AGS - Parametric Curves

One of the More Famous Curves, the Butterfly Curve, and its Parametric Function

Parametric curves are the representation of curvilinear extrapolations over the course of an interval.  Thus, the coordinates of a point "p" of a parametric curve "c" are expressed as functions of parameter, "t."  This means that a spatial curve c is represented by c(t) = ( x(t), y(t), z(t) ), where x(t), y(t) and z(t) are known as the coordinate functions.  Thereby, every parameter t is mapped to a curve point p(t).  Often, it is helpful to think about t as time, although t may not be time.  An interval t mapped over over the curve of three-dimensional space would look thusly:

Spatial Three-Dimensional Curve Showing Length of an Interval for Parameterization of t

By restricting the interval of a parameter, t, one obtains a subset of curve c (sometimes referred to as the curve segment).  A parametric curve defined by polynomial functions is called a polynomial curve.  The highest order of the parameter t in any of the three coordinate functions is called the degree of the polynomial curve.  So for a third order equation (or cubic root function), the degree of a polynomial curve would be three.  For a second order equation, the degree would be two and so on.

A Circle Represented as a Parametric Curve

Oftentimes, typical functions such as the equation of a circle, are represented as parametrics.  In the above photo, one can see that c(t) or p is represented by the variables x and y, which are dependent on the origin,  m(t), and the radius, n(t).  Rather than a unit circle, where cosines and sines determine a constant radius of 1 toward the perimeter, this circle is far more useful when delineating the most common curve used in architecture: namely, the circular curve.