Cardioid
Polar curvesThe heart-shaped curve r = a(1 + cos θ) traced by a point on a circle rolling around an equal circle.
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Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
What is the Cardioid?
The cardioid is the path of a point on a circle rolling around a second, equal circle — the simplest epicycloid. Its polar equation r = a(1 + cos θ) makes the heart shape explicit: the radius doubles at θ = 0 and collapses to a cusp at θ = π. The name, from the Greek for “heart”, was given by Giovanni Salvemini de Castillon in 1741.
The cardioid keeps appearing where circles interact: it is the envelope of light rays reflected inside a coffee cup (a caustic), the boundary of the main bulb of the Mandelbrot set, and the pickup pattern of the microphones named after it.
Key properties
- Arc length: 8a. Enclosed area: (3/2)πa².
- It is the epicycloid with one cusp (rolling circle equal to the fixed circle).
- It is also the special limaçon r = b + a·cos θ with b = a.
- The caustic of a circle with a light source on its rim is a cardioid.
- Maximum width 2a occurs along the axis of symmetry.
Where you'll see it
Cardioid microphones, coffee-cup caustics, the Mandelbrot set’s main bulb, and cam profiles in machinery.