Representation and manipulation of curves
Human Centered CAD Lab.
B-Rep Structure – review
Body Face Edge
Vertex Vertex
List of faces List of edges End vertices
< Topology >
< Geometry >
Surface Eqn.
Curve Eqn.
X,Y,Z Position
Geometry vs. Topology
Types of curve equations
Parametric equation
x=x(t), y=y(t), z=z(t)
Ex) x=Rcos, y=Rsin, z=0 (02)
Implicit nonparametric
F(x, y, z)=0, G(x, y, z)=0
Intersection of two surfaces
Ambiguous independent parameters
Explicit nonparametric
Should choose proper neighboring point during curve generation
,
2 0
2
2 y R
x z 0
2 ,
2 x
R
y z 0
x
z
θ y
Conic curves
Curves obtained by intersecting a cone with a plane
Circle (circular arc), ellipse, hyperbola, parabola
Ex) Circle (circular arc)
Circle in xy-plane with center (xc, yc) and radius R
x = Rcos + xc
y = Rsin + yc
z = 0
Points on the circle are generated by incrementing by △θ from 0, points are connected by line segments
Equation of a circle lying on an arbitrary plane can be derived by transformation
Conic curves – cont’
Hermite curves
Parametric eq. is preferred in CAD systems
Polynomial form of degree 3 is preferred :
C2 continuity is guaranteed when two curves are connected
Impossible to predict the shape change from change in coefficients⇒ not intuitive
Bad for interactive manipulation
eq.
algebraic :
) 1 0
(
(1)
)]
( ) ( ) ( [ )
(
0 1 2 2 3 3
u
u u
u u
z u v u x
u a a a a
P
Hermite curves – cont’
Apply Boundary conditions to replace algebraic coefficients
Use ⇒ Substitute in Eq(1)
(1) (0)
(1)
(0)
, P , P , P
P
1 0
1
0 , P , P , P
P
3 2
1 1
(1)
1 0
(0)
3 2
1 0
1 (1)
0 0
(0)
a a
a P
P
a P
P
a a
a a
P P
a P
P
3 2
(2)
Hermite curves – cont’
Solve for in Eq (2)
3 2
1
0
, a , a , a a
1 0
1 0
1 0
0
P - P P
- P a
P P
P P
- a
P a
P a
2 2
- 2
- 3
3
3
0 1
0 2
0
1 (3)
Hermite curves – cont’
Substitute (3) into (1)
It is possible to predict the curve shape change from the change in P0, P1, P0, P1 to some extent
equation curve
Hermite
u u
u 2u
u 2u
3u 2u
3u 1
(u)
tt coefficien geometric 1 0 1 0
3 2
3 2
3 2
3 2
P P P P P
Hermite curves – cont’
Effect of P0’ and P1’ on curve shape
Hermite curves – cont’
determine the curve shape by blending the effects of P0, P1, P0, P1→ blending function
3 2
3 2
3 2
3
2
2 , 3 2 , 2 ,
3
1 u u u u u u u u u
Bezier curves
It is difficult to realize a curve in one’s mind by changing size and direction of P0, P1 in Hermite curves
Bezier curves
Invented by Bezier at Renault
Use polygon that enclose a curve approximately
Control polygon, control point
Bezier curves – cont’
Passes through 1st and last vertex of control polygon
Tangent vector at the starting point is in the direction of 1st segment of control polygon
Tangent vector at the ending point is in the direction of the last segment
Useful feature for smooth connection of two Bezier curves
The n-th derivative at starting or ending point is
determined by the first or last (n+1) vertices of control polygon
Bezier curve resides completely inside its convex hull
Useful property for efficient calculation of intersection points
Bezier curves – cont’
Bezier curves – cont’
1) u
(0 u)
(1 i u
(u) n n i n i i
0
i
P
P
1
u P(u)
)
1 ( )
( P0 P1
P u u u
1 )
1 ( 2 )
1 (
)
1 ( 2 )
1 ( ) (
2 2
2 2 1
0 2
u u u u
u u
u u
u P P P
P
↑ Control Point
: Straight line from P0 to P1 satisfies the desired qualities including convex hull property
satisfies the desired qualities
Bezier curves – cont’
Highest term is for the curve defined by (n+1) control points
Polynomial of degree n
Degree of curve is determined by number of control points
Large number of control points are needed to represent a curve of complex shape → high degree is necessary.
Heavy computation, oscillation
Better to connect multiple Bezier curves
Global modification property (not local modification)
Difficult to result a curve of desired shape by modifying portions
u
nBlending functions in Bezier curve
for degree 3
Bezier curves – cont’
Bezier Curve does NOT have local modification property
Derivative vectors
1 ,
, 1 , 0 )
1 1 (
) (
1 1 1
0
n i
where
t i t
n n dt
t d
i i
i
i i n n i
i
Ρ Ρ
a r a
i n
j
i n i
n
i j
j n j
n
i
i i
n i
i i n i
n
i n
i
n
i
i i
n i
i i n i
t i t
i n n t
j t j n
dt t d
i j let
t i t
i n n t
i t i n
t i t
i n n t
i t i n dt
t d pf
Ρ Ρ
Ρ Ρ
Ρ Ρ
r r
1
0
1 1
0 1
1 1
0
1 1
1
0 0
1 1
) 1 ( )
( )
1 1 (
) 1 ) (
(
1
) 1 ( )
( )
1 (
) 1 ( )
( )
1 ) (
( )
Derivatives at the starting and the ending points
Derivative vectors – cont’
) ( )
1 1 (
) (
1 )!
1 (
!
)!
1 (
)!
(
!
! ) ) (
(
1 )!
1 (
!
)!
1 (
)!
1 (
)!
1 (
! ) 1 (
1 j 1) n (j
1 1
0
a t
i t n n
dt t d
i n n
i n i
n n i
n i
n i n i
i n n
j n n
j n j
n n j
n j
n j
i i n i
n
i
r a
) (
) 1 (
) (
) 0 (
1 0 1
n
n n
n
Ρ Ρ
r
Ρ Ρ
r
Derivative vectors – cont’
Control polygon determines tangent vectors at the ends
Differentiating (a) in the same way gives
) 2 ,
, 0 (
) 1 2 (
) 1 ) (
(
1 2
0
2 2
2
n i
where
t i t
n n dt n
t d
i i
i
n
i i
i n i
a a
b r b
Elevation of a degree
Conversion of Bezier curve of degree n into degree (n+1), i.e. derivation of (n+1) control points into (n+2) control points
) ( ) 1 ( ) ( )
(
, , )
1 ( )
(
0
1 0
t t t
t t
Express
t i t
t n
n
i
n i
i n i
r r
r
Ρ Ρ
Ρ Ρ
r
Elevation of a degree – cont’
) ( )
1 ( )
(
)!
(
!
! )!
( )!
1 (
)!
1 (
1 1
) 1 1 (
) 1 (
1 ,
1
) 1 ( )
(
0
1 1
1 0
1
m a t k
k t t m
t
k m m
k k
m k
m m
k k
m k
m k
m
t k t
t m t
m n
k i
let
t i t
t n t
m
k
k k
m k
m
k
k k m k
n
i
i i n i
Ρ r
Ρ r
Ρ r
Elevation of a degree – cont’
) ( )
1 (
) 1 ( )
( ) 1 (
)!
(
!
! )!
1 (
!
)!
1 1 (
) 1 1 (
) ( ) 1 (
1 ,
) 1 ( )
( ) 1 (
0 1
0 1
0 0
1
m b k t m
k t m
m k t m
k t t m
t
k m m
k m k
m k
m m
k m k
m k
m k
m
t k t
t m t
n m i
k let
t i t
t n t
k m
k
k m k
k m
k
k m k
m
k
k k m k
n
i
i i
n i
Ρ Ρ r
Ρ r
Ρ r
Elevation of a degree – cont’
Control points of the Bezier curve of degree (n+1) can be derived from (c) as below
n n
n
n n n
n n
nΡ Ρ Ρ Ρ Ρ Ρ
Ρ Ρ ,
, 1 1 ,
) 1 (
, 2
, 0 1 1 1 2 1
0
) 1 (
) 1
) ( 1 1 (
) ( ) 1 ( ) (
) (
&
) (
1
0
1 1
n c
k n
t k k t
t n t
t t
b a
From
n
k
k k k
n
k
Ρ Ρ
r r
de Casteljau algorithm
Bezier Cubic
For
r n i
n r
t t
u B
u
i i
r i r
i r
i
n i
n n
i i
Ρ Ρ
Ρ Ρ
Ρ Ρ
r
, , 0
, , 1 ) 1
(
below.
calculated as
is ) ( )
(
0
1 1 1
0 0 ,
The Value of curve at t of Bezier Curve
de Casteljau algorithm – cont’
3 )
1 (
2 )
1 (
1 )
1 (
2 1 2
3
1 1 1
2
1 1
r t
t
r t
t
r t
t
i i
i
i i
i
i i
i
Ρ Ρ
Ρ
Ρ Ρ
Ρ
Ρ Ρ
Ρ
de Casteljau algorithm – cont’
