## 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