  # Epicycloid vs. Hypocycloid: What's the Difference? Edited by Sumera Saeed || By Sawaira Riaz || Updated on October 3, 2023
An epicycloid is a curve traced by a point on the circumference of a circle rolling on the outside of another circle, while a hypocycloid is traced by a point on a circle rolling inside another circle. ## Key Differences

An epicycloid is a captivating curve that emerges when a circle rolls externally along the perimeter of another circle without slipping. The tracing point on the rolling circle's circumference delineates the epicycloid. On the other hand, a hypocycloid results when a circle rolls on the inside of another circle, with the path being described by a point on the rolling circle.
The creation of an epicycloid and hypocycloid depends on the relative sizes of the circles involved. For an epicycloid, if the fixed and rolling circles have the same radius, the resultant curve resembles a straight line. With hypocycloids, when the rolling circle's radius is half that of the fixed circle, the resulting curve is a straight line, known as a diameter of the fixed circle.
Both epicycloid and hypocycloid have unique mathematical properties and equations that define their shapes. The epicycloid has a larger spread due to its generation outside the fixed circle, whereas the hypocycloid is confined within the bounds of the larger circle since it's generated inside.
Epicycloids and hypocycloids can be found in various practical applications. While epicycloids can be observed in the design of certain gear teeth, hypocycloids play a role in devices like the Spirograph, which uses rotating gears to create intricate patterns. Both curves, with their mesmerizing patterns, have also inspired art and design through the ages.

## Comparison Chart

### Definition

Curve from a circle rolling externally
Curve from a circle rolling internally

### Position

Outside the fixed circle
Inside the fixed circle

Confined within the larger circle

### Example Usage

Certain gear teeth designs
Spirograph designs

## Epicycloid and Hypocycloid Definitions

#### Epicycloid

A curve with a spread larger than the fixed circle.
The wide design on the paper is an epicycloid.

#### Hypocycloid

Curve made by a circle rolling inside another.
The spirograph toy creates patterns similar to a hypocycloid.

#### Epicycloid

Can be described using parametric equations.
The mathematical model for the epicycloid is intriguing.

#### Hypocycloid

Often seen in artistic patterns and designs.
Artists sometimes use the hypnotic shape of the hypocycloid.

#### Epicycloid

A curve formed by a circle rolling outside another.
The gear's teeth were shaped like an epicycloid.

#### Hypocycloid

Can be represented through specific equations.
Graphing a hypocycloid requires knowledge of its mathematical formula.

#### Epicycloid

The trace of a point on the moving circle's circumference.
The point on the smaller circle traces the epicycloid.

#### Hypocycloid

Trace of a point on a rolling circle's edge.
The point's movement describes the hypocycloid.

#### Epicycloid

Seen in specific mechanical applications.
The epicycloid curve is vital in this machine's design.

#### Hypocycloid

Curve stays within the bounds of the larger circle.
The shape inside the circle is a hypocycloid.

#### Epicycloid

The curve described by a point on the circumference of a circle as the circle rolls on the outside of the circumference of a second, fixed circle.

#### Hypocycloid

The plane locus of a point fixed on a circle that rolls on the inside circumference of a fixed circle.

#### Epicycloid

(geometry) The locus of a point on the circumference of a circle that rolls without slipping on the circumference of another circle. Category:en:Curves

#### Hypocycloid

(geometry) The locus of a point on the circumference of a circle that rolls without slipping inside the circumference of another circle. Category:en:Curves

#### Epicycloid

A curve traced by a point in the circumference of a circle which rolls on the convex side of a fixed circle.

#### Hypocycloid

A curve traced by a point in the circumference of a circle which rolls on the concave side in the fixed circle. Cf. Epicycloid, and Trochoid.

#### Epicycloid

A line generated by a point on a circle rolling around another circle

#### Hypocycloid

A line generated by a point on a circle that rolls around inside another circle

## FAQs

#### Can a hypocycloid curve ever be outside the larger circle?

No, a hypocycloid remains within the larger circle's boundary.

#### What toy creates patterns resembling hypocycloids?

The Spirograph toy creates patterns similar to hypocycloids.

#### Where can we see epicycloids in real-life applications?

Epicycloids are often seen in certain gear teeth designs.

#### What's the main difference between an epicycloid and a hypocycloid?

An epicycloid is formed outside a circle, while a hypocycloid is inside.

#### Do epicycloids always form outside the circle?

Yes, epicycloids are always formed when a circle rolls externally.  