defined
by:
. Given two points, p1
and p2 in the plane as well as two slopes m1 and m2 and an
interval [u.v], define a
general procedure which will define a curve 
such that
and
which has some prescribed nice property between neighbouring points
and 
. One of the simplest such proeprties is that of
being quadratic; so a quadratic spline for a sequence of points is a function f which has different
definitions on each of the intervals 
such that on
each of these intervals the function is quadratic. It is also required
that at each of the points the transition from one
definition to
the other is smooth -- namely the tangents from both the left and right
have the same slope.
Write a procedure -- TwoPointSpline -- which, when given the inputs (m, P1, P2), will find a quadratic function passing through the two points P1 and P2 and having slope m at the first point P1. Using this procedure find a quadratic spline through the points (1,1), (2,2), (4, 5/2), (5,0) and having slope -2 at the point (1,1). Using the knowledge you have gained in this exercise, write a procedure that will find a quadratic spline through any given sequence (list) of points with a given slope at the first point. Test your procedure on the sequence which can be viewed by clicking here and then downloaded using Netscape's "save" command.
So far, only splines of points forming a function have been discussed,
However, it is also possible to create more general curves with
splines. Consider the quadratic parametric curve
defined
by:
. Given two points, p1
and p2 in the plane as well as two slopes m1 and m2 and an
interval [u.v], define a
general procedure which will define a curve such that
