Canal Surfaces and Inversive Geometry
Univ.Ass. Dipl.-Math. Katja Bühler
in cooperation with Prof. Marco Paluszny,
CGGA, Universidad Central de Venezuela, Caracas.
Introduction
Canal surfaces are an important tool in geometric
modeling, specially in blending and joining
surface primitives along circular profiles. These include planes, cones,
Dupin cyclides and arbitrary surfaces of revolution.
Applying the bijective generalized
stereographic projection onto its generating family of spheres, canal
surfaces can be represented as curves in (projective) 4-space and vice
versa. This methode reduces the problem of connecting two circular
contacts or segments of canal surfaces in euclidean 3-space to connect
two curve segments in (projective) 4-space. Studying a curve in 4-space
gives in an easy way information on basic properties of the represented
canal surface like algebraic degree, algebraic genus, (optical) connectivity,
singularities and shape.
Some basic agreements and definitions
In the following considerations 3-space denotes the euklidean 3-space
projectively closed by the plane at infinity and 4-space the projective
4-space.
Planes are spheres with infinit radius and lines are circles
with infinite radius. Thus, talking about spheres and circles means talking
about spheres, planes, circles and lines.
A canal surface is the envelope
of a 1-parametric family of spheres. The envelope is defined as the union
of all circles of intersection of infinitessimally neighbouring pairs of
spheres. These circles are called composing circles/lines.
Well known canal surfaces:
cone, cylinder, torus, dupin cyclide, surfaces of
revolution,.....
The
classic stereographic projection
introduced by Moebius maps the Moebius plane (the x-y-plane projectively
close by one point at infinity) onto the unit sphere of 3-space or -one
dimension higher- the Moebius space (3-space projectively closed by one
point at infinity) onto the unit hypersphere of 4-space. The unit (hyper)sphere
is called Moebius (hyper)sphere.
The generalized stereographic projection is is introduced to
give a relation between the spheres and planes of 3-space and the points
of 4-space as follows:
An arbitrary sphere or a plane of 3-space denote by S is mapped by
stereographic projection onto the intersection of the moebius hypersphere
with a hyperplane H. The generalized sterographic projection maps S onto
the pole of the hyperplane H with respect to the Moebius Hypersphere.
Generating canal surfaces in 3-space
using curves in 4-space
The simplest curves in 4-space are lines. These are mapped onto
to linear families of spheres or planes which are classically known as
coaxial pencils. A pencil is called elliptic (parabolic, hyperbolic) if
all spheres have one circle (one point, one complex circle) in common.
This circle is called carrying circle. It is real iff the line does
not intersect the Moebius hypersphere.
An arbitrary (C1-)curve in 4-space corresponds to a one parametric
family of spheres. Its envelope is a canal surface by definition.
The tangents of the curve correspond to coaxial pencils of spheres
whose carrying circles are equivalent to the composing circles of
the envelope.
Studying the algebraic degree and genus, the position and the shape
of the curve in 4-space gives information about an upper bound for the
algebraic degree of the canal surface generated by this curve, its genus
and some informations about its shape. Studying the position of the tangents
of the curve gives information about connectivity, singularities and self
intersections of the generated surface.
Constructing tubes
A circular contact in 3-space is a pair of a sphere and a circle lying
on it. Its corresponding figure in 4-space is a point and a line through
it. The problem of constructing piecewise canal surfaces consists of interpolating
circular contacts given in pairs, i.e. constructing a surface that is tangent
to pairs of lines at given points. Higher order continuity between the
curves will be translated into higher order smoothness of the tube at the
joins.
