Page 299 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes be two such spheres with centers p, q. According to (11), the images b(T) of the tangent
planes T satisfy the relations
H : –r + u p + u p + u p + u = 0,
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1
1
2
1
4
H : –r + u q + u q + u q + u = 0.
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3 3
1 1
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Since p  q, the image curve b(D) lies in the plane P = H  H and b(D) is a conic.
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1
Given a cloud of data points p , this section discusses the recognition and classification of
 i
developable surfaces according to their Blaschke images. The algorithm contains the
following steps:
Estimation of tangent planes T at data points p and computation of the image points
 i i
b(T ).
i
Analysis of the structure of the set of image points b(T ).
 i
If the set b(T ) is curve-like, classification of the developable surface which is close to
 i
p. i
D is a smooth surface not carrying singular points. D is not necessarily exactly developable,

but one can run the algorithm also for nearly developable surfaces (one small principal
curvature).
The density of data points pi has to be approximately the same everywhere.

The image b(T ) of the set of (estimated) tangent planes T has to be a simple, curve-like
 i i
region on the Blaschke cylinder which can be injectively parameterized over an interval.
According to the made assumptions, the reconstruction of a set of measurement point p of
 i
a developable surface D can be divided into the following tasks:
Fitting a curve c(t)  B to the curve-like region formed by the data points b(T ).
 i
Computation of the one-parameter family of planes E(t) in  and of the generating
3

lines L(t) of the developable D* which approximates measurements p . i
Computation of the boundary curves of D* with respect to the domain of interest

in  .
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23.7 Keywords
Cylinder: D is a general cylinder if all its tangent planes T(u) are parallel to a vector a and thus
its normal vectors n(u) satisfy n . a = 0. This implies that the image curve b(T(u)) = b(D) is
contained in the three-space H : a u + a u + a u = 0.
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Cone: D is a general cone if all its tangent planes T(u) pass through a fixed point p = (p , p , p ).
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This incidence is expressed by p n + p n + p n + n = 0. Thus, the Blaschke image curve
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b(T(u)) = b(D) is contained in the three space H : p u + p u + p u + u = 0.
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23.8 Self Assessment
1. Let S be the oriented sphere with center m and signed radius r, ................ = 0. The tangent
planes T of S are exactly those planes, whose signed distance from m equals r.
S
2. The surface D is a ................ of constant slope, if its normal vectors n(u) form a constant
angle  with a fixed direction vector a.
3. Let D be a cylinder of revolution with axis A and radius r. The ................ T of D are tangent
to all spheres of radius r, whose centers vary on A.
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