Journal of Information & Computational Science 2: 2 (2005) 415–419
Available at http://www.joics.com
Poisson Developable Surfaces ?
Xingwang Zhang a,b , Guojin Wang a,b,∗
a Institute
b State
of Computer Images and Graphics, Zhejiang University, Hangzhou 310027, China
Key Laboratory of CAD & CG, Zhejiang University, Hangzhou 310027, China
Received 9 April 2005; revised 13 June 2005
Abstract
Recurring to some excellent geometric and algebraic properties of Poisson curves, we present an algorithm
that generates a developable Poisson surface through a Poisson curve of arbitrary degree. The technique
is based on necessary and sufficient conditions for a developable surface, degree elevation formula of
a Poisson curve and linear independence of Poisson basis. Some common transcendental surfaces are
represented as Poisson developable surfaces accurately.
Keywords: Poisson basis; Developable; Transcendental; Linear independence
1
Introduction
Developable surfaces are a type of important and fundamental surfaces universally used in industry design [4]. Different methods have been presented for the design of developable surfaces
[1, 2, 3, 6, 8]. Previous techniques can’t exactly represent a tangent surface or a cone through
a helix frequently appearing in industry design. However, transcendental surfaces are frequently
met in revolution cutting and are some important parts of the surfaces of milling cutters, helicoids and precision machinery parts. Noting that some common transcendental curves can be
represented as Poisson curves, this paper introduces Poisson curves to the field of the design of
developable surfaces, and successfully generates some developable transcendental surfaces. We
represent some common transcendental surfaces as Poisson developable surfaces precisely. Then
we approximate to the Poisson surfaces using the Poisson subdivision algorithm. The above results could be applied to the geometric configuration design and machining, as well as the plate
expansion of spiral-like pipe surfaces.
?
Supported by the National Grand Fundamental Research 973 Program of China (No. 2004CB719400), the
National Natural Science Foundation of China (No. 60373033, No. 60333010) and the National Natural Science
Foundation for Innovative Research Groups (No. 60021201).
∗
Corresponding author.
Email address: wgj@math.zju.edu.cn (Guojin Wang).
1548–7741/ Copyright °
c 2005 Binary Information Press
June 2005
416
X. Zhang et al. /Journal of Information & Computational Science 2: 2 (2005) 415–419
2
Properties of a Poisson Curve
The parametric form of a degree n Poisson curve can be expressed as
P(u) =
∞
X
[n]
bnk (u)pk ,
0 ≤ u < R,
(1)
k=0
where
(nu)k
exp(−nu), k = 0, 1, · · · ,
(2)
k!
µm
¶+∞
P n
[n]
are degree n Poisson basis. The sequences
bk (u)pk
are assumed to converge on the
bnk (u) =
k=0
m=1
interval [0, R), and P(u) is called a degree n Poisson parameter curve with its control points
[n]
[n]
[n]
{p0 , p1 , p2 , · · · }. In this paper, we mainly employ the degree 1 Poisson curve. In the sequel,
[1]
[1]
[1]
for the sake of brevity, we denote its control points {p0 , p1 , p2 , · · · } as {p0 , p1 , p2 , · · · }, and
its basis b1k (u) (k = 0, 1, · · · ) as bk (u) (k = 0, 1, · · · ).
Some properties of a Poisson curve are similar to those of a Bézier curve. It is worth pointing
out the high-order derivatives formulae,
dl P(u)
dul
=
∞
P
bk (u)∆l pk ,
k=0
l−1
∆l pk = ∆
0 ≤ u < R,
l = 0, 1, · · · ,
pk+1 − ∆l−1 pk , ∆0 pk = pk ,
k = 0, 1, · · · ,
of a degree 1 Poisson curve [7], where ∆ is the forward difference operator. And its degree
elevation formula [5] is,
∞
+∞
X
X
[n]
bk (u)pk =
bnk (u)pk ,





[n]
P0
[n]
P1
..
.
[n]
Pk

k=0

 
 
=
 
k=0
B00 ( n1 )
0
B10 ( n1 ) B11 ( n1 )
..
..
.
.
0 1
1 1
Bk ( n ) Bk ( n )

···
···
..
.
0
0
0
···
Bkk ( n1 )




P0
P1
..
.



,

Pk
where Bkj (t) is the Bernstein basis function.
A Poisson curve has de Casteljau-like corner cutting algorithm and the subdivision scheme has
the uniform convergence. So some common transcendental curves can be approximated using
Poisson subdivision algorithm with a small approximate error as well as a high approximating
speed.
3
Fundamental Form of a Poisson Developable Surface
Blending P(u) =
∞
P
k=0
bk (u)pk with Q(u) =
+∞
P
bk (u)qk , we construct a ruled surface
k=0
r(u, v) = (1 − v)P(u) + vQ(u) = P(u) + vT(u),
(u, v) ∈ [0, R) ⊗ [0, 1],
(3)
X. Zhang et al. /Journal of Information & Computational Science 2: 2 (2005) 415–419
417
where P(u) and T(u) = Q(u) − P(u) are called a directrix and a generator of the ruled surface
respectively. r(u, v) is developable if and only if for any u ∈ [0, R), there exist three scalar
functions D(u), E(u) and F (u) that are not all zeros simultaneously such that DP0 +ET+F T0 = 0
[9]. There are only three types of developable surfaces: cylinders (including planes), cones and
tangent surfaces.
4
Geometric Representation of a Poisson
Developable Surface
Case A D(u) ≡ 0. In this case, E(u), F (u) are not both zeroes simultaneously such that
ET + F T0 = 0. Let e(u) be a unit vector paralleled to T(u). Define l(u) = kT(u)k , then
T0 = l0 e + le0 . Therefore T × T0 = le × le0 = 0, i.e., e × e0 = 0. From the Lagrange’s identity,
it follows that e0 · e0 = 0, i.e., e0 = 0. This indicates that the direction of T(u) does not change.
i l(0)
i T(0)
= ∆i (q0 − p0 ) = ddu
Consequently r(u, v) is a cylinder. Since d du
i
i e, (qi − pi ) k e, i = 0, 1, · · · .
³ k
´
k
kπ
2 sin
Particularly, P(u) = (cos u, sin u, 0), T(u) = (0, 0, 1), pk = 2 2 cos kπ
,
2
,
0
, qk =
4
4
³ k
´
k
2 2 cos kπ
, 2 2 sin kπ
, 1 , k = 0, 1, · · · . Using the Poisson subdivision algorithm, we obtain a
4
4
surface approximating to the right circular cylinder surface.
E(u)
F (u)
Case B D(u) 6= 0. In this case, there exist two functions α(u) = − D(u)
and β(u) = − D(u)
such
0
0
that P = αT + βT . Thus r(u, v) can be expressed as
r(u, v) = A(u) + (v + β)T,
(4)
where A(u) = P(u) − βT(u) and A0 = P0 − β 0 T − βT0 = (α − β 0 )T.
Case B.1 α = β 0 . In this case, A0 = 0; if we introduce a parameter transformation vb = v+β(u) ∈
[β(u), β(u)], then r(u, v) can be represented as
r(u, v) = A + vbT(u).
(5)
It means that r(u, v) is a cone with its constant vertex A.
Let α = − exp(−u), β = exp(−u), then
+∞
X
bi (u)∆pi = − exp(−u)
i=0
+∞
X
bi (u)(qi − pi ) + exp(−u)
i=0
+∞
X
bi (u)∆(qi − pi ).
i=0
From degree elevation formula and linear independence of Poisson basis, we can obtain
qi+1
¶
i µ
X
i
= pi+1 + 2(qi − pi ) +
∆pj ,
j
i = 0, 1, 2, · · · .
(6)
j=0
Specially, p(u) = (exp(−u) cos u, exp(−u) sin u, exp(−u))³, α = − exp(−u) and β = exp(−u). So´
¡
¢
k
k
p0 = (1, 0, 1), pk = cos kπ
, sin kπ
, 0 , q0 = (2, 0, 2), qk = cos kπ
+ 2 2 cos kπ
, sin kπ
+ 2 2 sin kπ
,1 ,
2
2
2
4
2
4
k = 1, 2, · · · . Fig. 1 illustrates the corresponding conical surface through the conical spiral curve
(conical logarithmic curve) P(u).
418
X. Zhang et al. /Journal of Information & Computational Science 2: 2 (2005) 415–419
q
0
q3
Q(u)
q
1
q2
q2
Q(u)
q3
p0
p3
P(u)
p2 q
1
P(u)
q
p1 0
p3
p2
p1
p
0
Fig. 1: A conical Poisson surface
Fig. 2: A Poisson tangent surface
through a conical spiral.
through a cylinder spiral.
1
0
Case B.2 α 6= β 0 . Hence T = α−β
b=
0 A . We introduce a new parameter v
is a tangent surface of the curve A = A(u), whose expression is
v+β(u)
,
α(u)−β 0 (u)
then r(u, v)
r(u, v) = A(u) + v̂A0 (u).
(1) Let α = 1, β = 0, then we get
+∞
P
bi (u)∆pi =
i=0
+∞
P
(7)
bi (u)(qi − pi ). Then we obtain
i=0
qi = pi+1 , i = 0, 1, · · · .
(8)
´
³ k
k
kπ
2 sin
,
2
,
k
, qk =
Particularly, P(u) = (cos u, sin u, u), α = 1, β = 0. pk =
2 2 cos kπ
4
4
³ k+1
´
k+1
2 2 cos (k+1)π
, 2 2 sin (k+1)π
, k + 1 , k = 0, 1, · · · . So we get a Poisson tangent surface through
4
4
a cylinder spiral, as depicted in Fig. 2.
(2) Let α = σ2 , β = σu , σ 6= 0, then we get
+∞
X
bi (u)∆pi =
i=0
+∞
X
i=0
+∞
X
2
i
bi (u) (qi − pi ) +
bi (u) ∆(qi−1 − pi−1 ).
σ
σ
i=1
Based on linear independence of Poisson basis, we obtain
½
0
q0 = p0 + σ∆p
, i = 0;
2
σ∆pi +i(qi−1 −pi−1 )
, i = 1, 2, · · · .
qi = pi +
i+2
(9)
³
k
In particular, P(u) = (cos u − u sin u, sin u + u cos u, 2u), α = 2, and β = u. So pk = 2 2 cos kπ
−
4
´
³ k+1
k−1
k−1
k−1
k+1
k
2 2 k sin (k−1)π
, 2 2 sin kπ
+ 2 2 k cos (k−1)π
, 2k , qk = 2 2 cos (k+1)π
− 2 2 k sin (k−1)π
,2 2
4
4
4
4
4
´
k−1
+ 2 2 k cos (k−1)π
, 2k + 1 , k = 0, 1, · · · . Now making v̂ = v + u, A = (cos u, sin u, u)
sin (k+1)π
4
4
is an edge of regression. Thus r(u, v) = (cos u − (u + v) sin u, sin u + (u + v) cos u, 2u + v) is a
tangent surface of the cylinder spiral A(u), as shown in Fig. 3.
(3) Let α = u1 , β = 1, then
+∞
X
i=1
bi (u)i∆pi−1 =
+∞
X
i=0
bi (u)(qi − pi ) +
+∞
X
i=1
bi (u)i∆(qi−1 − pi−1 ).
X. Zhang et al. /Journal of Information & Computational Science 2: 2 (2005) 415–419
q
419
q3
3
p3
Q(u)
q2
Q(u)
p3
q
p2
2
P(u)
P(u)
p2
q1
A(u)
A(u)
p1
p
q0
q1
1
p0 q0
p0
Fig. 3: A Poisson tangent surface
Fig. 4: A Poisson tangent surface
of a cylinder spiral.
through an involute spiral.
According to linear independence of Poisson basis, we obtain
½
q0 = p0 , i = 0;
−2pi−1 )
qi = pi + i(pi +qi−1
, i = 1, 2, · · · .
i+1
(10)
³
k
k−1
−2 2 k
Specially, P(u) = (cos u−u sin u, sin u+u cos u, 2u), α = and β = 1. So pk = 2 2 cos kπ
4
´
³
k−1
k+1
k+1
k
k
k
sin (k−1)π
, 2 2 sin kπ
+ 2 2 k cos (k−1)π
, 2k , qk = 2 2 cos kπ
− 2 2 k sin (k−1)π
, 2 2 sin kπ
+2 2 k
4
4
4
4
4
4
´
cos (k−1)π
,
3k
, k = 0, 1, · · · . We draw a tangent surface through the involute spiral P(u) of
4
a circle, as depicted in Fig. 4. The edge of regression is a cylinder spiral A = P − T =
(cos u, sin u, u).
1
u
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