## 1. Introduction spline curves

Every graphics system has some form of primitive to draw lines. Using these
primitives we can draw many complex shapes. However, as these shapes get
ever more complex and finely detailed, so does the data needed to describe
them accurately.

The worst case scenario is the curve. A curve can be described by a finite
number of short straight segments. However, on close inspection this is
only an approximation. To get a better approximation we can use more
segments per unit length. This increases the amount of data required to
store the curve and makes it difficult to manipulate. We clearly need a way
of representing these curves in a more mathematical fashion. Ideally, our
descriptions will be:

- Reproducable - the representation should give the same curve every time;
- Computationally Quick;
- Easy to manipulate, especially important for design purposes;
- Flexible;
- Easy to combine with other segments of curve.

The types of curve I will discuss fall into two broad categories:
interpolating or approximation curves. Interpolating curves will pass
through the points used to describe it, whereas an approximating curve will
get near to the points, though exactly what is called near will be
discussed later. The points through which the curve passes are known as
knots; the curve described by the equation is often referred to as a
spline. This term originated in manual design, where a spline is a thin
strip. This strip was held in place by weights to create a curve which
could then be traced. In the same way we now use knots to describe a
curve.

Next: Explicit curves

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