A typical problem which arises in designing an object is that of
connecting a sequence of points by a smooth curve. For example in
designing a car body engineers may have determined several points
on the body and the designer must connect them with an esthetically
pleasing curve. Computer graphics and type face design offer similar
challenges. A spline is the mathematical term for a curve which passes
through a sequence of given points and
which has some presrcibed 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 intervak the function is quadratic. It is als required
that at each of the points the transition from ne defitnion to
the other is smooth -- namely the tangent 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 *P*1 and *P*2 and
having slope *m* at the first point *P*1. 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 aprocedure that will find a quadratic
spline through any given sequence of points with a given slope at the
first point. Test your procedure on a sequence of 10 points.

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, *p*1
and *p*2 in the plane as well as two slopes *m*1 and *m*2 and an
interval [*u*.*v*], define a
general procedure which will define a curve such that

Thu Mar 27 17:09:43 EST 1997