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Hello, I am looking for someone to write an essay on On the geometry of piecewise circular curves T.F.Banchoff and P.J.Giblin, American Mathematical Monthly, (101),1994,403-416, In the plane a smooth
Hello, I am looking for someone to write an essay on On the geometry of piecewise circular curves T.F.Banchoff and P.J.Giblin, American Mathematical Monthly, (101),1994,403-416, In the plane a smooth piecewise circular curve is a curve made from arcs of circles such that at points where two arcs join the. It needs to be at least 1500 words.
While they are almost as easy to define as polygons, PCCs are more versatile as they have a well defined tangent at every point. They are said to be ‘smooth’ if ‘the directed tangent line at the end of one arc coincides with the directed tangent line at the beginning of the next.’ Also no arc degenerates to a single point in a smooth PCC.
When PCCs are used to approximate smooth curves, not only is the approximation point wise close but also the tangent lines at the points of the smooth curve are approximated by the tangent lines of the PCC.
A classical example is finding the shortest length enclosing a fixed area. The result here is a circle. When the curve is required to surround a fixed pair of points, then the curve of shortest length enclosing a given area will be either a circle or a lens.
PC ellipse If it is required that a curve of fixed length L surround a given pair of discs of the same radius, then, for a certain range of values of L, the curve that encloses the greatest area is a smooth convex PC curve consisting of two arcs on the boundary circles of the discs and two arcs of equal radius tangent to both discs. Such four-arc convex PC curves have long been used in engineering drawing for approximating
The parallel curves of a PCC are PCCs since the parallel curves of circular arcs are circular arcs. This is especially important in computer graphics. Parallel curves are also called offset curves as they are obtained by moving away fro the curve. However, if the radii are decreased a set of inner parallel curves will be obtained.
A remarkable example of a symmetry set of a PCC is seen in a transition called a moth in the study of symmetry sets of one-parameter families of plane curves. Bitangent circles either grow out or come together and disappear when a four-arc PCC is perturbed.
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