Epitrochoid

Definition

An epitrochoid is a type of roulette curve generated by tracing a point attached to a circle as it rolls around the outside of a fixed circle. The parametric equations for an epitrochoid involve trigonometric functions and depend on the radii of the rolling and fixed circles, as well as the distance from the tracing point to the center of the rolling circle.

5 Must Know Facts For Your Next Test

1. The parametric equations for an epitrochoid are $x(\theta) = (R + r) \cos(\theta) - d \cos((R+r)/r \cdot \theta)$ and $y(\theta) = (R + r) \sin(\theta) - d \sin((R+r)/r \cdot \theta)$ where $R$ is the radius of the fixed circle, $r$ is the radius of the rolling circle, and $d$ is the distance from the tracing point to the center of the rolling circle.
2. Epitrochoids can produce various shapes depending on values of R, r, and d; when $d = r$, they form epicycloids.
3. An epitrochoid is related to but distinct from a hypotrochoid, which involves rolling inside rather than outside a fixed circle.
4. The complexity and number of cusps or loops in an epitrochoid depend on whether $(R+r)/r$ is rational or irrational.
5. Epitrochoids have applications in gear design and art (e.g., Spirograph toy patterns).

Review Questions

Related terms

Hypotrochoid: A roulette curve generated by tracing a point attached to a circle that rolls inside another fixed circle.

Epicycloid: A special case of an epitrochoid where the distance from the point to center equals radius of rolling circle ($d = r$).

Parametric Equations:

$x(t)$ and $y(t)$ equations expressing coordinates as functions of one or more parameters, often used to describe curves.