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Epicycloid vs. Hypocycloid: Know the Difference

Shumaila Saeed
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.
Epicycloid vs. Hypocycloid

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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Shumaila Saeed
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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Shumaila Saeed
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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Shumaila Saeed
Dec 07, 2023
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Comparison Chart

Rolling Motion

Outside another circle
Inside another circle
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Curve Shape

Extended, convoluted curves
Star-like shapes with inward cusps
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Cusp Direction

Outward pointing
Inward pointing
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Appearance

Varies from elongated loops to flower-like
Ranges from simple ellipses to complex stars
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Applications

Gear design with non-standard tooth profiles
Compact and efficient gear systems
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Shumaila Saeed
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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Shumaila Saeed
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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Shumaila Saeed
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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Shumaila Saeed
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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Shumaila Saeed
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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Shumaila Saeed
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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Shumaila Saeed
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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Shumaila Saeed
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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Shumaila Saeed
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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Shumaila Saeed
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 about the cusps of a hypocycloid?

They point inward.
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Shumaila Saeed
Dec 07, 2023

What defines a hypocycloid?

A curve traced by a point on a circle rolling inside another circle.
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Shumaila Saeed
Dec 07, 2023

What is an epicycloid?

A curve traced by a point on a circle rolling outside another circle.
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Shumaila Saeed
Dec 07, 2023

Can an epicycloid be simple in shape?

It tends to be more complex.
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Dec 07, 2023

Is a hypocycloid always star-like?

Often, but it can also be an ellipse.
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Shumaila Saeed
Dec 07, 2023

Are epicycloids used in gear design?

Yes, for non-standard tooth profiles.
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Shumaila Saeed
Dec 07, 2023

What about hypocycloids in gears?

Used in compact and efficient gears.
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Shumaila Saeed
Dec 07, 2023

How do the cusps of an epicycloid point?

Outward.
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Shumaila Saeed
Dec 07, 2023

Does the size ratio affect hypocycloids?

Yes, it alters their shape too.
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Shumaila Saeed
Dec 07, 2023

Can hypocycloids be seen in everyday objects?

Sometimes, in specific gears.
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Shumaila Saeed
Dec 07, 2023

Do epicycloids vary with circle size ratio?

Yes, they change shape with the ratio.
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Shumaila Saeed
Dec 07, 2023

Do children's toys use epicycloids?

Some drawing toys do.
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Dec 07, 2023

What is a complex epicycloid example?

A curve with many loops or petals.
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Dec 07, 2023

Can epicycloids be symmetric?

They often have symmetrical patterns.
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What's a simple hypocycloid example?

A three-lobed curve.
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Dec 07, 2023

Do epicycloids have practical applications?

Yes, in specific mechanical designs.
Shumaila Saeed
Shumaila Saeed
Dec 07, 2023

Are hypocycloids in children's toys?

Yes, in some spirograph toys.
Shumaila Saeed
Shumaila Saeed
Dec 07, 2023

Are hypocycloids always symmetrical?

Generally, but depends on the circle ratio.
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Shumaila Saeed
Dec 07, 2023

Are epicycloids found in nature?

Not typically, they're more mathematical.
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What's a unique use of hypocycloids?

In rotary engines for compact design.
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Dec 07, 2023

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About Author
Shumaila Saeed
Written by
Shumaila Saeed
Shumaila 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.

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