Hellmuth Stachel

Zur Einzigkeit der Bricardschen Oktaeder

In 1896 R. BRICARD gave a complete description of all flexible octahedra; a different approach is due to R. CONNELLY (1978). In this paper a new proof is given using methods of classical projective geometry. The proof is based on the observation that by a converse of the theorem of IVORY each pair of incongruent octahedra with the same edge lengths is connected with a certain pair of confocal guadrics.

Flexible Octahedra in the Hyperbolic Space

This paper treats flexible polyhedra in the hyperbolic 3-space ℍ3. It is proved that the geometric characterization of octahedra being infinitesimally flexible of orders 1 or 2 is quite the same as in the Euclidean case. Also Euclidean results concerning continuously flexible octahedra remain valid in hyperbolic geometry: There are at least three types of continuously flexible octahedra in #x210D;3; the line-symmetric Type 1, Type 2 with planar symmetry, and the non-symmetric Type 3 with two flat positions. However, Type 3 can be subdivided into three subclasses according to the type of circles in hyperbolic geometry. The flexibility of Type 3 octahedra can again be argued with the aid of Ivory’s Theorem.

Circular pipe-connections

Abstract Technical designers sometimes need specially formed pipe bends to connect given openings. Choosing a curve of constant curvature for the central curve of such a pipe-connection improves its aerodynamical properties. Further pipe bends with circular central curve are much easier to produce than that based on arbitrary spatial curves. Hence the following geometric problem arises: Let (A, a), (B, b) be given line elements with skew oriented tangent lines a, b. How to connect these two elements by a smooth curve? How to fulfill additional requirements? Within the continuum of possible connections we pay attention only to the following four cases: The connecting curve is a helix or it consists of two circles, of n ≧ 2 congruent circular arcs with congruent twist angles or of n ≧ 3 congruent circular arcs with almost congruent twist angles. Solutions of the first and third kind turn out to exist only if the given line elements are symmetric in a certain sense. We present an algorithm for generating curves of the fourth type with the aid of computers.

Ivory's Theorem in Hyperbolic Spaces

According to the planar version of Ivorys theorem, the family of confocal conics has the property that in each curvilinear quadrangle formed by two pairs of conics the diagonals are of equal length. It turned out that this theorem is closely related to selfadjoint affine transformations. This point of view opens up a possibility of generalizing the Ivory theorem to the hyperbolic and other spaces.

Zwei bemerkenswerte bewegliche Strukturen

Two overconstrained mechanisms are presented that both are related to regular polyhedra in the Euclidean 3-space. The first example, the HEUREKA-polyhedron, is a modification of BUCKMINSTER-FULLERS Jitterbug [1]. The spherical joints at the vertices of 8 regular triangles are replaced by particular cardan joints. A 15m high model of this polyhedron was exhibited at the national research exposition of Switzerland 1991 in Zürich. Further the GRÜNBAUM-framework is discussed. Here the 10 regular tetrahedra inscribed to a regular pentagon-dodecahedron are linked together at the common vertices. This framework allows at least two types of constrained motions. The first was found by R. Connelly [2]. These motions preserve the fivefold symmetry with respect to any face axis. Motions of the two second type preserve the symmetry with respect to any vertex axis.

HIGHER ORDER FLEXIBILITY OF OCTAHEDRA

More than hundred years ago R. Bricard determined all continuously flexible octahedra. On the other hand, also the geometric characterization of first-order flexible octahedra has been well known for a long time. The objective of this paper is to analyze the cases between, i.e., octahedra which are infinitesimally flexible of order n > 1 but not continuously flexible. We prove explicit necessary and sufficient conditions for the orders two, three and even for all n < 8, provided the octahedron under consideration is not totally flat. Any order ≥ 8 implies already continuous flexibility, as the configuration problem for octahedra is of degree eight.

The Computational Fundamentals of Spatial Cycloidal Gearing

The tooth flanks of bevel gears with involute teeth are still cut using approximations such as Tredgold’s and octoid curves, while the geometry of the exact spherical involute is well known. The modeling of the tooth flanks of gears with skew axes, however, still represents a challenge to geometers. Hence, there is a need to develop algorithms for the geometric modeling of these gears. As a matter of fact, the modeling of gears with skew axes and involute teeth is still an open question, as it is not even known whether it makes sense to speak of such tooth geometries. This paper is a contribution along these lines, as pertaining to gears with skew axes and cycloid teeth. To this end, the authors follow and extend results reported by Martin Disteli at the turn of the 20th century concerning the general synthesis of gears with skew axes. The main goal is to shed light on the geometry of the tooth flanks of gears with skew axes. The dualization of the tooth profiles of spherical cycloidal gears leads to ruled surfaces as conjugate tooth flanks such that at any instant the contact points are located on a straight line. A main result reported herein is Theorem 5, which is original. All results are proven by means of a consistent use of dual vectors representing directed lines and rigid-body twists.

On the flexibility and symmetry of overconstrained mechanisms

In kinematics, a framework is called overconstrained if its continuous flexibility is caused by particular dimensions; in the generic case, a framework of this type is rigid. Famous examples of overconstrained structures are the Bricard octahedra, the Bennett isogram, the Grünbaum framework, Bottemas 16-bar mechanism, Chasles’ body–bar framework, Burmesters focal mechanism or flexible quad meshes. The aim of this paper is to present some examples in detail and to focus on their symmetry properties. It turns out that only for a few is a global symmetry a necessary condition for flexibility. Sometimes, there is a hidden symmetry, and in some cases, for example, at the flexible type-3 octahedra or at discrete Voss surfaces, there is only a local symmetry. However, there remain overconstrained frameworks where the underlying algebraic conditions for flexibility have no relation to symmetry at all.

Composition of Spherical Four-Bar-Mechanisms

We study the transmission by two consecutive four-bar linkages with aligned frame links. The paper focusses on so-called “reducible” examples on the sphere where the 4-4-correspondance between the input angle of the first four-bar and the output-angle of the second one splits. Also the question is discussed whether the components can equal the transmission of a single four-bar. A new family of reducible compositions is the spherical analogue of compositions involved at Burmester’s focal mechanism.

Coordinates—a survey on higher geometry

Abstract Under the name “Higher Geometry” usually those different geometries are summarized which in the sense of F. Kleins Erlangen program (1872) are isomorphic to subgeometries of projective geometry. In the following we give a brief survey on such geometries like multi-dimensional projective, affine and Euclidean geometry, the geometry of lines and the geometry of oriented spheres in the three-dimensional case. It is a goal of this paper to demonstrate both the elegance of the classical analytical treatment and its applicability for various tasks, e.g. in the field of CAGD. The latter however has been reduced to the presentation of references only.