G2-cubic contours

Abstract

Let b 0, b 2,…, b 2n be the vertex sequence of a convex polygon and through each b 2j choose a line such that b 2j−2 and b 2j+2 lie on one of its sides, and let b 2j+1 be the intersection point of line through b 2j and b 2j+2 . For any choice q 2j+1 of an interior point in each triangle b 2j b 2j+1 b 2j+2 we construct G 2-cubic algebraic splines which interpolate the vertices b 0, b 2,…, b 2n and the points q 2j+1 . At each b 2j the spline is tangent to the prescribed line at this point and it is contained in the union of the triangles b 2j b 2j+1 b j+2 . For any j=0,1,…, n we show how the choice of q 2j+1 limits the range of variation of the curvatures at the vertices b 2j and b 2j+2 . We study the conditions for the curvatures at the specific vertices to vary arbitrarily, hence allowing for the construction of G 2-interpolating cubic splines which are as flat or as sharp, as desired at these points. A generalization for nonconvex data sequences is given by breaking the polygon into maximal monotonically convex subsequences. The resulting spline has inflections at user controlled points.