Computational Geometry for Design and Manufacture by I.D. Faux, M.J. Pratt

By I.D. Faux, M.J. Pratt

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A film spanning a separating solution of a Pfaff equation is also a film spanning a solution for any equivalent Pfaff equation. , the points in the tangent space in which a is identically zero) form a closed set. In the complementary open set, the Pfaff equation a = 0 determines a distribution of cooriented hyperplanes. For equivalent Pfaff equations, the sets of singular points, and the distribution of the cooriented hyperplanes in the complementary open sets, coincide. x. , if in the neighbourhood of each point there is a 1-form a whose set of singular points coincides with the singularity set in that neighbourhood, and the distribution of the cooriented hyperplanes in the complement is determined 111.

But a sufficiently small neighbourhood in M of a E U n H does not contain points of W. and contains only points of U at which f has large values (as f (x) +oc when x tends in U to a). Therefore the point (a, y), where y is an arbitrary number, is an interior point of M_ . (2) We finish the proof that M_ is a spanning film. The only boundary points of M_ are the points of the graph of the function. Around each boundary point, M_ is determined by the inequality y < f(x) . Therefore M_ is a manifold with the graph of the function as boundary.

The intersection of the curve with the separating solution is a closed set. The complement of that set decomposes into (finite or infinite) intervals. 2. ALGEBRAIC PROPERTIES OF P-CURVES 23 finite intervals contains a contact point. Let t, and t2 be the parameters of the ends of such an interval and let t, precede t2 in the sense of the orientation of the curve. Assume without loss of generality that the curve lies in UI for all intermediate values of the parameter. We shall show that there is an instant in time close to t, when the ordered pair of vectors, consisting of the tangent to the curve and the vector of the dynamical system, gives the same orientation of the plane as when the curve enters U, .

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