# Hypotrochoid vs. Epitrochoid: What's the Difference?

By Aimie Carlson & Janet White || Updated on May 22, 2024
Hypotrochoid is a curve traced by a point attached to a circle rolling inside another circle, while epitrochoid is a curve traced by a point attached to a circle rolling outside another circle.

## Key Differences

Hypotrochoid is a type of roulette curve generated when a fixed point on a smaller circle rolls inside a larger, stationary circle. This point traces out a path that depends on the relative sizes of the circles and the distance of the point from the center of the rolling circle. Epitrochoid, on the other hand, is formed when a fixed point on a smaller circle rolls around the outside of a larger, stationary circle. The path traced by the point also depends on the size ratio of the circles and the distance from the center of the rolling circle to the tracing point.
Hypotrochoids are usually more complex and can create intricate internal patterns, whereas epitrochoids often result in smoother, outward-facing curves. Both types of curves are used in various fields, including mechanical engineering, art, and mathematics, but they have distinct applications due to their differing geometric properties.
While hypotrochoids involve internal rolling and tend to stay within the confines of the larger circle, epitrochoids involve external rolling, resulting in curves that extend outward beyond the boundary of the larger circle.
Hypotrochoids can create star-like or flower-like patterns based on the fixed point's position relative to the center, while epitrochoids tend to produce more looped and circular patterns, often seen in cycloidal gear designs.

## Comparison Chart

### Definition

Curve traced by a point on a circle rolling inside another circle
Curve traced by a point on a circle rolling outside another circle
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### Rolling Mechanism

Inside a larger circle
Outside a larger circle

### Common Applications

Spirograph designs, mathematical curves
Gear mechanisms, rotary engine designs

### Pattern Complexity

Typically more intricate and internal
Generally smoother and outward-facing

### Path Tracing

Stays within the larger circle
Extends beyond the larger circle

## Hypotrochoid and Epitrochoid Definitions

#### Hypotrochoid

A curve formed by a point on a circle rolling inside another circle.
The artist used a hypotrochoid pattern to create the intricate design.
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#### Epitrochoid

Curve type used in cycloidal gears.
Epitrochoid patterns are common in gear mechanisms.

#### Hypotrochoid

A type of roulette curve with complex, star-like patterns.
The Spirograph toy utilizes hypotrochoid principles.

#### Epitrochoid

A curve formed by a point on a circle rolling outside another circle.
The epitrochoid curve was crucial in the rotary engine design.

#### Hypotrochoid

Geometric path dependent on circle size ratio.
Different radii combinations produce unique hypotrochoid shapes.

#### Epitrochoid

Produces looped and circular paths.
The epitrochoid design created a smooth, looping pattern.

#### Hypotrochoid

Curve used in decorative and artistic designs.
The bracelet's pattern was inspired by hypotrochoid curves.

#### Epitrochoid

Dependent on the ratio of circle sizes.
The epitrochoid's shape varied with different circle radii.

#### Hypotrochoid

Mathematical curve explored in engineering.
Engineers study hypotrochoids for mechanical designs.

#### Epitrochoid

Seen in various mechanical and engineering applications.
The epitrochoid geometry optimized the machine's efficiency.

#### Hypotrochoid

(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

#### Epitrochoid

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

#### Hypotrochoid

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.

#### Epitrochoid

A kind of curve. See Epicycloid, any Trochoid.

## FAQs

#### What applications utilize epitrochoids?

Epitrochoids are used in gear mechanisms and rotary engine designs.

#### How is an epitrochoid different from a hypotrochoid?

An epitrochoid is formed by a point on a circle rolling outside another circle, unlike a hypotrochoid, which rolls inside.

#### Can hypotrochoids be used in engineering?

Yes, hypotrochoids are studied for mechanical and engineering applications.

#### Where are hypotrochoids commonly used?

Hypotrochoids are commonly used in Spirograph toys, art designs, and mathematical studies.

#### What is a hypotrochoid?

A hypotrochoid is a curve formed by a point on a circle rolling inside another circle.

#### What shapes do hypotrochoids create?

Hypotrochoids often create intricate, star-like patterns.

#### Do epitrochoids have practical uses in machinery?

Yes, epitrochoids are integral in the design of certain gears and rotary engines.

#### What kind of patterns do epitrochoids form?

Epitrochoids usually form smoother, looped, and circular patterns.

#### Do both curves depend on circle size ratios?

Yes, the shapes of both hypotrochoids and epitrochoids depend on the relative sizes of the circles involved.

#### Can epitrochoids extend beyond the larger circle's boundary?

Yes, epitrochoids extend beyond the boundary of the larger circle.

#### Can epitrochoid curves be seen in engine designs?

Yes, epitrochoid curves are important in certain rotary engine designs.

#### What kind of mathematical curves are hypotrochoids?

Hypotrochoids are a type of roulette curve.

#### What is the main difference in the rolling mechanism between the two curves?

Hypotrochoids roll inside a larger circle, whereas epitrochoids roll outside a larger circle.

#### Do epitrochoids feature in cycloidal gear design?

Yes, epitrochoids are often used in cycloidal gear design.

#### Are hypotrochoids used in artistic designs?

Yes, hypotrochoids are frequently used in decorative and artistic designs.

#### Are hypotrochoids more complex than epitrochoids?

Typically, yes, hypotrochoids can create more intricate internal patterns compared to epitrochoids.

#### What defines the complexity of a hypotrochoid pattern?

The complexity of a hypotrochoid pattern is defined by the circle size ratio and the point's distance from the center.

#### Which curve type involves rolling inside a stationary circle?

Hypotrochoids involve rolling inside a stationary circle.

#### Which curve is characterized by outward-facing loops?

Epitrochoids are characterized by outward-facing loops and circular paths.

#### Which curve is associated with Spirograph toys?

Hypotrochoids are associated with Spirograph toys.
About Author
Written by
Aimie Carlson
Aimie Carlson, holding a master's degree in English literature, is a fervent English language enthusiast. She lends her writing talents to Difference Wiki, a prominent website that specializes in comparisons, offering readers insightful analyses that both captivate and inform.
Co-written by
Janet White
Janet White has been an esteemed writer and blogger for Difference Wiki. Holding a Master's degree in Science and Medical Journalism from the prestigious Boston University, she has consistently demonstrated her expertise and passion for her field. When she's not immersed in her work, Janet relishes her time exercising, delving into a good book, and cherishing moments with friends and family.