cse590b lecture 5 cubic bezier curves...2d=p1 point sets on line linear quadratic cubic quartic....
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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/
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Cubic Bezier Curves
When does cusp/loop happen?
Parameters at self intersection?
Implicit Equation?
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The Grid
1
4D=P3
Surfaces
in space
3D=P2
Curves
in plane
2D=P1
Point sets
on line
Linear
Quadratic
Cubic
Quartic
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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
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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
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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
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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
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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
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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
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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
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2D Epsilon
0 1
1 0e
K L K L
QQ-2detQ =
QQ*
Adjugate of Matrix
Determinant of Matrix
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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
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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
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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
=
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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
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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
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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
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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
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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
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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
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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
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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
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Other Hessians
C CH
Higher orders
F FH
N NH
Higher Dimension
C C
C
H
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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=
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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
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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
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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
?
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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
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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
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Types of Cubics
11
1Type 3Type
21Type
111Type
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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
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Less obvious property
Triple root line is tangent to plane at infinity
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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
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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
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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
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Discriminant in EFGH space Zeeman 1976
Helamon Fergusen
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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
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Transforming Deltoid to Cardioid
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Degree 3 Diagrams
p
p
p
C C
= 0C
C
C = J
C
p
p p
p
C
C
= 0C
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Proof of 0
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C C
C
C
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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
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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
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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
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Terminology
Q C
Q C
Jacobian of Q,C
= First Transvectant of Q,C
Second Transvectant of Q,C
= Apolar Covariant of Q,C
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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
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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
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