Parametric Equations

Graphing parametric equations on the Desmos Graphing Calculator, Geometry Tool, or the 3D Calculator  is as easy as plotting an ordered pair. 

Instead of numerical coordinates, use expressions in terms of the special parameter \(t\), like (cos \(t\), sin\(t\)). In the 3D Calculator, you can also use expressions in terms of the parameters \(u\) and \(v\) like (cos\(u\), sin\(v\), \(u\)).

Watch the video for an introduction to parametrics, and read on to learn more.

To graph a parametric curve, simply create an ordered pair where one or both coordinates are defined in terms of the parameter \(t\).

By default, parametric curves are plotted for values of \(t\) in the interval \([0, 1]\). However, you can manually adjust the domain using the inputs in the expression line.

Just like other expressions, you can use dynamic variables in parametric equations, including in the bounds for the interval of \(t\).

Parametric curves can be represented in a single coordinate expression or can reference other defined component functions using the parameter \(t\).

In Desmos, since lowercase \(x\) and \(y\) are reserved variables, you must use uppercase \(X\) and \(Y\) if you want to define functions for \(x\) and \(y\).

Click and hold the Color icon to open the Options menu. From there, you can toggle Fill to visualize the area enclosed by the parametric curve.

When you fill a parametric curve, its interior is determined according to the non-zero winding rule.

Graph a parametric surface in 3D with the parameters \(u\) and \(v\).

Consider the parametric surface \((u, f(u)\)cos\(v\), \(f(u)\)sin\(v)\) where the parameter \(u\) represents the \(x\)-values, and \(v\) represents the angle of rotation. This formula rotates any function \(f(x)\) around the \(x\)-axis.

For instance, try rotating \(f(x) = \sqrt(x)\) with this parametric surface. Adjust the inputs beneath the expression to \(0 \le u \le 5\) and \(0 \le v \le 2\pi\). To connect this rotation to a slider, create a slider \(b\) from \(0\) to \(2\pi\) and adjust the input for \(v\) to \(0 \le v \le b\).