CSE590B Lecture 5

Cubic Bezier Curves

Moving up in degree,

Moving up in dimension

James F. Blinn JimBlinn.Com

http://courses.cs.washington.edu/courses/cse590b/13au/

http://courses.cs.washington.edu/courses/cse590b/13au/

Cubic Bezier Curves

When does cusp/loop happen?

Parameters at self intersection?

Implicit Equation?

The Grid

1

4D=P3

Surfaces

in space

3D=P2

Curves

in plane

2D=P1

Point sets

on line

Linear

Quadratic

Cubic

Quartic

Other Dimensions

x y w 3D algebra Points in P2

4D algebra

Points in P3

x y z w

2D algebra

Points in P1

x w

P

1

4D=P3

Surfaces

in space

3D=P2

Curves

in plane

2D=P1

Point sets

on line

Linear

Quadratic

Cubic

QuarticxXw

x y

X Ww w

x y z

X Y Zw w w

Other Dimensions

, ,A

f x y w Ax By Cw x y w B

C

Lines in P2

3D algebra

Planes in P3

4D algebra , , ,

A

Bf x y z w Ax By Cz Dw x y z w

C

D

Point in P1

2D algebra ,

Af x w Ax Bw x w

B

P L1

4D=P3

Surfaces

in space

3D=P2

Curves

in plane

2D=P1

Point sets

on line

Linear

Quadratic

Cubic

Quartic

Other Orders

1

4D=P3

Surfaces

in space

3D=P2

Curves

in plane

2D=P1

Point sets

on line

Linear

Quadratic

Cubic

Quartic

2 2, 2x w Ax Bxw Cw Q

3 2 2 3, 3 3x w Ax Bx w Cxw Dw C

4 3 2 2 3 4, 4 6 4x w Ax Bx w Cx w Dxw Ew F

,x w Ax Bw L

5 4 3 2 2 3 4 5, 5 10 10 5x w Ax Bx w Cx w Dx w Exw Fw N

Other Orders

1

4D=P3

Surfaces

in space

3D=P2

Curves

in plane

2D=P1

Point sets

on line

Linear

Quadratic

Cubic

Quartic

,A B x

x w x wB C w

Q p Q p=

,A B B C x x

x w x wB C C D w w

C p= C

p

p

p L=

,

A B B C

B C C D x xx w x w x w

w wB C C D

C D D E

F F

p p

pp

,A

x w x wB

L

Quadratic,

Different dimensions 1

4D=P3

Surfaces

in space

3D=P2

Curves

in plane

2D=P1

Point sets

on line

Linear

Quadratic

Cubic

Quartic

,A B x

x w x wB C w

Q

p Q p=

, ,A B C x

x y w x y w B D E y

C E F w

Q

, , ,

A B C D x

B E F G yx y z w x y z w

C F H J z

D G J K w

Q

Diagram Same Across Dimensions

P L = 0

P Q P = 0

P = 0C

P

P

= C'CT*

T*

T*

F

P

PP

P

P Q = L

Dimensionality and Epsilon

ijke

i

k

j

l

ijkleijke

i

kj

ije

i j

3D algebra

2D geometry

2D algebra

1D geometry

4D algebra

3D geometry

2D Epsilon

0 1

1 0e

K L K L

QQ-2detQ =

QQ*

Adjugate of Matrix

Determinant of Matrix

3D Epsilon

0 0 0 0 0 1 0 1 0

0 0 1 0 0 0 1 0 0

0 1 0 1 0 0 0 0 0

e

P S S P

1 if is an even permutation of 012

1 if is an odd permutation of 012

0 otherwise

ijk

ijk

ijk

ijk

ijk

e

e

e

3D Epsilon Usage

P

S

= L

Q

Q

Q* 2

= det Q

Q

Q

Q

Adjugate of Matrix

Determinant of Matrix

Line thru two points

4D Epsilon

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0

0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 1 0 0 0e

0 0 1 0

0 0 0 0

1 0 0 0

0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0

0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0

0 0 1 0 0 0 0 0

0 1 0 0 1 0 0 0

0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

1 if is an even permutation of 0123

1 if is an odd permutation of 0123

0 otherwise

ijkl

ijkl

ijkl

ijkl

ijkl

e

e

e

=

4D Epsilon Usage

= E

P

S

T

=S

T

L

Q

Q

Q

Q

= 24 det Q

3 Points = A Plane

2 Points = A Line Determinant of Matrix

Q

Q

Q = 6 adj Q

Adjugate of Matrix

P1 Quadratic Polynomial

2 2, 2x w Ax Bxw Cw Q

p Q p=

QQ

discriminant

1

4D=P3

Surfaces

in space

3D=P2

Curves

in plane

2D=P1

Point sets

on line

Linear

Quadratic

Cubic

Quartic

x

w

f(x,w)

x

w

f(x,w)

x

w

f(x,w)

x

w

f(x,w)

x

w

f(x,w)

2- 11 2+1

1

1

1

Q

I

Q

Q 0 0

P1 Cubic Polynomial

3 2 2 3, 3 3x w Ax Bx w Cxw Dw C

p= C

p

p

A B B C x x

x wB C C D w w

Type 1

11

Type 12Type 111Type 3

1

4D=P3

Surfaces

in space

3D=P2

Curves

in plane

2D=P1

Point sets

on

Linear

Quadratic

Cubic

Quartic

Note: can always flip

sign with 180o rotation

2D Cubic Polynomial Covariants

p C

p

pp

p

p

p

C C = 0

= Hpp C C

C C = 0

p

p

p

C C

= 0C

C

C = J

C

p

p p

Derivatives

x,w= C

x,w

3 2 2 3, 3 3x w Ax Bx w Cxw Dw

A B B C x xx w

B C C D w w

C

x,w

2 2

2 2

23

2

x

w

Ax Bxw Cw

Bx Cxw Dw

C

C= 3

x,w

C

x,w

Derivatives

1,0 C

x,w

1

0

A B B C xx w

B C C D w

x,w

A B x

x wB C w

3

21 1 1 3

d d d d dx xxx x xx x x x xx x

dx dx dx dx dx

xx x x xx x

x,w =C

x,w

x

x,w

x,w +C

x,w

x

x,w

x,w C

x,wx

x,w

+x,w C

x,w

x

x,w

= 1,0 +C

x,w

x,w

x,w C

1,0

x,w

+x,w C

x,w

1,0

= 1,0 C

x,w

x,w

3

Derivatives

x,w= C

x,w

3 2 2 3, 3 3x w Ax Bx w Cxw Dw

A B B C x xx w

B C C D w w

C

x,w

2 2

2 2

23

2

x

w

Ax Bxw Cw

Bx Cxw Dw

C

C= 3

x,w

C

x,w

x,w6 6xx xw

xw ww

Ax Bw Bx Cw

Bx Cw Cx Dw

C C

C CC

Hessian of cubic

x,w Hessian det 18xx xw

xw ww

C CC

C CC C x,w

C CºH

x,w6 6xx xw

xw ww

Ax Bw Bx Cw

Bx Cw Cx Dw

C C

C CC

2 2det 6 36

218

2

Ax Bw Bx CwAC BB x AD BC xw DB CC w

Bx Cw Cx Dw

AC BB AD BC xx w

AD BC DB CC w

Hessian det xx xw

xw ww

º

C CC

C C

Other Hessians

C CH

Higher orders

F FH

N NH

Higher Dimension

C C

C

H

Geometric Meaning of Hessian

C CH

H

H

2det 2 H

C C

C C

2

LCL

L H L

L

LL

L

L

0

MCL

L L

M

L L

L

M

Type 3

Type 21

H LL

ML

L

M

C

C

0

H

H

C

C=

Geometric Meaning of Discriminant

H

H

2det 2 H

C C

C C

L K

L M

M

K

K

M

M

L

L

K

= +...

MCL

K L

M

K K

L

M

LK

M K

M

LM

K

L

Type 111

=

K K

K KL L

L L

M M

M M

Negative

Types of Cubic

C CH

H

H

C C

C C

2

H

H

0

Type 1

11

Type 12Type 111Type 3

H 0

H

H

0

H

H

0

Discriminant Surfaces

C C

C C

2 2

3 3 2 2

62

4 4 3

A D ABCD

AC B D B C

F/E

B/E

QQ 22 AC B

Quadratic Cubic

A

C

B

Q

?

Discriminant Surface 2 2 3 3 2 26 4 4 3 0A D ABCD AC B D B C

, ,B C D

X Y ZA A A

º º º

2 3 3 2 2

6 4 4 3 0D B C D C B D B C

A A A A A A A A A

2 3 3 2 26 4 4 3 0Z XYZ Y X Z X Y

Discriminant Surfaces

C C

C C

2 2

3 3 2 2

62

4 4 3

A D ABCD

AC B D B C

B/A

C/A

D/A

F/E

B/E

QQ

Quadratic Cubic

Types of Cubics

11

1Type 3Type

21Type

111Type

Locus of Cubics with

one (or more) roots at (x,w)=(0,1)

Plane rolls along surface as root=0 changes

B/A

(0,d

,d)

(s,0,0)(0,0,0)

Slice thru space at D/A

(a,b,0)

(0,c,c)

C/A

3 2 2 3, 3 3 0f x w Ax Bx w Cxw w

Tangent

Three roots: (r1,r2,r3),

r1=0

Less obvious property

Triple root line is tangent to plane at infinity

A New Coordinate System

cos sinx w r r

3 34 43 2 2 3 3

1 14 4

cos sin3 3

3 cos3 3 sin3

A C B DAx Bx w Cxw Dw r

A C B D

/ 4

/ 4

3 / 4

3 / 4

E A C

F B D

G A C

H B D

C(x,w)

33 cos 3 sin

cos3 sin3

E Fr

G H

Discriminant in new system

C C

C C

2 2

3 3 2 2

62

4 4 3

A D ABCD

AC B D B C

3

3

E G A

E G C

F H B

F H D

4 2 2 4

2 2 3 3

2 2 2 2 2 2 2 2

4 2 2 4

3 6 3

24 24 8 8

6 6 6 6

2

4

F F E E

FHE GEF GE HF

F G F H G E H E

G G H H

22 2

2 2 2

2 2 2 2

22 2

3

8 3

64

8 3

F E

HF E F GE FF E

F E G H

G H

Discriminant Plot in new system

E/G

H/G

F/G

C C

C C

22 2

2 2 2

2 2 2 2

22 2

3

8 3

64

8 3

F E

HF E F GE FF E

F E G H

G H

Discriminant in EFGH space Zeeman 1976

Helamon Fergusen

Another View of Discriminant

E/G

H/G

F/G

F/E

H/E

G/E

C C

C C

22 2

2 2 2

2 2 2 2

22 2

3

8 3

64

8 3

F E

HF E F GE FF E

F E G H

G H

Transforming Deltoid to Cardioid

Degree 3 Diagrams

p

p

p

C C

= 0C

C

C = J

C

p

p p

p

C

C

= 0C

Proof of 0

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C C

C

C

Proof of 0

p

p

p

C C

C

p

p

p

C C

C

Mirror

p

r

q

C C

C

q

r

p

C C

C

q

p

p

C C

C b

q

q

p

C C

C

p p p p

p p p p

Bx Cw Ax Bw

Cx Dw Bx Cw

C

p

What about?

q

r

p

C C

C

Write r in terms of p,q

Or: brute force expansion

Skew covariant

2 3

2 2

2 2

2 3

3 2

2

2

3 2

J

J

J

J

A A D ABC B

B AC ABD B C

C ACD B D BC

D AD BCD C

C C Cx,w

x,w

x,w

3 2 2 33 3J J J JA x B x w C xw D w ,x w J

Cx,w

x,w

x,w

3 2 2 33 3Ax Bx w Cxw Dw ,x w C

2 2

2 2

2 0 1 2

2 1 0 2

TAC BB AD BC Ax Bxw Cw

x wAD BC DB CC Bx Cxw Dw

Syzygy of J

C

C

C

C

C

p p

p

pC

p p

C

C

C

C

C

p p

p

pC

p p

C

p p

p

C

C

C

C

C

p p

p

C

CC

pp

C

p p

C

C

p

p

1

2= +

1

2

C

C

C

p p

p

+=C

C

C

C

C

p p

p

pC

p p

C

C

C

C

C

p p

p

pC

p p

Terminology

Q C

Q C

Jacobian of Q,C

= First Transvectant of Q,C

Second Transvectant of Q,C

= Apolar Covariant of Q,C

Terminology

A B n2

......

m2

Jacobian of A,B

= First Transvectant of A,B

Second Transvectant of A,B

= Apolar Covariant of A,B

A B n1

......

m1

A B nm

......

m

mth Transvectant of A,B

Terminology using H

H C

H C

Skew = Jacobian of H,C

= First Transvectant of H,C

Second Transvectant of H,C

= Apolar Covariant of H,C = 0