Epitrochoid
Rolling curves (roulettes)The outer-rolling trochoid family whose two-lobed member shapes the Wankel rotary engine housing.
Adjust parameters
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 Epitrochoid?
The epitrochoid is the outside-rolling counterpart of the hypotrochoid: a circle of radius r rolls around a fixed circle of radius R while a point rigidly attached at distance d from its center draws the curve. On the rim (d = r) it reduces to the epicycloid; off the rim it produces waves or loops.
Its most famous member has R = 2r with a suitable d: a smooth two-lobed peanut shape. Rotate a Reuleaux-style triangular rotor inside it and all three rotor tips stay in contact with the wall — the working principle of the Wankel rotary engine that powered Mazda’s RX series.
Key properties
- d = r recovers the epicycloid; d = 0 gives a circle of radius R + r.
- Closes after r/gcd(R, r) revolutions around the fixed circle.
- R = 2r with moderate d gives the two-lobed Wankel housing profile.
- Ptolemy’s planetary model of deferents and epicycles traces epitrochoids.
- The curve stays within radius R + r + d of the center.
Where you'll see it
Wankel rotary engine housings, Ptolemaic planetary paths, spirograph outer-ring patterns, and guilloché ornament.