I Cardioid Math

When you think of Valentine’s Day, you think of hearts. The shape you often associate with the heart is a mathematical shape. It can be easily graphed on a graphing calculator. In this lesson we’ll explore these graphs.

The Polar Coordinate System

These heart-shaped graphs, or cardioids as they’re called, are graphed on a polar coordinate system. Unlike a rectangular system, like the Cartesian system, the polar system looks circular. You can compare the two systems below.

Whatever system you use, coordinates determine the location of a point. In the case of the rectangular system, use (x, y) coordinates. In the case of the polar system use (r, 𝛳). See the examples below. 

As you can see with a polar system, you use a distance and angle to determine location.

Graphs in a Polar Coordinate System

Because of the different ways of graphing points, the graphs of equations look different, too. Take a look at this graph. This is the graph of r = cos(𝛳). Its shape is circular.

Compare the polar graph of r = cos(𝛳) and y = cos(x).

They are both periodic, but in different ways. For the Cartesian graph, the graph repeats for every cycle of 2π. The circular graph repeats the circular shape for every increment of 360°.

Graphing Cardioid Shapes

By modifying the polar graph, we can graph cardioid shapes, even with minor changes to the graph. Take a look.

We’ve created a Desmos graphing calculator activity below. See if you can change the parameters on the cardioid to see if you can get heart shape in its usual orientation. Good luck!