Page 284 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 23: Developable Surface Fitting to Point Clouds

The basic properties concerning the Blaschke image (Blaschke model) of the set of planes in R3 Notes
which is relevant for the implementation of the intended reconstruction. Section 23.3 tells about
a classification, and Section 23.4 discusses the recognition of developable surfaces in point
clouds using the Blaschke image of the set of estimated tangent planes of the point set.
Section 23.5 describes the concept of reconstruction of these surfaces from measurements. Finally,
we present some examples and discuss problems of this approach and possible solutions.

23.2 The Blaschke Model of Oriented Planes in R 3

Describing points x by their Cartesian coordinate vectors x = (x, y, z), an oriented plane E in
Euclidean space  can be written in the Hesse normal form,
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E : n x + n y + n z + d = 0, n  n  n  1. (5)
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We note that n x + n y + n z + d = dist(x, E) is the signed distance between the point x and the
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plane E. In particular, d is the origin’s distance to E. The vector n = (n , n , n ) is the unit normal
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vector of E. The vector n and the distance d uniquely define the oriented plane E and we also use
the notation E : n . x + d = 0.
The interpretation of the vector (n , n , n , d) as point coordinates in  , defines the Blaschke
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mapping
b : E  b(E) = (n , n , n , d) = (n, d). (6)
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In order to carefully distinguish between the original space  and the image space  , we
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denote Cartesian coordinates in the image space  by (u , u , u , u ). According to the
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normalization n = 1 and (6), the set of all oriented planes of  is mapped to the entire point set
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of the so-called Blaschke cylinder,
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B : u + u + u = 1. (7)
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Thus, the set of planes in  has the structure of a three-dimensional cylinder, whose cross
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sections with planes u = const. are copies of the unit sphere S (Gaussian sphere). Any point
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U  B is image point of an oriented plane in  . Obviously, the Blaschke image b(E) = (n, d) is
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nothing else than the graph of the support function d (distance to the origin) over the Gaussian
image point n.
Let us consider a pencil (one-parameter family) of parallel oriented planes E(t) : n . x + t = 0. The
Blaschke mapping (6) implies that the image points b(E(t)) = (n, t) lie on a generating line of B
which is parallel to the u -axis.
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23.2.1 Incidence of Point and Plane
We consider a fixed point p = (p , p , p ) and all planes E : n . x + d = 0 passing through this point.
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The incidence between p and E is expressed by
p n + p n + p n + d = p . n + d = 0, (8)
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and therefore, the image points b(E) = (n , n , n , d) in  of all planes passing through p lie in the
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three-space
H : p u + p u + p u + u = 0, (9)
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passing through the origin of  . The intersection H  B with the cylinder B is an ellipsoid and
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any point of H  B is image of a plane passing through p. Fig. 23.1 shows a 2D illustration of this
property.
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