Course Contents

Affine Transformations: Translation, Rotation, Scaling, Reflections, Shear, Homogeneous coordinates and concatenations of transformations, Inverse of transformations, Applications of affine transformations. Spline Curves: Convex and barycentric coordinates and combinations, Convex hull. Different forms of cubic curves: Algebraic form, control point form, Ball form and Timmer form. The idea of spline function, Conic sections and subsplines, Cubic and quintic Hermite and their approximation properties, Cubic and quintic splines, Spline in tension: Exponential splines polynomial, spline in tension. Hermite, Natural and Periodic splines and related results, Odd degree interpolating splines, Existence theorems, Minimum properties. Bézier Curves: Bernstein polynomials and their properties, Integral and rational Bézier, Tangent properties of Bézier curves, Bézier curves construction algorithms: de Casteljau algorithm, subdivision algorithm, degree raising algorithm. Conversion between ordinary polynomial curves and Bézier curves, Barycentric co-ordinates w.r.t a triangle, Relation between conics and rational Bézier curves, Bézier splines. B-splines: Integral B-splines curves, Properties of B-splines basis functions and B-splines curves, Types of B-splines: Open B-splines, Uniform B-splines, Periodic B-splines, Closed Periodic B-splines, Open Uniform B-splines. Applications of B-splines, N-step deformation of B-spline, The de Boor algorithm, Tangent properties of B-spline curves, NURBS, Tangent properties of NURBS, Rational de Boor algorithm. Knot insertion, Boehm’s Algorithm. Geometric spline curves: Geometric spline curves with minimal norm property. Tangent continuous spline curves. Curvature continuous spline curves.

Course Learning Outcomes

The students should come away from this class with ability to apply knowledge of relevant mathematics to solve domain design engineering problems