A typical equation of a curve describes a relationship between two coordinates, x and y. For example, the following equation is an equation of an ellipse with half axes a and b,
An example, with a=1 and b=2:
When we write parametric equations, we introduce a new variable, let's say t, and we express both x and y in terms of t. We also specify the range of t. For example, the following parametric equations describe the ellipse mentioned above:
x = a*cos(t), y = b*sin(t), for 0 ≤ t ≤ 2π.
Remark
The variable t is very often used because Newton thought about curves as trajectories of moving points. So x(t) and y(t) show the position of a point at time t. But the variable t, also called a parameter, can have any meaning. In the example above it represents an angle measured in radians.
Task
Set MODE to Par, ZOOM to ZStandard, ZDecimal, 2nd FORMAT to AxesOff.
Define,
\X1T=9cos(T)
\Y1T=9sin(CT)
Graph the curve for several whole values of C = 1, 2, 3, ...
Define and graph the ellipse described above for several values of a and b, using parametric equations.
