Hypotrochoid vs. Epitrochoid — What's the Difference?

A hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle. The parametric equations for a hypotrochoid are: x ( θ ) = ( R − r ) cos ⁡ θ + d cos ⁡ ( R − r r θ ) {\displaystyle x(\theta )=(R-r)\cos \theta +d\cos \left({R-r \over r}\theta \right)} y ( θ ) = ( R − r ) sin ⁡ θ − d sin ⁡ ( R − r r θ ) {\displaystyle y(\theta )=(R-r)\sin \theta -d\sin \left({R-r \over r}\theta \right)} where θ {\displaystyle \theta } is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because θ {\displaystyle \theta } is not the polar angle).

An epitrochoid ( or ) is a roulette traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is at a distance d from the center of the exterior circle. The parametric equations for an epitrochoid are x ( θ ) = ( R + r ) cos ⁡ θ − d cos ⁡ ( R + r r θ ) , {\displaystyle x(\theta )=(R+r)\cos \theta -d\cos \left({R+r \over r}\theta \right),\,} y ( θ ) = ( R + r ) sin ⁡ θ − d sin ⁡ ( R + r r θ ) .

(geometry) A geometric curve traced by a fixed point on the radius line outside one circle which rotates inside the perimeter of another circle. Category:en:Curves

A geometric curve traced by a fixed point on one circle which rotates around the perimeter of another circle. Examples include the shape of the Wankel engine Category:en:Curves

A curve, traced by a point in the radius, or radius produced, of a circle which rolls upon the concave side of a fixed circle. See Hypocycloid, Epicycloid, and Trochoid.

A kind of curve. See Epicycloid, any Trochoid.