Although it seems not to be possible to deduce the fact from Vogt's theorem (at least in its present formulation), it turns out that Ho( Top B) is also equivalent to a category H B whose objects are spaces over B and whose morphisms are coherent homotopy classes of homotopy equivalent triangles with sink vertex B. Some interest is attached to the associated homotopy category Ho( Top B), which is obtained by inverting the maps over B that are homotopy equivalences in their own right. The category Top B of spaces over a fixed space B can be regarded as a (nonfull) subcategory of the category of pairs Top 1. It seemed that it would be advantageous to embark on a systematic study of (homotopy theoretic aspects of) the simplest diagram categories, partly with the view to the light their properties might throw on more complex categories, but also with a view to studying their interrelations in the hope that feedback of results would eventually yield new techniques for study of Ho( Top) itself. It is obvious that, yielded a simple and natural alternative to the pair homotopy theory of Eckmann and Hilton. Robert Gardner, in Real Analysis with an Introduction to Wavelets and Applications, 2005 Proof Hoschek, to introduce duality and dual BГ©zier curves and surfaces to CAGD.Дon Hong. Cusps and inflections are dual to each other, but cusps are sometimes easier to detect than inflections. To achieve inflections, it is best to locate them at end points of the BГ©zier curve segments (see ).

For many applications, cusps are not desirable and therefore the convexity condition plays an important role. In the dual representation, very special conditions on the control structure must be met to design an inflection, but it is easy to get cusps. By using the standard representation it is difficult to design cusps but quite easy to achieve inflections of the curve segment. However, their behavior when used for design purposes is different. There are simple formulae for conversion between the two forms (see, e.g., ,). Now the convexity condition reads: If the BГ©zier lines U i and the frame lines F i of a dual BГ©zier curve c are among the edges and support lines of a convex domain D, and the points U i в€© U i +1 are among D's vertices, then c is convex and lies completely outside D.ДЄ planar rational curve segment, or more precisely, a rational parameterization of it, possesses two BГ©zier representations: the usual, point-based form, and the dual line-based representation. A support line L of a convex domain D is a line through a point of the boundary of D such that D lies entirely on one side of L. By a convex curve we understand part of the boundary of a convex domain. This result easily implies a sufficient condition for convexity of a dual BГ©zier curve. The one which contains the frame line is part of the complete dual control structure (see Figure 3.3).Дual to the variation diminishing property of a rational BГ©zier curve with respect to its projective control polygon we can state the following result: If c is a planar rational BГ©zier curve, the number of c's tangents incident with a given point P does not exceed the number of lines of the complete dual control structure which are incident with P (if no control line has zero weight).

We dualize the geometric control polygon: The line pencil spanned by lines U i and U i +1 is divided into two subsets, bounded by U i and U i +1. Dual to this, the end tangents of a dual BГ©zier curve are U 0 and U m, and their points of contact are given by U 0 в€© U 1 and U m в€’1 в€© U m, respectively. An invariant statement of theorems is also important for their dualization.ДЇor a BГ©zier curve, the control points B 0 and B m are the end points of the curve segment, and the lines B 0V B 1 and B m в€’1V B m are the tangents there. Because weights are no projective invariants, it is preferrable to use frame lines and frame points. We could also use weights instead of frame lines, just as we could have used weights instead of frame points. This is dual to the collinearity of a frame point with its two adjacent BГ©zier points. (3.5) f i = u i + u i + 1, i = 0, вЂ¦, m в€’ 1.ДЇrame line F i is concurrent with the BГ©zier lines U i, and U i +1.

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