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ing on whether their spines are central conics, parabolas, straight lines and circles,
or degenerate conics. Each of these types of cyclides can be further subdivided into
three subtypes. We describe that subdivision in the case of central cyclides. Assume
that the coordinate system is again chosen so that the cyclide is in standard position
as shown in Figure 12.22.
Definition. The central cyclide is called a ring cyclide if 0 £ c < k £ a. It is called a
horned cyclide if 0 < k £ c < a. It is called a spindle cyclide if 0 £ c £ a < k.
Visually, a horned cyclide looks like two crescent-shaped solids that meet at their
pinched points. A spindle cyclide has two pinch points at which the two parts of the
entire surface meet with one part looking like a spindle inside the other. Ring cyclides,
such as the one shown in Figure 12.21, are the easiest to draw because they do not
have any of these degeneracies. Another way to describe these subtypes is in terms of
two important values
ka
c
kc
a
s
=
and
t
=
.
Let L s be the line in the x-y plane that is parallel to the y-axis and passes through the
point (s,0,0) on the x-axis. Let L t be the line in the x-z plane that is parallel to the z-
axis and passes through the point (t,0,0) on the x-axis. We will have a ring cyclide if
L s does not intersect the focal ellipse E and L t does not intersect the focal hyperbola
H . We will have a horned cyclide if L s intersects the ellipse E and a spindle cyclide if
L t intersects the hyperbola H . The lines L s and L t are called the characteristic lines of
the cyclide ([ChDH89]).
Some properties of cyclides that make them attractive to CAGD are (see [Prat90]):
(1) Each curvature line is a circle and cyclides are the only fourth-degree surfaces
whose curvature lines are circles. The curvature lines split into two families similar
to what happens in the case of a torus.
(2) The planes of each family of curvature lines meet in a line.
(3) One gets offset surfaces for a cyclide by changing the parameter k. Furthermore,
the offset of a cyclide is a cyclide.
(4) For each curvature line, the angle between the principal normal of that curve and
the surface normal is constant. Hence there is a right circular cone tangent to any
circular curvature line of the cyclide.
(5) The inversion in a sphere of a cyclide with respect to a point not on it is again a
cyclide.
One can show that the change of parameters
q
y
u
=
tan
,
v
=
tan
2
2
produces a rational biquadratic parameterization of the cyclide. The lines of constant
u and v correspond to lines of curvature, namely circles. If we call a region in a cyclide
bounded by lines of curvature a principal cyclide patch , then we can use the rational
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