5th january: intro.pdf-what is cagd. geometrical elements and
an introduction to an optimization problem.
9th january: implicit versus parametric forms of curves and
surfaces. the resultant calculation and converting
parametric to implicit. tangent spaces for curves
and surfaces in both forms. equivalence of both
the definitions when a surface/curve is given as implicit
as well as parametric.
12th january:the representation of a solid model. topological data and
geometric data. the basic curve and surface definitions
as box and mapping function. the VList,EList, FList
and SMList. pointers to geometry. parametrization of a curve.
the usefulness of the weierstrass approximation heorem.
polynomials of a given degree and bases. the taylor and
lagrange basis. The sohoni-approximation theorem, the bernstein
basis. the beta-function probability connection.
16th january:Basic properties of the bernstein basis. proof that its a basis.
the bernstein approximation theorem. 1-d to 3-d and the bezier
curves construction. control points and greville abscissa.
the end conditions and derivatives at ends.
evaluation of bezier curves, the de-castelljeu algorithm.
degree elevation.
19th january: Tutorial sheet.
23rd january:Review of the Bezier curve data-structure, i.e., degree, grevilles and
control polygon. review of evaluation and
its graphical meaning. review of differentiation and degree elevation.
degree elevation on control polygon. subdivision and its
graphical interpretation. proof of the result.
splines as piece-wise polynomial
functions and junction continuity. knots and their meaning as
(i) interval, (ii) sub-intervals, (iii) degree and (iv) junction
continuity. the space V(alpha) for a knot alpha, and its dimension.
an example of knots and containments of spaces V(alpha).
grevilles for splines.
30th january:The space V(alpha) and the choice of a basis and coefficients
so that consistency under repeated knot-insertions is maintained.
The knot insertion algorithm and proof of the consistency condition.
Evaluation of B-splines as repeated knot insertions. Lab-demo on MATLAB
of B-splines. Linear operations on B-splines.
2nd february:The 2d function approximation problem and the 2d-approximation theorem.
The 2D bernstein basis and the general tensor-product basis. Interpretation
of tensor-product surfaces as curves with moving control points.
Evaluation of surfaces as a multiplication of 3 matrices. Associativity
of matrix multiplication and its interpretation. Lab-demo of MATLAB
spline surfaces.
6th february:The general tensor product surface. End curves and End points.
Iso-parametric curves and partial derivatives. The surface normals
and a normal system. Specification of a normal system on a surface.
Junctions between surfaces and their parametric continuity. Geometric
continuity between surfaces.
13th february:Tutorial on splines.
19th february: ACIS lab session.