Page 287 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Since p  q, the image curve b(D) lies in the plane P = H  H and b(D) is a conic.
1 2
Cones of revolution D can be obtained as envelopes of the common tangent planes of two oriented
spheres S , S with different radii r  s. Thus, b(D) is a conic contained in the plane
1
2
P = H  H which is defined by
1
2
H : –r + u p + u p + u p + u = 0, (6)
1
2
2
3
3
4
1
1
H : –s + u q + u q + u q + u = 0.
1 1
2
4
3 3
2 2
Conversely, if the Blaschke image b(D) of a developable surface is a planar curve  P, how can
we decide whether D is a cone or cylinder of revolution?
Let b(D) = b(T(u)) be a planar curve  P and let P be given as intersection of two independent
three-spaces H , H , with
1
2
H : h + h u + h u + h u + h u = 0. (7)
i
i1
i0
2
i3
3
4
1
i2
i4
Using the results of Section 23.2.2, the incidence relation b(T(u))  H implies that T(u) is tangent
1
to a sphere, or is passing through a point (h = 0), or encloses a fixed angle with a fixed direction
10
(h = 0). The same argumentation holds for H .
14
2
Thus, by excluding the degenerate case h = h = 0, we can assume that P = H  H is the
1
2
14
24
intersection by two three-spaces H , H of the form (5) or (6).
2
1
(1) Let the plane P = H  H be given by equations (5). Then, the developable surface D is a
1
2
cylinder of revolution. By subtracting the equations (5) it follows that the normal vector
n(u) of T(u) satisfies
n . (p – q) = 0.
Thus, the axis A of D is given by a = p – q and D’s radius equals r.
(2) Let the plane P = H  H be given by equations (6). The pencil of three spaces H + H 2
2
1
1
contains a unique three-space H, passing through the origin in  , whose equation is
4
3
H :  u (sp – rq ) + u (s – r) = 0.
i
i
4
i
i=1
Thus, the tangent planes of the developable surface D are passing through a fixed point
corresponding to H, and D is a cone of revolution. Its vertex v and the inclination angle 
between the axis A : a = p – q and the tangent planes T(u) are
1 s  r
V  (sp  rq ), and sin  
s  r q  p
23.4 Recognition of Developable Surfaces from Point Clouds
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:
(1) Estimation of tangent planes T at data points p and computation of the image points b(T ).
i
i
i
(2) Analysis of the structure of the set of image points b(T ).
i
(3) If the set b(T ) is curve-like, classification of the developable surface which is close to p . i
i
280 LOVELY PROFESSIONAL UNIVERSITY