Math illustrations planes
So each family contains exactly one of it and any circular cone of the other family contains that line as a ruling. This follows from Theorem 5 since the axes a 1and a 2of a parabolic cyclide (lie on it and) are degenerated curvature circles. Only a few remarks should be added in this context: Remark 1Ī parabolic Dupin cyclide occurs if and only if the two cones or cylinders C * 1and C * 2have a common ruling. The proof of this last assertion as well as the many details and special cases must be omitted for brevity. Now going back to the outer offsets of C ⋄ 1, C ⋄ 2and Ψ ⋄ yields the desired cyclide Ψ. An other way to recognize its validity is to use Theorem 3 on offsets of cyclides: Since the common inscribed sphere Shas its midpoint at the intersection of the two axes, one can replace C * 1and C * 2by their inner offsets C ⋄ 1 C ⋄ 2at a distance d equal to the radius r of S then C ⋄ 1 C ⋄ 2have their apexes in common and so they can always be blended by a Dupin cyclide Ψ ⋄. This theorem could be easily proved by methods of Lie geometry (see section 23.5) however these are beyond the scope of this volume. Then, in general, it does exist and is unique. This is a third condition to be imposed on the “geometric data” C 1, C * 1and C 2, C * 2for the existence of a blending cyclide. So one of them, say C 1can be prescribed (for instance by choosing its mid point on the axis b 1of C * 1) then the position of C 2is determined.
#Math illustrations planes free#
On the other hand it is to be seen that one parameter remains free when the previous two steps are done. The theorem does not say anything about the position of the circles C 1and C 2.
Two right circular cones or cylinders with different axes in the same plane can be blended by a Dupin cyclide if and only if they have a common inscribed sphere. This problem is solved by a theorem of Sabin 12 Theorem 10 In the second step, one looks for the possibilities to blend a pair of different cones C * 1, C * 2with coplanar axes b 1, b 2in E, say, by a Dupin cyclide (neglecting the positions of the circles C 1, C 2at the moment). However the theoretical results dealt with in section 23.2 are able to solve it also in this case.įirst, one concludes from the one-sided conditions that the two axes b 1and b 2of C * 1and C * 2must be coplanar and not identical, hence E, respectively F, is uniquely determined. The problem of existence becomes less trivial. Usually one wants to span a part of a cyclide between these two geometric objects so that the circles belong to the same family 11.
Next we consider two-sided blendings: Then there are given a pair of objects C 1, C * 1and C 2, C * 2of the same kind as before. In the case of a cylinder the radii of the other circles of the same family as C must either be constant (Ψ then being a torus) or extremal at C.
The plane A contains the other axis ( a 2for E and a 1for F). The apex of C * lies on the axis of that symmetry plane ( a 1for E and a 2for F). The axis b of C * is contained in one of the symmetry planes E or F Obviously, the cyclide Ψ always exists and is by far not unique due to Theorem 4, the only conditions are as follows that there is no other collision of those parts of the surfaces, which are of further interest.) (We assume without further mentioning that both surfaces lie on different sides of the plane A of C and omit additional technical “side conditions”, e.g. The “one-sided” blending consists of a G 1-continuous transition from a surface Φ to a Dupin cyclide Ψ along one of its curvature circles C. In this subsection, we deal with Dupin cyclides only, referring to the next subsection for the case of supercyclides. Usually, blending is done along prescribed curves on the surfaces to be blended however, also “free blendings” where only the surfaces and some regions on it (where the transition curve is desired to lie in) are given, were discussed.īecause of the very extensive work on blending, only the principal methods can be quoted here, leaving the details for further reading in the literature. Wendelin Degen, in Handbook of Computer Aided Geometric Design, 2002 23.4.2 Using cyclides as blendingsīlending is the most useful application of cyclides to CAGD and geometric modeling many papers are devoted to that topic (,, ,, ,, ,, ) and various generalizations have been found (,, ,, ) in the past.