Showing papers on "Regular polygon published in 1974"

Reentrant polygon clipping

Abstract: A new family of clipping algorithms is described. These algorithms are able to clip polygons against irregular convex plane-faced volumes in three dimensions, removing the parts of the polygon which lie outside the volume. In two dimensions the algorithms permit clipping against irregular convex windows.Polygons to be clipped are represented as an ordered sequence of vertices without repetition of first and last, in marked contrast to representation as a collection of edges as was heretofore the common procedure. Output polygons have an identical format, with new vertices introduced in sequence to describe any newly-cut edge or edges. The algorithms easily handle the particularly difficult problem of detecting that a new vertex may be required at a corner of the clipping window.The algorithms described achieve considerable simplicity by clipping separately against each clipping plane or window boundary. Code capable of clipping the polygon against a single boundary is reentered to clip against subsequent boundaries. Each such reentrant stage of clipping need store only two vertex values and may begin its processing as soon as the first output vertex from the preceeding stage is ready. Because the same code is reentered for clipping against subsequent boundaries, clipping against very complex window shapes is practical.For perspective applications in three dimensions, a six-plane truncated pyramid is chosen as the clipping volume. The two additional planes parallel to the projection screen serve to limit the range of depth preserved through the projection. A perspective projection method which provides for arbitrary view angles and depth of field in spite of simple fixed clipping planes is described. This method is ideal for subsequent hidden-surface computations.

Regular Complex Polytopes

Abstract: Frontispiece Preface to the second edition Preface to the first edition 1. Regular polygons 2. Regular polyhedra 3. Polyhedral kaleidoscopes 4. Real four-space and the unitary plane 5. Frieze patterns 6. The geometry of quaternions 7. The binary polyhedral groups 8. Unitary space 9. The unitary plane, using quaternions 10. The complete enumeration of finite reflection groups in the unitary plane 11. Regular complex polygons and Cayley diagrams 12. Regular complex polytopes defined and described 13. The regular complex polytopes and their symmetry groups Tables Reference Index Answers to exercises.

Polyhedral convexity cuts and negative edge extensions

Abstract: This note shows that convexity cuts defined relative to polyhedral convex sets can utilize negative as well as positive edge extensions under appropriate circumstances, yielding stronger cuts than customarily available. We also show how to partially collapse the polyhedron to further improve these cuts.

On Dvoretzky’s theorem on almost spherical sections of convex bodies

Abstract: A complete and rather simple proof of the famous Dvoretzky’s theorem is presented.

Some Remarks on Random Sets Mosaics

Abstract: Given a set of points S having no accumulation points contained in a subset T of the plane, associate with each element a of S the subset Sa of points in T closer to a than the other points of S. The subsets Sa are convex polygons and partition T. T is not necessarily closed or bounded, but of course, if T is compact then S is finite. These polygons are variously referred to as Dirichlet cells, Meijering cells (1953), Voronoi regions (1908), Gilbert cells (1961), cells or tiles, and the partition is called an S-mosaic or mosaic. If the elements of S are sprinkled randomly in T call the partition a random S-mosaic. The point a is called an S-point for the tile Sa. Generalizations of this idea to higher dimensional spaces (even to metric spaces) are apparent. We have not looked into any of these in detail, although metric space (using just metric convexity?) or normed linear space generalizations of known En results might be interesting.

On ordinary points in arrangements

Abstract: We show that in an arrangement ofn curves in the plane (or on the sphere) there are at leastn/2 points where precisely 2 curves cross (ordinary points). Furthermore there are at least (4/3)n triangular regions in the complex determined by the arrangement. Triangular regions and ordinary vertices are both connected with boundary vertices of certain distinguished subcomplexes. By analogy with rectilinear planar polygons we distinguish concave and convex vertices of these subcomplexes. Our lower bounds arise from lower bounds for convex vertices in the distinguished subcomplexes.

On Polars of Convex Polygons

Abstract: (1974). On Polars of Convex Polygons. The American Mathematical Monthly: Vol. 81, No. 9, pp. 1016-1018.

On the construction of split-face topologies

Abstract: We give a general theorem to facilitate the construction of interesting examples of split-face topologies of compact, convex sets. Introduction. Let K be a compact convex set in a locally convex topological vector space. Let F be a closed face of K and F be the union of all faces of K disjoint from F. It is always true that K = co(F U F') [1, Proposition 11.6.5]. The face F is said to be split if F' is a face and K is the direct convex sum of F and F' [1, p. 133], i.e. if each x E K can be expressed by a unique convex combination