Page 282 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 23: Developable Surface Fitting to Point Clouds

Discuss the Classification of Developable Surfaces according to their Image on B Notes

Describe Cones and Cylinders of Revolution

Explain Recognition of Developable Surfaces from Point Clouds

Describe Reconstruction of Developable Surfaces from Measurements

Introduction

Given a cloud of data points p in  , we want to decide whether p are measurements of a
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i
i
cylinder or cone of revolution, a general cylinder or cone or a general developable surface. In
case, where this is true we will approximate the given data points by one of the mentioned
shapes. In the following, we denote all these shapes by developable surfaces. To implement this
we use a concept of classical geometry to represent a developable surface not as a two-parameter
set of points but as a one-parameter set of tangent planes and show how this interpretation
applies to the recognition and reconstruction of developable shapes.
Points and vectors in  or  are denoted by boldface letters, p, v. Planes and lines are displayed
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as italic capital letters, T,L. We use Cartesian coordinates in  with axes x, y and z. In  , the axes
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of the Cartesian coordinate system are denoted by u , . . . , u .
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1
Developable surfaces shall briefly be introduced as special cases of ruled surfaces. A ruled
surface R carries a one parameter family of straight lines L. These lines are called generators or
generating lines. The general parametrization of a ruled surface R is
x(u, v) = c(u) + ve(u), (1)
where c(u) is called directrix curve and e(u) is a vector field along c(u). For fixed values u, this
parametrization represents the straight lines L(u) on R.
The normal vector n(u, v) of the ruled surface x(u, v) is computed as cross product of the partial
derivative vectors x and x , and we obtain
v
u
n(u, v) = c(u) × e(u) + ve(u) × e(u). (2)


For fixed u = u , the normal vectors n(u , v) along L(u ) are linear combinations of the vectors
0
0
0
c(u ) × e(u ) and e(u ) × e(u ). The parametrization x(u, v) represents a developable surface D if


0
0
0
0
for each generator L all points x  L have the same tangent plane (with exception of the singular
point on L). This implies that the vectors c× e and e × e are linearly dependent which is expressed


equivalently by the following condition
det(c, e, e)  = 0. (3)

Any regular generator L(u) of a developable surface D carries a unique singular point s(u) which
does not possess a tangent plane in the above defined sense, and s(u) = x(u, v ) is determined by
s
the parameter value
(c e) (e e)


 

v =  (e e) 2 . (4)
s

If e and e  are linearly dependent, the singular point s is at infinity, otherwise it is a proper point.
In Euclidean space  , there exist three different basic classes of developable surfaces:
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(1) Cylinder: the singular curve degenerates to a single point at infinity.
(2) Cone: the singular curve degenerates to a single proper point, which is called vertex.
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