M.PHIL IN MATHEMATICS

Course Structure

M-701: BEZIER TECHNIQUES IN COMPUTER
AIDED DESIGN

M-702: COMBINATORICS OF DESIGN OF
EXPERIMENTS

M-703: HOMOTOPY AND SINGULAR HOMOLOGY
THEORY

M-704: RATIONAL FINITE ELEMENTS

M-705: SPLINE APPROXIMATION THEORY


M-706: THEORY OF DISTRIBUTION


M-707: FUZZY TOPOLOGY

Unit-I: Affine map, Linear interpolation, piecewise linear interpolation, the de Casteljau algorithm, properties of Bezier curves, The Bernstein form of Bezier curves, Definition of Bernstein Polynomials, their derivatives, further properties of Bezier curves including their derivatives, Bezier curves : degree elevation and reduction variation diminishing property, nonparametric curves, integerals Bezier and barycentric forms of Bezier curves.

Unit-II: Weierstrass approximation theorem for parametric curves, Aitkin's algorithm, Lagrange Polynomials, limits of Lagrange interpolation, Cubic and quintic Hermite interpolation, Spline curves in Bezier forms, Global and local parameters, subdivision, domain transformation, smoothing conditions, C1 and C2 continuity, C1 Parametrization, C2 quadratic B-spline curves, C2 cubic B-spline curves, C1 piecewise cubic interpolations.

Unit-III: Cubic spline interpolation: end conditions, parametrization, minimum property, B-spline revisited, De-Boor algorithm, smoothness or B-spline curves.
Two recursion formulae and repeated subdivision, Some properties of B-spline curves. Geometric continuity, motivation, characterization of G2 curves and Nu splines, G2 piecewise Bezier curves, Gamma splines, Local basis functions for G2 splines, Beta-splines, another characterization for G2 - continuity.

Unit-IV: Projectives maps of real line, conics and rational quadratics, Rational Bezier and B-spline curves of arbitrary degree. Tensor product Bezier surfaces, the direct de-Casteljau algorithm, tensor product approach, degree elevation, derivatives. Matrix form of Bezier Patch, non parametric patches.

Unit-V: composite surfaces and interpolation, subdivision and smoothness, bicubic B-spline surfaces, tensor product interpolants, Bezier triangles, Barycentric coordinates, de Castel jau algorithm, Bernstein polynomials, derivatives, subdivision, differentiability, degree elevation, nonparametric patches.