Page 290 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 23: Developable Surface Fitting to Point Clouds

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According to the normalization e  e  e  1, the distance of the Blaschke image b(E) to the Notes
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origin in  is bounded by 1. This is also important for a discretization of the Blaschke cylinder
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which we discuss in the following.
23.4.3 A Cell Decomposition of the Blaschke Cylinder
For practical computations on B, we use a cell decomposition of B to define neighborhoods of
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image points b(T) of (estimated) tangent planes T. We recall that B’s equation is u  u  u  1.
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Any cross section with a plane u = const. is a copy of the unit sphere S in  . In order to obtain
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a cell decomposition of B, we start with a triangular decomposition of S and lift it to B.
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A tessellation of S can be based on the net of a regular icosahedron. The vertices v , i = 1, . . . , 12,
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with v = 1 of a regular icosahedron form twenty triangles t and thirty edges. All edges have
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same arc length. This icosahedral net is subdivided by computing the midpoints of all edges
(geodesic circles). Any triangle t is subdivided into four new triangles. The inner triangle has
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equal edge lengths, the outer three have not, but the lengths of the edges to not vary too much.
By repeated subdivision, one obtains a finer tessellation of the unit sphere.
The cell decomposition of the Blaschke cylinder consists of triangular prismatic cells which are
lifted from the triangular tessellation of S in u -direction. Since we measure distances according
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to (1), the height of a prismatic cell has to be approximately equal to the edge length of a
triangle. When each triangle of the tessellation is subdivided into four children, each interval in
u -direction is split into two subintervals.
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According to the scaling of the data points p , the coordinates of the image points b(E) on B are
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bounded by ±1. We start with 20 triangles, 12 vertices and 2 intervals in u -direction. The test-
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implementation uses the resolution after three subdivision steps with 1280 triangles, 642 vertices
and 16 intervals in u -direction. In addition to the cell structure on B, we store adjacency
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information of these cells.
Remark concerning the visualization: It is easy to visualize the spherical image (first three
coordinates) on S , but it is hard to visualize the Blaschke image on B. We confine ourself to plot
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the spherical image on S , and if necessary, we add the fourth coordinate (support function) in a
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separate figure. This seems to be an appropriate visualization of the geometry on the Blaschke
cylinder, see Figures 23.3 to Figure 23.7.
23.4.4 Analysis and Classification of the Blaschke Image
Having computed estimates T of the tangent planes of the data points and their images b(T ), we
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check whether the Blaschke image of the considered surface is curve-like. According to
Section 23.4.3, the interesting part of the Blaschke cylinder B is covered by 1280 × 16 cells C . We
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compute the memberships of image points b(T ) and cells C and obtain a binary image on the
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cell structure C of B. Let us recall some basic properties of the Blaschke image of a surface.
(1) If the data points p are contained in a single plane P, the image points b(T ) of estimated
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tangent planes T form a point-like cluster around b(P) on B.
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(2) If the data points p are contained in a developable surface, the image points b(T ) form a
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curve-like region in B.
(3) If the data points p are contained in a doubly curved surface S, the image points b(T ) cover
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a two-dimensional region on B.
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