Developable surfaces are deﬁned as zero gaussian curvature surfaces (intrinsically ﬂat). That is, plane patches that are curved by just folding, rolling or cutting, but without stretching or combing. Useful for depicting steel plates in naval industry, cloth in textile industry. . . But they are difﬁcult to include in the NURBS formulation for the zero curvature requirement

Fernández Jambrina, Leonardo

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

Triangular Bézier Developable Patches
Leonardo Fernández-Jambrina
Matemática Aplicada
E.T.S.I. Navales
Universidad Politécnica de Madrid, Spain
leonardo.fernandez@upm.es
2009 SIAM/ACM Joint Conference on Geometric and Physical
Modeling
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 1 / 17
Sketch
1 Introduction
2 Triangular developable surfaces
3 Results
4 Conclusions
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 2 / 17
Motivation
Developable surfaces are defined as zero gaussian curvature
surfaces (intrinsically flat).
That is, plane patches that are curved by just folding, rolling or
cutting, but without stretching or combing.
Useful for depicting steel plates in naval industry, cloth in textile
industry. . .
But they are difficult to include in the NURBS formulation for the
zero curvature requirement.
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 3 / 17
Description
Developable surfaces are a special case of ruled surfaces,
c(u, v) = (1− v)c(u) + vd(u),
with null gaussian curvature,
d ′(u) · c′(u)×
(
d(u)− c(u)
)
= 0.
This amounts to requiring that vectors w(u) = d(u)− c(u), c′(u), d ′(u)
are coplanary.
And the tangent plane is the same for points on the same ruling.
c2(u)
c1(u)
c'2(u0)
c'1(u0)
c2(u0)-c1(u0)
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 4 / 17
Description
Developable surfaces are a special case of ruled surfaces,
c(u, v) = (1− v)c(u) + vd(u),
with null gaussian curvature,
d ′(u) · c′(u)×
(
d(u)− c(u)
)
= 0.
This amounts to requiring that vectors w(u) = d(u)− c(u), c′(u), d ′(u)
are coplanary.
Or put in another way,
c′(u) = λ(u)w(u) + µ(u)w′(u),
except in the case w ‖ w′ (cylinders).
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 4 / 17
Description
Developable surfaces are a special case of ruled surfaces,
c(u, v) = (1− v)c(u) + vd(u),
with null gaussian curvature,
d ′(u) · c′(u)×
(
d(u)− c(u)
)
= 0.
This amounts to requiring that vectors w(u) = d(u)− c(u), c′(u), d ′(u)
are coplanary.
Or put in another way,
c′(u) = λ(u)w(u) + µ(u)w′(u),


c˜(u) = c(u)− µ(u)w(u),
c˜′(u) = λ˜(u)w(u),
λ˜(u) = λ(u)− µ′(u).
except in the case w ‖ w′ (cylinders).
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 4 / 17
Classification of developable surfaces
Planar surfaces: pieces of planes.
Cylindrical surfaces: The rulings are parallel.
c0,0
c1,0
c2,0
c0,1
c1,1
c2,1
Conical surfaces: The rulings have a common point, the vertex.
c0,1 c1,1
c2,1
c0,0
c1,0
c2,0
a
Tangent surfaces: Tangent lines to a curve (edge of regression).
c0,0
c1,0
c3,0
c0,1
c1,1
c2,1
c3,1
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 5 / 17
Developability condition in polar form
Let us consider two n-degree rational curves c(u), d(u) with control
polygons {c0, . . . , cn}, {d0, . . . ,dn} and weights {w0, . . . ,wn},
{ω0, . . . , ωn}.
The derivative of a rational curve c(u) is
c′(u) =
nwn−1)0 (u)w
n−1)
1 (u)
w
n)
0 (u)
2
(
cn−1)1 (u)− c
n−1)
0 (u)
)
,
as a difference between the two last-but-one points in the de Casteljau
algorithm.
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 6 / 17
Developability condition in polar form
Let us consider two n-degree rational curves c(u), d(u) with control
polygons {c0, . . . , cn}, {d0, . . . ,dn} and weights {w0, . . . ,wn},
{ω0, . . . , ωn}.
The surface is then developable iff cn−1)0 (u), c
n−1)
1 (u), d
n−1)
0 (u),
dn−1)1 (u) are coplanary.
cm-10(u)
cm-11(u)
dm-11(u)dm-10(u)
c(u)
d(u)
c(u,v)
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 6 / 17
Developability condition in polar form
Let us consider two n-degree rational curves c(u), d(u) with control
polygons {c0, . . . , cn}, {d0, . . . ,dn} and weights {w0, . . . ,wn},
{ω0, . . . , ωn}.
The surface is then developable iff cn−1)0 (u), c
n−1)
1 (u), d
n−1)
0 (u),
dn−1)1 (u) are coplanary.
Or using the corresponding 4D vectors,
p
c
q
x0 = 0
x0 = 1d
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 6 / 17
Developability condition in polar form
Let us consider two n-degree rational curves c(u), d(u) with control
polygons {c0, . . . , cn}, {d0, . . . ,dn} and weights {w0, . . . ,wn},
{ω0, . . . , ωn}.
The surface is then developable iff cn−1)0 (u), c
n−1)
1 (u), d
n−1)
0 (u),
dn−1)1 (u) are coplanary.
Equivalently, iff the 4D vectors pn−1)0 (u), p
n−1)
1 (u), q
n−1)
0 (u), q
n−1)
1 (u)
are linearly dependent,
pn−1)i (u) =
(
w
n−1)
i (u),w
n−1)
i (u)c
n−1)
i (u)
)
,
qn−1)i (u) =
(
ω
n−1)
i (u), ω
n−1)
i (u)d
n−1)
i (u)
)
.
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 6 / 17
Developability and linear dependence
The linear dependence condition for pn−1)0 (u), p
n−1)
1 (u), q
n−1)
0 (u),
qn−1)1 (u) may be written as
(1− Λ)pn−1)0 (u) + Λp
n−1)
1 (u) = σ
(
(1−M)qn−1)0 (u) + Mq
n−1)
1 (u)
)
,
with rational Λ, M, σ.
Or using the de Casteljau algorithm in terms of blossoms,
p[u<n−1>,Λ] = σq[u<n−1>,M],
taking into account that
pn−1)0 (u) = p[u
<n−1>,0], pn−1)1 (u) = p[u
<n−1>,1].
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 7 / 17
Developability and blossoms
Theorem
Two rational curves c(u), d(u) with control polygons {c0, . . . , cn},
{d0, . . . ,dn} and weights {w0, . . . ,wn}, {ω0, . . . , ωn} define a generic
developable surface iff their respective 4D polar forms are related by
p[u<n−1>,Λ] = σq[u<n−1>,M].
c(u)c[u
<n-1>,0]
c[u<n-1>,1]
d[u<n-1>,0]
d[u<n-1>,M]
d(u)
c[u<n-1>,Λ]
d[u<n-1>,1]
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 8 / 17
Interpretation of Λ, M , σ
We know that c(u), w(u) = d(u)− c(u) satisfy
c′(u) = λ(u)w(u) + µ(u)w′(u).
In the polinomial and σ = 1 case,
λ =
n
Λ−M , µ =
M − u
Λ−M .
p[u<n−1>,Λ] = q[u<n−1>,M] is then c′ = λw + µw′ in blossom form!
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 9 / 17
Interpretation of Λ, M , σ
We know that c(u), w(u) = d(u)− c(u) satisfy
c′(u) = λ(u)w(u) + µ(u)w′(u).
In the polinomial and σ = 1 case,
λ =
n
Λ−M , µ =
M − u
Λ−M .
p[u<n−1>,Λ] = q[u<n−1>,M] is then c′ = λw + µw′ in blossom form!
The rational case can be done either, but introducing third function ν,
λ =
nσ
Λ− σM + u(σ − 1) , µ =
σ(M − u)
Λ− σM + u(σ − 1) , ν =
n(σ − 1)
Λ− σM + u(σ − 1)
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 9 / 17
Edge of regression
We may obtain the curve of singular points of the developable surface,
just requiring
cu(u, v) ‖ cv (u, v).
Singular points are located along the edge of regression
v =
σ(u −M)
σ(u −M) + Λ− u (u = (Λ−M)v + M if σ = 1) ,
which is a rational curve on the developable surface.
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 10 / 17
Tangent surfaces
In particular we may put the developable surface in the form of a
surface tangent to its edge of regression,
c(u, v) = c(u) + v c′(u).
Functions Λ, M and σ are remarkably simple in this case,
Λ = u + n, M = u, σ = 1.
Since all generic rational developable surfaces have a rational edge of
regression c(u), this means we may construct all rational developable
surfaces by enlarging these singular patches.
c˜(u, v) = c(u) + c′(u) {v a(u) + (1− v)b(u)} .
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 11 / 17
Constant Λ, M , σ
For instance, extending the tangent surface with
a(u) = l − u
n
, b(u) = m − u
n
,
we get the constant functions case,
Λ = (n + 1)l , M = (n + 1)m, σ = 1,
which is a totally equivalent representation to singular Bézier patches
for all generic developable surfaces.
This construction has been used for generating developable surfaces
due to its simplicity, since it provides planar control cells,
(1− Λ)pj + Λpj+1 = (1−M)qj + Mqj+1, j = 0, . . . ,n − 1,
for both rational and polynomial developable surfaces (and even
spline).
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 12 / 17
Changes of parametrizations
u = f (u˜), f of degree m: Λ (f (u˜)) = f˜
(
u˜, Λ˜
)
.
e.g. u = u˜2 implies Λ˜u˜ = Λ.
Change of weights: {w0, . . . ,wn} → {αw0, . . . , αwn},
{ω0, . . . , ωn} → {βw0, . . . , βwn} implies σ → αβσ.
Reparametrization: u = u˜(1−ρ)u˜+ρ ⇔
{w0, . . . ,wn} → {ρ
nw0, . . . , ρ
0wn},
{ω0, . . . , ωn} → {ρ
nω0, . . . , ρ
0ωn}
implies Λ = Λ˜
(1−ρ)Λ˜+ρ , M =
M˜
(1−ρ)M˜+ρ , σ = σ˜
(1−ρ)M˜+ρ
(1−ρ)Λ˜+ρ .
. . .
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 13 / 17
Summary
We have shown that every generic rational developable surface
may be generated from a singular Bézier patch (explicit tangent
surface).
Or equivalently from a constant Λ, M, σ patch.
By extending the patch along the rulings of the surface.
Usual transformations of weights and coordinates are easily
implemented.
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 14 / 17
Some references
G. Aumann.
Interpolation with developable Bézier patches.
CAGD 20, 601-619 (2003).
G. Aumann.
B-spline control nets for developable surfaces.
CAGD 21, 667-670 (2004).
L. F.-J.
Degree elevation and developable Bézier surfaces.
CAGD 24, 189-196 (2007).
L. F.-J.
(in preparation)
Leonardo Fernández-Jambrina (U.P.M.) Triangular Bézier Developable Patches 2009 SIAM/ACM 15 / 17


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