Singularity of Cubic Bézier Curves and Surfaces Edmond Nadler Eastern Michigan University . joint work with Tae-wan Kim Min-jae Oh Sung-ha Park Seoul National University SIAM Conference on Geometric and Physical Modeling October 24, 2011 E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 1/35
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Singularity of Cubic Bézier Curves andSurfaces
Edmond Nadler
Eastern Michigan University.
joint work with
Tae-wan KimMin-jae Oh
Sung-ha ParkSeoul National University
SIAM Conference on Geometric and Physical Modeling
October 24, 2011
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 1/35
Outline
Introduction
Singularity of a Parametric Curve
Bézier Curves
Singularity of Bézier Curves
Tangent at Bézier Curve singularity
Surfaces
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 2/35
Parametric Cubic Curve
C(t) = a0 + a1t + a2t2 + a3t3
Example (“twisted cubic”): C(t) =⟨
t, t2, t3⟩
-2
-1
0
1
2
x
01
23
4
y
-5
0
5
z
:
-2-1012
x
0 1 2 3 4
y
-5
0
5
z
,
01234 y-2-1012
x
-5
0
5
z
,
-2-1012
x
0
1
2
3
4
y
-505z
>
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 3/35
Singularity of a Parametric Curve
Singularity of a curve C(t): t∗ where C′(t∗) = ~0
Geometrically, can be a cusp
Example: C(t) =⟨
4t3 − 3t2 + 1, 4t3 − 9t2 + 6t⟩, t ∈ [0, 1]
C′(t) =⟨
12t2 − 6t, 12t2 − 18t + 6⟩
C′(12) = 〈 0, 0〉 =⇒ t∗ = 1
2
C(t∗) =⟨
34, 5
4
⟩
0.5 1. 1.5 2.x0.
0.5
1.
1.5y
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 4/35
Bézier Curves
A representation of parametric polynomial curves
Geometric and intuitive, facilitating creative design process
Computationally efficient and stable
At the core of Computer Aided Geometric Design (CAGD)
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 5/35
Bézier Curves of degree 1
C(t) = (1− t)P0 + t P1, t ∈ [0, 1]
P0
P1
C H0.25L
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 6/35
Bézier Curves of degree 1
C(t) = (1− t)P0 + t P1, t ∈ [0, 1]
P0
P1
C H0.5L
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 6/35
Bézier Curves of degree 1
C(t) = (1− t)P0 + t P1, t ∈ [0, 1]
P0
P1
C H0.75L
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 6/35
Bézier Curves of degree 1
C(t) = (1− t)P0 + t P1, t ∈ [0, 1]
P0
P1
C H1.L
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 6/35
Bézier Curves of degree 2
P01 = (1− t)P0 + t P1 ; P12 = (1− t)P1 + t P2
C(t) = (1− t)P01 + t P12, t ∈ [0, 1]
P0
P1
P2
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 7/35
Bézier Curves of degree 2
P01 = (1− t)P0 + t P1 ; P12 = (1− t)P1 + t P2
C(t) = (1− t)P01 + t P12, t ∈ [0, 1]
P0
P1
P2
P01
P12
C H0.25L
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 7/35
Bézier Curves of degree 2
P01 = (1− t)P0 + t P1 ; P12 = (1− t)P1 + t P2
C(t) = (1− t)P01 + t P12, t ∈ [0, 1]
P0
P1
P2
P01
P12
C H0.25L
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 7/35
Bézier Curves of degree 2
P01 = (1− t)P0 + t P1 ; P12 = (1− t)P1 + t P2
C(t) = (1− t)P01 + t P12, t ∈ [0, 1]
P0
P1
P2P01
P12
C H0.5L
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 7/35
Bézier Curves of degree 2
P01 = (1− t)P0 + t P1 ; P12 = (1− t)P1 + t P2
C(t) = (1− t)P01 + t P12, t ∈ [0, 1]
P0
P1
P2
P01
P12
C H0.75L
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 7/35
Bézier Curve – DefinitionDegree 3:
C(t) =
3∑
i=0
(3i
)(1− t)3−it i Pi
=
3∑
i=0
B3i (t) Pi , t ∈ [0, 1]
where
B3i (t) =
(3i
)(1− t)3−it i is
the ith Bernstein (basis) polynomial of degree 3, and
Pi are known as (Bézier) control points.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 8/35
Bézier Curve – DefinitionDegree n:
C(t) =
n∑
i=0
(ni
)(1− t)n−it i Pi
=
n∑
i=0
Bni (t) Pi , t ∈ [0, 1]
where
Bni (t) =
(ni
)(1− t)n−it i is
the ith Bernstein (basis) polynomial of degree n, and
Pi are known as (Bézier) control points.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 8/35
de Casteljau algorithmthis method of recursive convex combinations to evaluate theBézier function from its control points
C(t) = (1− t)[(1− t)[(1− t)P0 + t P1] + t [(1− t)P1 + t P2]
]
= 1- + t[(1− t)[(1− t)P1 + t P2] + t [(1− t)P2 + t P3]
Singularity of Bézier Curve of degree 3 – part 4bExample: Interval interior acts like endpoints
Convergence of de Casteljau points P012 and P123 as t → t∗:
P0 P1
P2
P3
t_sing=0.55
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 24/35
Singularity of Bézier Curve of degree 3 – part 4bExample: Interval interior acts like endpoints
Convergence of de Casteljau points P012 and P123 as t → t∗:
P0 P1
P2
P3
P01
P12
P23
P012
P123
C H0.35L
t=0.35
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 24/35
Singularity of Bézier Curve of degree 3 – part 4bExample: Interval interior acts like endpoints
Convergence of de Casteljau points P012 and P123 as t → t∗:
P0 P1
P2
P3
P01
P12
P23P012
P123
C H0.45L
t=0.45
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 24/35
Singularity of Bézier Curve of degree 3 – part 4bExample: Interval interior acts like endpoints
Convergence of de Casteljau points P012 and P123 as t → t∗:
P0 P1
P2
P3
P01
P12
P23P012
P123
C H0.5L
t=0.5
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 24/35
Singularity of Bézier Curve of degree 3 – part 4bExample: Interval interior acts like endpoints
Convergence of de Casteljau points P012 and P123 as t → t∗:
P0 P1
P2
P3
P01
P12
P23
P012P123
C H0.525L
t=0.525
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 24/35
Singularity of Bézier Curve of degree 3 – part 4bExample: Interval interior acts like endpoints
Convergence of de Casteljau points P012 and P123 as t → t∗:
P0 P1
P2
P3
P01
P12
P23
P012P123
C H0.55L
t=0.55
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 24/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
P0
P1
P2
P3
t_sing=0.05
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
P0P1
P2
P3
t_sing=0
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
P0P1
P2
P3
t_sing=0
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
P0
P1
P2
P3
t_sing=0.05
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
P0
P1
P2
P3
t_sing=0.1
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
P0
P1
P2
P3
t_sing=0.2
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
P0
P1
P2
P3
t_sing=0.3
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
P0
P1
P2
P3
t_sing=0.4
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
P0
P1
P2
P3
t_sing=0.5
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
P0
P1
P2
P3
t_sing=0.6
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
P0
P1
P2 P3
t_sing=0.7
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
P0
P1P2 P3
t_sing=0.8
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
P0
P1P2 P3
t_sing=0.9
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
P0
P1P2 P3
t_sing=0.95
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Singularity of Bézier Curve of degree 3 – part 4cInterval endpoints act like interior – cusp
Singularity at ends acts like one in the interior: exhibits acusp . . .. . . if parameter interval is extended beyond [0, 1]:
P0
P1P2P3
t_sing=1.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 25/35
Geometric tangent at singularity of Bézier Curve
Recall definition of Bézier curve of degree n :
C(t) =
n∑
i=0
Bni (t) Pi
Suppose m coincident control points at endpoint P0 :
P0 = P1 = · · · = Pm−1, m ≤ n
Suppose so, as we have seen, singularity at t = 0.
Then geometric tangent at P0 is in the direction of Pm − P0
Proof: Under the reparameterization C̃(τ) = C(τ1m ),
C̃′(0) =(n
m
)(Pm − P0) .
And symmetrically for m control points coincident at other endpoint Pn.
To obtain geometric tangent at interior singularity,first apply de Casteljau Subdivision at the singularity,then reparameterize either curve segment as above.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 26/35
Geometric tangent at singularity of Bézier Curve
Recall definition of Bézier curve of degree n :
C(t) =
n∑
i=0
Bni (t) Pi
Suppose m coincident control points at endpoint P0 :
P0 = P1 = · · · = Pm−1, m ≤ n
Suppose so, as we have seen, singularity at t = 0.
Then geometric tangent at P0 is in the direction of Pm − P0
Proof: Under the reparameterization C̃(τ) = C(τ1m ),
C̃′(0) =(n
m
)(Pm − P0) .
And symmetrically for m control points coincident at other endpoint Pn.
To obtain geometric tangent at interior singularity,first apply de Casteljau Subdivision at the singularity,then reparameterize either curve segment as above.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 26/35
Geometric tangent at singularity of Bézier Curve
Recall definition of Bézier curve of degree n :
C(t) =
n∑
i=0
Bni (t) Pi
Suppose m coincident control points at endpoint P0 :
P0 = P1 = · · · = Pm−1, m ≤ n
Suppose so, as we have seen, singularity at t = 0.
Then geometric tangent at P0 is in the direction of Pm − P0
Proof: Under the reparameterization C̃(τ) = C(τ1m ),
C̃′(0) =(n
m
)(Pm − P0) .
And symmetrically for m control points coincident at other endpoint Pn.
To obtain geometric tangent at interior singularity,first apply de Casteljau Subdivision at the singularity,then reparameterize either curve segment as above.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 26/35
Geometric tangent at singularity of Bézier Curve
Recall definition of Bézier curve of degree n :
C(t) =
n∑
i=0
Bni (t) Pi
Suppose m coincident control points at endpoint P0 :
P0 = P1 = · · · = Pm−1, m ≤ n
Suppose so, as we have seen, singularity at t = 0.
Then geometric tangent at P0 is in the direction of Pm − P0
Proof: Under the reparameterization C̃(τ) = C(τ1m ),
C̃′(0) =(n
m
)(Pm − P0) .
And symmetrically for m control points coincident at other endpoint Pn.
To obtain geometric tangent at interior singularity,first apply de Casteljau Subdivision at the singularity,then reparameterize either curve segment as above.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 26/35
Singularity of a Parametric Surface
Local parametric surface: S(u, v) : (D ∈ R2) → R
3
Example:S(u, v) =
⟨u2 + v2, u, v
⟩: paraboloid with axis the x-axis
Normal vector N̂(u, v) is unit vector ⊥ S at (u, v), if this exists,
usually computed as N̂(u, v) =Su × Sv
‖Su × Sv‖(u, v) .
Surface is regular at any (u, v) where N̂(u, v) exists,equivalently, where Su,Sv are linearly independent.Otherwise, singular.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 27/35
Singularity of a Parametric Surface
Local parametric surface: S(u, v) : (D ∈ R2) → R
3
Example:S(u, v) =
⟨u2 + v2, u, v
⟩: paraboloid with axis the x-axis
Normal vector N̂(u, v) is unit vector ⊥ S at (u, v), if this exists,
usually computed as N̂(u, v) =Su × Sv
‖Su × Sv‖(u, v) .
Surface is regular at any (u, v) where N̂(u, v) exists,equivalently, where Su,Sv are linearly independent.Otherwise, singular.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 27/35
Singularity of a Parametric Surface
Local parametric surface: S(u, v) : (D ∈ R2) → R
3
Example:S(u, v) =
⟨u2 + v2, u, v
⟩: paraboloid with axis the x-axis
Normal vector N̂(u, v) is unit vector ⊥ S at (u, v), if this exists,
usually computed as N̂(u, v) =Su × Sv
‖Su × Sv‖(u, v) .
Surface is regular at any (u, v) where N̂(u, v) exists,equivalently, where Su,Sv are linearly independent.Otherwise, singular.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 27/35
Example of surface with singularity
S(u, v) =⟨
u3, v3, uv⟩, u ∈ [−1, 1], v ∈ [−1, 1]
-1.0
-0.5
0.0
0.5
1.0
x
-1.0 -0.5 0.0 0.5 1.0
y
-1.0
-0.5
0.0
0.5
1.0
z
Singularity at (u, v) = (0, 0)
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 28/35
Example of surface with singularity
S(u, v) =⟨
u3, v3, uv⟩, u ∈ [0, 1], v ∈ [0, 1]
0.0
0.5
1.0
x
0.0 0.5 1.0
y
0.0
0.5
1.0
z
Singularity at (u, v) = (0, 0)
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 28/35
Rectangular Bézier Surface – DefinitionBivariate Polynomial of Degree (m, n) in (u, v),i.e., S(u, v0) degree m in u, S(u0, v) degree n in v :
S(u, v) =
m∑
i=0
n∑
j=0
ai,j ui vj
Better to represent as . . .Bézier Surface of degree (m, n) in (u, v):
S(u, v) =m∑
i=0
n∑
j=0
Bmi (u) Bn
j (v) Pi,j , u ∈ [0, 1], v ∈ [0, 1]
where, as before,
wher Bpk(t) =
(pk
)(1− t)p−kt k : Bernstein (basis) polynomial
wher Pi,j : Bézier control points.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 29/35
Rectangular Bézier Surface – DefinitionBivariate Polynomial of Degree (m, n) in (u, v),i.e., S(u, v0) degree m in u, S(u0, v) degree n in v :
S(u, v) =
m∑
i=0
n∑
j=0
ai,j ui vj
Better to represent as . . .Bézier Surface of degree (m, n) in (u, v):
S(u, v) =m∑
i=0
n∑
j=0
Bmi (u) Bn
j (v) Pi,j , u ∈ [0, 1], v ∈ [0, 1]
where, as before,
wher Bpk(t) =
(pk
)(1− t)p−kt k : Bernstein (basis) polynomial
wher Pi,j : Bézier control points.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 29/35
Bézier Surface
S(u, v) =
m∑
i=0
n∑
j=0
Bmi (u) Bn
j (v) Pi,j , u ∈ [0, 1], v ∈ [0, 1]
Bézier surface inherits many properties from Bézier curves, includingconvex hull property.Its boundary curves are Bézier curves, with control points from theboundary of the surface control net [Pi,j], i = 0..m, j = 0..n,e.g., curve S(u, 0) has control point sequence 〈Pi,0〉, i = 0..m
Hence, S singular at (0, 0) if P0,0, P1,0, P0,1 are collinear, whichincludes the cases of P1,0 = P0,0 and P0,1 = P0,0.
Consider the tangential bilinear surface: degree (1, 1) surface withP0,0, P0,1, P1,0, P1,1 from a higher order surface.This bilinear surface shares N̂(0, 0) with the higher order surface,if both exist.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 31/35
Singularity of a Bézier SurfaceConsider singularity at a corner, say, (u, v) = (0, 0) and find thegeometric normal, in case one exists.
Note: Singularity at points other than the corners can be handled using de Casteljau
subdivision in each variable, in analogy with the curve case, allowing treatment of any
point in the domain as a corner of a rectangular sub-domain.
Recall
S singular at (0, 0) iff Su(0, 0),Sv(0, 0) linearly dependent
Hence, S singular at (0, 0) if P0,0, P1,0, P0,1 are collinear, whichincludes the cases of P1,0 = P0,0 and P0,1 = P0,0.
Consider the tangential bilinear surface: degree (1, 1) surface withP0,0, P0,1, P1,0, P1,1 from a higher order surface.This bilinear surface shares N̂(0, 0) with the higher order surface,if both exist.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 31/35
Singularity of a Bézier SurfaceConsider singularity at a corner, say, (u, v) = (0, 0) and find thegeometric normal, in case one exists.
Note: Singularity at points other than the corners can be handled using de Casteljau
subdivision in each variable, in analogy with the curve case, allowing treatment of any
point in the domain as a corner of a rectangular sub-domain.
Recall
S singular at (0, 0) iff Su(0, 0),Sv(0, 0) linearly dependent
Hence, S singular at (0, 0) if P0,0, P1,0, P0,1 are collinear, whichincludes the cases of P1,0 = P0,0 and P0,1 = P0,0.
Consider the tangential bilinear surface: degree (1, 1) surface withP0,0, P0,1, P1,0, P1,1 from a higher order surface.This bilinear surface shares N̂(0, 0) with the higher order surface,if both exist.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 31/35
Cases for Tangential Bilinear SurfaceLet N̂m,n denote N̂(0, 0) of a degree (m, n) surfaceLet N̂1,1 denote N̂(0, 0) of its tangential bilinear surface
0 P0,0, P1,0, P0,1 not collinearN̂1,1 = unit ((P1,0 − P0,0) × (P0,1 − P0,0))N̂m,n = N̂1,1
1 P0,0, P1,0, P0,1 collinear, and P1,1 not on this lineTangential bilinear surface is planar.N̂1,1 = unit ((P0,1 − P1,1) × (P1,0 − P1,1))N̂m,n may not exist ; if exists, = N̂1,1.
2 P0,0, P1,0, P0,1, P1,1 all collinearN̂1,1 does not exist, as bilinear surface degenerates to aline (or even a point).N̂m,n may exist.
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 32/35
Example of Planar (Case 1) Tangential Bilinear SurfaceDegree (1, 3) Bézier Surface with P0,1 = P0,0 =⇒ singularity
Geometric singularity:No N, as two possible ones:using geometric tangent alongboundary curve S(0, v) vs.using planar tangential bilinearsurface
No geometric singularitywith blue control points coplanar :adjusted P0,2 to achieve
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 33/35
Example of Planar (Case 1) Tangential Bilinear SurfaceDegree (1, 3) Bézier Surface with P0,1 = P0,0 =⇒ singularity
Geometric singularity:No N, as two possible ones:using geometric tangent alongboundary curve S(0, v) vs.using planar tangential bilinearsurface
No geometric singularitywith blue control points coplanar :adjusted P0,2 to achieve
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 33/35
SummaryParametric polynomial curves & surfaces of degree 3 areuseful.
Need to understand their singularities
Use Bézier form to describe singularities of parametricpolynomial curves of degrees 1,2,3.
Unify cases of endpoint and interior Bézier curvesingularity.
Express geometric tangent of Bézier curve singularity.
Analogous approach for Bézier surface singularity, butmuch more complex; use curve results with additionalconsiderations.
Current and future related workComplete analysis of singularity of Bézier surfaces.Analyze curvatureApplication: G1 surface fitting in the presence of T-junction
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 34/35
For Further Reading I
G. FarinCurves and Surfaces for CAGDAcademic Press, 1990
G. Farin, J. Hoschek, M.-S. Kim, eds.Handbook of Computer Aided Geometric DesignElsevier, 2002
R.T. FaroukiPythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Springer-Verlag, 2008
A. GrayModern Differential Geometry of Curves and SurfacesCRC Press, 1993
B.-Q. Su, D.-Y. LiuComputational Geometry – Curve and Surface ModelingAcademic Press, 1990
E. Nadler, Eastern Michigan University Bézier Curve & Surface Singularity 35/35