Geometry Topology

Download A Catalog of Special Plane Curves by J. Dennis Lawrence PDF

By J. Dennis Lawrence

Forty years after its preliminary ebook, this quantity maintains to rank one of the field's most-cited references. one of many greatest and most interesting on hand collections, the catalog covers common homes of curves and kinds of derived curves. The curves and the values in their parameters are illustrated by means of approximately ninety pictures from a CalComp electronic incremental plotter.
Suitable for college kids and researchers in geometry and machine technological know-how, the textual content starts off via introducing basic houses of curves and kinds of derived curves. next chapters observe those homes to conics and polynomials, cubic and quartic curves, algebraic curves of excessive measure, and transcendental curves. a complete of greater than 60 targeted curves are featured, every one illustrated with a number of CalComp plots containing curves in as much as 8 diverse editions. Indexes supply tables of derived curves, curve names, and a 95-item consultant to extra reading.

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Extra resources for A Catalog of Special Plane Curves

Example text

Proof. Notice that f is a local homeomorphism. Since X is compact, f is a covering map. Positive expansivity ensures that there exists a compatible metric D and constants 6 > O,X > 1 such that D(x, y) < 6 implies D( f (x), f (y)) 2 XD(x, y). If V is an open set of diameter less than 6 and if q E fP1(V), then there is a unique open set containing q of diameter less than (1/X)6, which is mapped homeomorphically onto V. This enables us to lift chains of small open sets. Let {V, : 1 5 i 5 n) be a fixed open cover of X so that the diameter of each is less than 6.

Thus we leave the proof to the readers. 9. (1) I f f : X -+ X is positively expansive and Y is a closed subset of X with f (Y) = Y, then f l y : Y -,Y is positively expansive. (2) If fi : Xi -+ Xi (i = 1,2) are positively expansive, then the continuous sujection fl x f2 : X1 x X2 -,X1 x X 2 defined by is positively expansive. Every finite direct product of positively expansive maps is positively expansive. (3) If X is compact and f : X -+ X is positively expansive, then h o f o h-I : Y -+ Y is positively expansive where h: X -+ Y is a homeomo~phism.

Let X and Y be compact metric spaces and let f: X -, X and g: Y -, Y be continuous surjections. Then f is said to be topologically conjugate to g if there exists a homeomorphism cp : Y 4 X such that f o cp = cp o g. If cp is a continuous surjection, then f is said to be toplogically semi-conjugate to g, in other words, f is called a factor of g. When f is topologically conjugate to g, let us define a relation f g. Then the relation is an equivalence relation on the class of all continuous surjections.

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