INTRODUCTION TO PARAMETRIC GRAPHS

Parametric graphing, like polar graphing, uses a different method of calculating points on the plane to come up with curves that may be difficult to compute using normal rectangular coordinates. They are unique in that the Cartesian `x` and `y` coordinates are calculated based on a third variable (the "parameter" of `x` and `y`) which is traditionally called `t` (not to be confused with the `t` used by Graphmatica to represent theta). `t` is allowed to increase from the start of the domain you specify to the end. At each value, the functions x(t) and y(t) are calculated to give an (x,y) coordinate which is graphed. Graphmatica then connects these points to form a smooth curve--if something you graph begins to look jagged, you probably need to adjust the fineness. (Parametric graph fineness is linked to the same fineness control as Cartesian and polar graphing, and should be decent at the default fineness value, but if you need to, you can increase or decrease this value. Be aware that this will affect the fineness of non-parametric graphs as well. See Adjusting the Fineness for details.)

To enter a parametric graph, you need to remember four basic parts: the x(t) and y(t) functions, the semicolon between them (this is how Graphmatica knows it's a parametric graph), and the domain for `t`.

```       semicolon
x-function | y-function  domain
|      |     |         |
x = 2t ;  y = 2t^2  {-10, 10}
```

Although as in all other Graphmatica equations you don't need to solve for x and y (i.e. `t = 5x` would be OK), only one x and one y can appear in the whole equation, and "double" equations like `x^2 = t` where Graphmatica would normally solve for both the positive and negative roots are not supported (you can enter them but only the positive root will be found). You can type the x and y equations in either order, as long as they are separated by a semicolon, and the domain will be recognized anywhere on the line. You must specify a domain for each parametric equation! The variety of curves that can be drawn with parametric equations is great and makes choosing an appropriate default domain impossible. Some curves (like those including the circular functions sine and cosine) tend to work best over a `{0,2pi}` domain, like polar graphs. Others will match up better with the default domain of the normal graphs, the size of the viewing area. Some have a very compact domain, between say 0 and 1, where they will appear on the screen. If you over- or under-estimate the domain, you can always abort the graph and edit your equation.

kSoft, Inc. ksoft@graphmatica.com Last updated: Sun 11 Jun 2017