Hypotrochoid vs. Epitrochoid — What's the Difference?

Used in mechanical engineering to design compact internal mechanisms.

Creates star-like or rounded patterns that are aesthetically distinct.

A geometric curve created by a point on a smaller circle rolling inside a larger stationary circle.

Featured in artistic applications like spirographs and visual arts.

Often found in educational tools for teaching geometry and design.

Used in the mathematical study of dynamical systems and curves.

Can generate patterns resembling flowers or complex ellipses.

A curve traced by a point attached to a circle rolling outside another circle.

Involves the subtraction of radii in its mathematical formulation.

Based on the sum of the radii involved in its generation.

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.

An epitrochoid is formed when a circle rolls outside another stationary circle, with a fixed point on the rolling circle tracing the path.

It is based on the sum of the radii of the rolling and fixed circles.

They are used in teaching geometric principles and design in educational settings.

It's a type of curve generated when a circle rolls inside another fixed, larger circle, with a point on the rolling circle tracing the path.

They are often used in gear systems and compact mechanical designs.

They are primarily used in the design of rotary engines.

Hypotrochoids tend to produce more symmetrical and closed designs, whereas epitrochoids generate more sprawling, star-like patterns.

Epitrochoids are used to create unique and extensive patterns in visual arts, particularly with spirographs.

Yes, they are popular in geometric art and can be created using tools like spirographs.

It involves subtracting the radius of the rolling circle from the radius of the fixed circle.

The main difference lies in the rolling circle's movement; epitrochoids are generated by a circle rolling outside another, while hypotrochoids involve inside rolling.

They typically create intricate, flower-like or elliptical patterns.

While mostly used in rotary engines, they also find applications in complex mechanical designs and artistic tools.

The mathematical formulation helps in accurately designing and applying these curves in various mechanical and artistic contexts.

They are known for producing star-like or rounded, open patterns.