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Unit 23: Developable Surface Fitting to Point Clouds
where T(u) denotes the derivative with respect to u. The generating lines themselves envelope Notes
the singular curve s(u) which is the intersection
s(u) = T(u) T(u) T(u).
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2
1
2
2
Taking the normalization n n n n(u) into account, the Blaschke image b(T(u)) =
3
2
1
b(D) of the developable surface D is a curve on the Blaschke cylinder B. This property will be
applied later to fitting developable surfaces to point clouds.
23.3 The Classification of Developable Surfaces according to their
Image on B
This section will characterize cylinders, cones and other special developable surfaces D by
studying their Blaschke images b(D).
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. (1)
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3
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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. (2)
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1
There exist other special types of developable surfaces. Two of them will be mentioned here.
The surface D is a developable of constant slope, if its normal vectors n(u) form a constant angle
with a fixed direction vector a. Assuming a = 1, we get cos() = a . n(u) = = const. This implies
that the Blaschke images of the tangent planes of D are contained in the three-space
H : + a u + a u + a u = 0. (3)
3
2
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1
The developable surface D is tangent to a sphere with center m and radius r, if the tangent planes
T(u) of D satisfy n + n m + n m + n m r = 0, according to (11). Thus, the image curve b(D) is
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contained in the three-space
H : r + u m + u m + u m + u = 0. (4)
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23.3.1 Cones and Cylinders of Revolution
For applications, it is of particular interest if a developable surface D is a cone or cylinder of
revolution.
Let D be a cylinder of revolution with axis A and radius r. The tangent planes T of D are tangent to
all spheres of radius r, whose centers vary on A. Let
S : (x p) r = 0, S : (x q) r = 0
2
2
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1
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, (5)
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2
3
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3
H : r + u q + u q + u q + u = 0.
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1 1
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3 3
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