Mike Raiford wrote:

> Well, of course a spline is easy. Right?

Heh. Not so much... ;-)

> I just wish I knew a bit more of the underlying theory of how the 
> coefficients were calculated.
> 
> The article speaks of derivatives, so calculus must be involved. I'm 
> gathering that the formulae in the algorithm were probably found by 
> applying calculus to a 3rd order polynomial.

Suppose you want a line that passes through points A and B. That is, you 
want a function f(x) such that

   f(0) = A
   f(1) = B

That's quite easy to arrange:

   f(x) = A(1-x) + Bx

This gives us a straight line between any two points. Notice that we can 
rearrange it as:

   f(x) = A(1-x) + Bx = A - Ax + Bx = (B-A)x + B

which is your usual "standard form" for a linear equation.

Now, suppose we want a curve going through *three* points, A, B and C?

   f(0.0) = A
   f(0.5) = B
   f(1.0) = C

How do we do that? Well, you can make a function

   g(0.0) = A
   g(0.5) = B

which gives us g(x) = A(1-2x) + B(2x), and another function

   h(0.5) = B
   h(1.0) = C

which gives h(x) = B(1-2(x-0.5)) + C(2(x-0.5)). If you sort out all the 
coefficients, those are both linear, and you can "blend" between them 
like this:

   f(x) = (1-x) g(x) + x h(x)

When you sort out all the algebra, you end up with a quadratic curve 
passing through all three points, as required. I've found from 
experience that this spline isn't very easy to control though!

Part of the reason for that is that if B is higher than A or C, B isn't 
necessarily the highest point on the curve. That could be to either side 
of B.

The more sophisticated splines are defined by stipulating specific 
properties the curve must have. As you have noticed, this has to do with 
calculus. In particular, the derivative of a 1D function is the *slope* 
of the graph at that point. And the second derivative is the *curvature* 
of the graph at that point. Obviously slope and curvature are useful 
things to want to control.

For example, take my defective quadratic spline above. I said that B 
isn't necessarily the highest point. In fact, if you draw a line AB and 
a line BC, you'll discover that the slope of the function doesn't match 
either of these lines at any point.

(I should perhaps point out that it's impossible for a quadratic curve 
to fit any better than this. Any three points in space uniquely define a 
single quadratic curve. For a more controllable curve, you'd need a 
higher-order polynomial.)

More sophisticated splines say "I want the slope of the graph to match 
the line AB at the start". Since "slope" is actually "first derivative", 
you end up using calculus. So you end up with something like

   f(0.0) = A
   f(0.5) = B
   f(1.0) = C
   f'(0.0) = B-A
   f'(1.0) = C-B

How you can from this system of constraints to a working function is 
just slightly beyond by level of skill. But you see where the calculus 
comes in...

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