hermite cubic curves

p u=a u3bu2c ud , u∈[0,1]

p u :{x u=a xu

3bxu

2c x ud x

y u=a yu3b yu

2c yud y

z u=a zu3bz u

2c z ud z

Simple to understand mathematically, but how to design it ?

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p u :{x u=a xu

3bxu

2c x ud x

y u=a yu3b yu

2c yud y

z u=a zu3bz u

2c z ud z

Constrain :● Position P0 at u = 0● Position P1 at u = 1● Tangent V0 at u = 0● Tangent V1 at u = 1

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{P0= p 0=dP1= p 1=abcdV 0= p 0=cV 1= p 1=3a2bc


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{d=P0c=V 0

b=−3P03P1−2V0−V 1

a=2P0−2P1V 0V 1


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{d=P0c=V 0

b=−3P03P1−2V0−V 1

a=2P0−2P1V 0V 1

p(u)=F1(u)P0+F2(u)P1+F3(u)V 0+F4(u)V 1

{F1(u)=2u

3−3u2+1

F2(u)=−2u3+3u2

F3(u)=u3−2u2+u

F4 (u)=u3−u2


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p u=UMB=[u3 u2 u 1 ] [2 −2 1 1

−3 3 −2 −10 0 1 01 0 0 0

][P0P1V 0

V 1]

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p u=UMB=[u3 u2 u 1 ] [2 −2 1 1

−3 3 −2 −10 0 1 01 0 0 0

][P0P1V 0

V 1]

p u=U M B=[u3 u2 u 1 ][0 0 0 06 −6 3 3

−6 6 −4 −20 0 1 0

] [P0P1V 0

V 1] Velocity

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p(u)=UMB=[u3 u2 u 1 ] [2 −2 1 1

−3 3 −2 −10 0 1 01 0 0 0

][P0P1V 0

V 1]

p(u)=U M B=[u3 u2 u 1 ] [0 0 0 06 −6 3 3

−6 6 −4 −20 0 1 0

] [P0P1V 0

V 1]

p(u)=U M B=[u3 u2 u 1 ] [0 0 0 00 0 0 012 −12 6 6−6 6 −4 −2

][P0P1V 0V 1

]

Velocity

Acceleration

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More complex curves

What is the regularity of the curve if we join two Hermite cubic curves ?

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More complex curves

What is the regularity of the curve if we join two Hermite cubic curves ?C1 in general. Conditions for C2 ?

p=6P0−6P1+2V 0+4V 1 p=−6P1+6P2−4V 1−2V 2

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Another form for the Hermite curve

P0

V0

P1

V1

P0

P1

P2

P3

p(u)=(2u3−3u2+1)P0+(−2u3+3u2)P1+(u3−2u2+u)V 0+(u3−u2)V 1

p(u)=(1−u)3P0+3u (1−u)2P1

+3u2(1−u)P2+u3 P3

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Bezier curves

P0

P1

P2

P3

p(u)=(1−u)3P0+3u (1−u)2P1

+3u2(1−u)P2+u3 P3

p(u)=∑k=0

n

C knuk (1−u)n−k Pk

=∑k=0

n

Bkn(u)Pk

P0

P1

P2

P3

P4

Bernstein polynomials

n !k !(n−k )!

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Bezier curves : properties

p(u)=∑k=0

n


=∑k=0

n

Bkn(u)Pk

P0

P1

P2

P3

P4

n !k !(n−k )!

∑k=0

n

C kn uk (1−u)n−k=(u+(1−u))n=1

C knuk (1−u)n−k⩾0

● Curve is inside the convex hull of the control points● Polynomial of order n → regularity when joining two curves ?

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Joining Bezier curves

P0

P1

P2

P3

P4

P’0

P’1

P’2

P’3

P’4

● Match position → constrain (P4 and P’

0)

● Match velocity → constrain (P3, P

4, P’

0 and P’

1)

● Match acceleration → constrain (P2, P

3, P

4, P’

0, P’

1 and P’

2)

● …→ Higher regularity comes at a price (less freedom)

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Bezier patch

p(u)=∑k=0

n


=∑k=0

n

Bkn(u)Pk

P0

P1

P2

P3

P4

n !k !(n−k )!

p(u , v)=∑k=0

n

∑m=0

n

Bkn(u)Bm

n(v )Pkm

What about a 3d Bezier volume ?

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Bezier volume (free form deformations)

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Bicubic Bezier quad patch surface

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Bezier triangles

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Bezier triangles

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Bezier triangles

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Bezier trianglesBezier triangle of order n :

{ Pijk , i, j, k ≥ 0, i + j + k = n }

S(u , v )= ∑i+ j+k=n

Bijkn (u , v )Pijk

Bijkn(u , v ):=

n !i! j !k !

ui v j(1−u−v)k

● Boundaries of the triangle limit surface coincide with the quad case → Mixed triangle/quad Bezier surfaces

● Normals can be derived simply● Same properties than the quad case

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The de Casteljau algorithm● So far, explicit formulas have been given● The de Casteljau algorithm is a simple recursive algorithm that

takes advantage of a geometric property of the Bezier shapes (curves and surfaces)

● For a parameter u :● For each segment [P

i , P

i+1], build P

i’ = (1-u)P

i + uP

i+1

● Create the polyline joining the {Pi’} (one point less)

● Iterate until there is only one point left → C(u)

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The de Casteljau algorithm

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Input: array P[0:n] of n+1 points and a real number u in [0,1]

Output: point on curve, C(u)

Working: point array Q[0:n]

for i := 0 to n do

Q[i] := P[i]; // save input

for k := 1 to n do

for i := 0 to n - k do

Q[i] := (1 - u)Q[i] + u Q[i + 1];

return Q[0];

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(Non uniform) B Splines (NUBS?)● Use a knot vector (t

0, t

1, …, t

n)

● Generalization of Bezier curves (Uniform B splines are Bezier curves)

● Resulting curve is of order p-1

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Shape of the uniform B splines

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(Non uniform) B Splines● When several t are the same (t

i+1 = t

i), we use 0/0 = 0 in the

construction● On which domain is the curve defined ??? (we need k-p ≥ 0)

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knot vectors

● Uniform : ti+1

– ti = const

[0,1,2,3,4,5,6,7], [0,0.25,0.5,0.75,1]

● Open: ti = t

0 for i < p, t

i = t

k+p for i > n+p :

[0,0,0,0,1,2,3,4,4,4,4], for k = 4[0.1,0.1,0.1,0.3,0.5,0.7,0.9,1.1,1.1,1.1] for k = 3

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Uniform

p=3 p=4

p=6 p=10

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Uniform, open

k=3 k=4

k=6 k=10

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The de Boor algorithm

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Non Uniform Rational B Splines (NURBs)

● Add another parameter : a weight wi for each point

B Spline of order nWeight of P

i

m knots

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Why NURBs ?● Previous approaches could not model important primitives, such

as for example a circle (you can only approximate it with Bezier curves or B Splines)

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Why NURBs ?● Previous approaches could not model important primitives, such

as for example a circle (you can only approximate it with Bezier curves or B Splines)

● Offers more control● Here : changing w

1

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NURBs and B Splines● NURBs are actually B Splines in homogeneous coordinates !!!

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NURBs surfaces

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A quick&dirty note on Blossoming● NURBs can be defined by the process called blossoming :

The Polar Form of a polynomial of P(t):R→R of degree n is a multiaffine symmetric function P(t

1,t

2,...,t

n):Rn→R such that

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A quick&dirty note on Blossoming● NURBs can be defined by the process called blossoming :

The Polar Form of a polynomial of P(t):R→R of degree n is a multiaffine symmetric function P(t

1,t

2,...,t

n):Rn→R such that

● f is multiaffine if it is affine in any of its arguments :

● f is symmetric if it is invariant under swapping of any of its arguments

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A quick&dirty note on Blossoming● For example, for n = 3 :

● The value of P(t) ( = P(t,t,t) ) can be found if we are given the values of P(0,0,0), P(0,0,1), P(0,1,1) and P(1,1,1) :

● Useful for defining and proving lots of things (de Casteljau for ex, but also knot insertion, degree elevation, ...)

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A quick&dirty note on Automata● Blossoming can also be understood as a form of automata

The quadratic NURBS CIFS-automaton

The cubic NURBS CIFS-automaton

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A quick&dirty note on Automata● Blossoming can also be understood as a form of automata● True for tensor surfaces as well (take the product of both

automata)

The biquadratic NURBS CIFS-automaton The bicubic NURBS CIFS-automaton

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A quick&dirty note on Automata● Blossoming can also be understood as a form of automata● True for tensor surfaces as well (take the product of both

automata)● « fun » applications :

● More in « Representation of NURBS surfaces by Controlled Iterated Functions System automata » [Morlet et al.]

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Summary● Curves

● Bezier● B Splines● NURBs

● Surfaces● Bezier (triangle, quad)● B Splines (quad)● NURBs (quad)● PN triangles (see later)● Phong patches (see later)● Coon patches (not seen here)● And more and more and more…

● Which model should we use ???

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Subdivision surfaces● All models previously seen are nice, but are complex to use, not

easy to implement efficiently, complex to interact with…● Splines were the major standard in the 70s● Subdivision surfaces appeared after that to propose a more simple

mechanism.

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Subdivision surfaces

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Subdivision mask

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Criteria & Classification

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Classification

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Loop Scheme

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Loop Scheme : masks

Crease / Boundary

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Catmull-Clark

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Catmull-Clark

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Catmull-Clark

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Modified Butterfly

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Modified Butterfly

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sqrt(3)

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sqrt(3)

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Subdivision matrix

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Adaptive subdivision

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Geometry Processing

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Implementation

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Implementation

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Implementation

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Real Time

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Curved PN triangles

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Curved PN triangles

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Curved PN triangles

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Phong tesselation

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Phong tesselation

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Phong tesselation

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Phong tesselation

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Phong tesselation

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Comparison

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Phong tesselation

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Multiresolution modeling

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Memo

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A glimpse at the related work● « Open questions » :

● uv texturing ?● Filtering ?● LoDs ?● Fitting subdivision surfaces to HD

scans / surfaces

● Demo

● Questions ??

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Curves & Surfaces in CG - Telecom Paris