Cycloid Curves #1

A cycloid is a curve produced by a point on the circumference of a circle, while it rolls, without slipping, on a flat surface.

The hypocycloid is produced by a point on the circumference of a circle, while it rolls, without slipping, on the inside of another circle.

An epicycloid is a curve generated by a point on a circle as it rolls, without slipping, on the outside of a circle.

This particular hypocycloid has 3 cusps, making it a Deltoid.

A Deltoid is produced when the inner circle radius (r) is 1/3 radius (R) the outer circle. This completes 3 revolutions.

Some Cycloid Maths

Definitions

• generatrix: point on moving circle producing the curve
• r: radius of smaller circle
• R: radius of larger circle
• cusp: the point on the generating circle that touches the determinant circle
• generator: circle on which the generatrix is contained
• k: number of cusps (where generatrix touches the stationary circle)
• a: area of the generator circle
• baseline (base): the straight line on which the generator rolls without slipping

Cycloid Formulae

• length of baseline: 2𝝅r (circumference of the generator circle)
• length of cycloid: 8r
• area under cycloid: 3𝝅r² (3 times area of generator circle)

Hypocycloid Formulae

• r = 1/3 R
• length of arc: 16r
• area inside the curve: 2πr²

For the Deltoid curve above, we have circles of 150px and 50px.

Length of 3 arcs: 16r = 800
Length of 1 arc: 800/3 = 266.66

Deltoid Area: 2πr² = 15707.96
∴ Deltoid area = 2 * area of rolling circle
Area of R: 141371.67
Ratio of 2 areas: 141371.67 / 15707.96 = 9.00000191