## 3. Parametric spline curves

Parametric equations can be used to generate curves that are more general
than explicit equations of the form y=f(x). A quadratic parametric spline
may be written as

where **P** is a point on the curve, **a**0, **a**1 and **a**2 are three vectors defining the curve and t is the
parameter. The curve passes through three points labelled **P**0, **P**1 and **P**2. By convention the curve starts from point **P**0 with parameter value t=0, goes through point **P**1 when t=t1 (0<t1<1) and finishes at **P**2 when t=1. Using these conventions we can solve for the three **a** vectors as follows:

and rearranging these equations we get:

We can now apply this to any set of three points, as shown in the diagram
below. It is easy to see the much higher degree of flexibility achieved
through the use of parametric equations, and we will see this exploited with
more advanced methods later on.

Figure 3.1 - Parametric Curve (interactive)

Although this method of creating curves is easy to use, it is not
immediately clear how these shapes come about. The curve is actually a
combination of two quadratic curves, one is y=f(t) and one is x=f(t). By
varying t between 0 and 1, x and y will both vary and create the curve.
This is shown below; create a curve and animate it to see how the three
curves relate.

Figure 3.2 - Detail of Parametric Curve (interactive)

### Further Information

Previous: Explicit curves

Next: Bezier Curves

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