(a) t is known that the uniform quadratic B-spline curve is continuous in its first derivative but..

(a) t is known that the uniform quadratic B-spline curve is continuous in its first derivative but that it is not guaranteed continuous in its second derivative. Prove that N1,3 is discontinuous in its second derivative at one or more points.

(b) The univariate Chaikin curve scheme described in part (c) can be generalised to a bivariate scheme that generates surfaces. Explain how it can be generalised to generate surfaces from an arbitrary mesh of control points, paying attention to both the regular and the extraordinary cases

(c) A general piecewise curve definition, whether B´ezier, B-spline, or NURBS, can be written as a sum of products of basis functions, Ai(t), and control points, Pi :

P(t) = XAi(t)Pi , tmin = t

Give the conditions on the functions Ai that are needed to ensure that

a)Translation of all of the points by some vector, P0 i = Pi + ?P, causes a translation of the curve by the same vector, P0 (t) = P(t) + ?P.

b) The curve lies within the convex hull of the control points

c) The curve passes through one of the control points, Pj