Epitrochoid
The outer-rolling trochoid family whose two-lobed member shapes the Wankel rotary engine housing.
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Formulas and key values
- Parametric equations
- Period · whole-number R, r
- Farthest from the center
R is the fixed circle, r the rolling circle, and d the pen’s distance from the rolling circle’s center.
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Start the video01 · Definition
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.
02 · Properties
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.
03 · Questions
Epitrochoid: common questions
- What is the equation of the Epitrochoid?
- Parametric equations: x = (R+r) cos t - d cos(((R+r)/r)t), y = (R+r) sin t - d sin(((R+r)/r)t). R is the fixed circle, r the rolling circle, and d the pen’s distance from the rolling circle’s center.
04 · Wonders
See it in a math wonder
05 · Keep exploring