A recurring property of coordinate systems is that in order to locate a point in an *n*-dimensional space, you need *n* numbers (or *n* independent pieces of information about that point). There is a way to “cheat” this using just one number (called a *parameter*) to locate an *n*-dimensional point.

This isn’t really cheating as you still have to initially provide the required information, but once done, one number will suffice to place a point.

An example of this is the equation of a circle of radius *r*:

This is the standard Cartesian equation but a parametric way of defining a circle, using a parameter *t*, is

Defined in this way, any value of *t* will generate a point on the same circle. We can generate the Cartesian equation from these two parametric equations, but I will leave that as a topic for a future post.

Parametric equations can be a much more useful way to represent a curve, especially curves that model a physical process.

If a projectile is launched with an initial speed of 31.6267 m/s at an angle of 50.78° from the ground, it will follow a parabolic trajectory which can be represented by the equation

\[y=-0.01225(x-50)^2+30.625\]where *y* is the height above the ground and *x* is the distance along the ground from the launch point. The trajectory of the projectile (the graph of the above equation) looks like

This graph is useful in that it tells us how high the projectile goes and how far. But it doesn’t tell us where the ball is at any time or how long it take to hit the ground.

The initial velocity can be broken up into a horizontal and a vertical component. These components can be treated separately:

If resistance due to the air is neglected, the horizontal distance at a given time *t* is *x* = 20*t*. The vertical distance cannot be treated as simply as the vertical velocity is constantly changing due to gravity. From physics and calculus, the vertical distance is *y* = -4.9*t*^{2} + 24.5*t*. These two equations are the parametric forms of the Cartesian trajectory equation. For any time *t*, a point on this trajectory, (20*t*, -4.9*t*^{2} + 24.5*t*), is located and represents where the projectile is at that time:

Notice how we get more information about what is going on with this way of representing a graph. We can now tell where the projectile is at any time *t*, that it takes 2.5 seconds to reach the top of the trajectory, and that it takes 5 seconds to return to the ground.

This is not a unique way to represent this trajectory parametrically, but this one conforms to the physics of the problem. In general, parametric equations make it possible to plot graphs that are difficult or even impossible to plot with a single Cartesian equation.

This can be extended to higher dimensions as well. There is a lot of mathematics around parametric equations and it adds to the wonder (and complexity) of maths.