Geometric approximation of curves and singularities of secant maps : a differential geometric approach

In this thesis we study: (i) geometric approximation of curves in the plane and in space, and (ii) singularities of secant maps of immersed surfaces from a geometric perspective. Geometric Curve Approximation. In the current thesis our focus is on problems relating to approximation of parametric curves in the plane with conic arcs and biarcs. Biarcs are curves formed by joining two circular arcs in a tangent continuous fashion. In case of space curves we consider approximation with bihelical arcs. A bihelical arc is formed by joining two circular helices in a tangent continuous manner. We compute the complexity (minimum number of elements) of approximating a sufficiently smooth curve, with non-vanishing curvature in the plane, with biarc, parabolic or conic splines. We solve the issue of complexity of approximation of space curves with bihelix splines. To compute complexity we use the notion of Hausdorff distance between two curves. Spirals are curves in the plane with monotonically increasing/decreasing curvature. We propose an algorithm for approximating a spiral arc with bitangent biarcs (tangent to the spiral at its endpoints) and show that in case of spirals there is a unique biarc (among a one-parameter family) which is closest to the curve. Like curvature there is the notion of affine curvature, and affine spirals are curves with monotonically increasing/decreasing affine curvature. In this thesis we propose an algorithm for approximating an affine spiral curve in the plane with bitangent conics. We prove that there is a unique bitangent conic (among a one-parameter family) closest to any affine spiral. In case of space curves with monotonically increasing/decreasing curvature, there exists a three parameter family of bitangent bihelical arcs. In this case the problem of finding a bitangent bihelix which is closest to the curve is complex and still remains open. But we find an asymptotically optimal bihelix spline approximating a space curve w.r.t. the Hausdorff distance. Singularities of Secant Maps. Singularity theory is a far-reaching generalization of the study of functions at maximum and minimum points. In Whitney's theory functions are replaced by mappings, i.e., collections of several functions of several variables. The maximum or minimum points in a general setting are known as critical points or singularities of a map. In this thesis we study the secant maps of a surface immersed in , where . A secant map acting on two points p, q on a surface, maps them to the direction p – q. Keeping p fixed, if we let q approach p, then the direction p – q tends to the directional derivative at p. We discuss in detail the singularities of such maps in this thesis.