# Epicycloid vs. Hypocycloid: Know the Difference

By Shumaila Saeed || Updated on December 25, 2023

**An epicycloid is a curve traced by a point on a circle rolling outside another circle, whereas a hypocycloid is traced by a point on a circle rolling inside another circle.**

## Key Differences

An epicycloid is formed when a circle rolls around the outside of another circle, with the tracing point on the circumference of the rolling circle. On the other hand, a hypocycloid is generated when a circle rolls inside another circle, with the tracing point also on the circumference of the rolling circle.

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Dec 07, 2023

In an epicycloid, the resulting curve is more extended and convoluted, often with cusps pointing outward. In contrast, a hypocycloid typically has inward-pointing cusps, creating a star-like shape when the rolling circle is significantly smaller than the fixed circle.

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Dec 07, 2023

The mathematical complexity of an epicycloid increases with the size of the rolling circle relative to the fixed one. For hypocycloids, the complexity is influenced by the ratio of the diameters of the inner and outer circles.

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Dec 07, 2023

Epicycloids can have varying appearances based on the ratio of the diameters of the two circles, from elongated loops to flower-like patterns. Hypocycloids can range from simple ellipses to complex star shapes, again depending on the diameter ratio.

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Dec 07, 2023

Epicycloids have applications in gear design, particularly in systems where non-standard gear tooth profiles are beneficial. Hypocycloids are also used in gear systems, especially in mechanisms like rotary engines where compact and efficient gear shapes are necessary.

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Dec 07, 2023

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## Comparison Chart

### Appearance

Varies from elongated loops to flower-like

Ranges from simple ellipses to complex stars

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Dec 07, 2023

### Applications

Gear design with non-standard tooth profiles

Compact and efficient gear systems

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Dec 07, 2023

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## Epicycloid and Hypocycloid Definitions

#### Epicycloid

Curve traced by a point on the rim of a circle rolling outside another circle.

The epicycloid drawn by the rolling gear was intricate.

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Nov 27, 2023

#### Hypocycloid

Curve formed by a point on a circle rolling inside another circle.

The children's toy created a perfect hypocycloid.

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Nov 27, 2023

#### Epicycloid

Used in gear design for creating specific tooth profiles.

Engineers used an epicycloid curve for the new gear's design.

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Nov 27, 2023

#### Hypocycloid

Varies from simple ellipses to complex shapes.

The hypocycloid shifted from an ellipse to a more intricate pattern.

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Nov 27, 2023

#### Epicycloid

Appears in various forms, from loops to flower-like shapes.

The epicycloid resembled a series of interconnected flowers.

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Nov 27, 2023

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#### Hypocycloid

Used in compact and efficient gear mechanisms.

The rotary engine's design incorporated a hypocycloid.

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Nov 27, 2023

#### Epicycloid

A mathematical curve with cusp points facing outward.

The epicycloid in the diagram had several outward-facing cusps.

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Nov 27, 2023

#### Hypocycloid

Dependent on the ratio of inner to outer circle diameters.

Adjusting the diameters resulted in a different hypocycloid shape.

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Nov 27, 2023

#### Epicycloid

A complex curve depending on the diameter ratio of the involved circles.

Changing the circle sizes altered the epicycloid's shape.

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Nov 27, 2023

#### Hypocycloid

Characterized by inward-pointing cusps, often star-like.

The hypocycloid in the machinery looked like a star.

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Nov 27, 2023

#### 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.

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Oct 19, 2023

#### Hypocycloid

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

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Oct 19, 2023

#### 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

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Oct 19, 2023

#### 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

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Oct 19, 2023

#### Epicycloid

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

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Oct 19, 2023

#### 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.

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Oct 19, 2023

#### Epicycloid

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

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Oct 19, 2023

#### Hypocycloid

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

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Oct 19, 2023

## Repeatedly Asked Queries

#### What defines a hypocycloid?

A curve traced by a point on a circle rolling inside another circle.

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Dec 07, 2023

#### What is an epicycloid?

A curve traced by a point on a circle rolling outside another circle.

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Dec 07, 2023

#### Can hypocycloids be seen in everyday objects?

Sometimes, in specific gears.

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Dec 07, 2023

#### Do epicycloids vary with circle size ratio?

Yes, they change shape with the ratio.

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Dec 07, 2023

#### Do epicycloids have practical applications?

Yes, in specific mechanical designs.

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Dec 07, 2023

#### Are hypocycloids always symmetrical?

Generally, but depends on the circle ratio.

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Dec 07, 2023

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About Author

Written by

Shumaila SaeedShumaila Saeed, an expert content creator with 6 years of experience, specializes in distilling complex topics into easily digestible comparisons, shining a light on the nuances that both inform and educate readers with clarity and accuracy.