Definition

Noun: A hypocycloid is a specific type of roulette curve. It is the path traced by a fixed point on the circumference of a smaller circle as that circle rolls without slipping along the inside of a larger, fixed circle.

Usage

The term is used in geometry and mathematics to describe this precise curve. It is a specific instance of a trochoid. * The path of that point on the gear forms a hypocycloid. * A hypocycloid with four cusps is called an astroid.

Advanced Usage

• Special Cases: When the radius of the rolling circle is an integer fraction of the fixed circle's radius (e.g., 1/2, 1/3, 1/4), the resulting hypocycloid is a closed curve with a number of cusps equal to that denominator.
• Applications: Hypocycloids and their related curves (epicycloids, trochoids) have applications in gear design, specifically in the shape of gear teeth for certain types of gears to ensure smooth transmission of motion.

Variants and Related Words

• Epicycloid (noun): The curve traced by a point on a circle rolling another circle. This is the direct counterpart to a hypocycloid.
• Trochoid (noun): A more general term for the curve traced by a point fixed to a circle rolling along a straight line or another circle. Hypocycloids and epicycloids are specific types of trochoids.
• Astroid (noun): A specific type of hypocycloid with four cusps, formed when the rolling circle's radius is 1/4 of the fixed circle's radius.
• Deltoid (noun): A specific type of hypocycloid with three cusps, formed when the rolling circle's radius is 1/3 of the fixed circle's radius.

Synonyms

• Roulette curve (general category)
• Trochoidal curve (general category)

Related Idioms or Phrases

• None. This is a technical mathematical term and is not used in idiomatic expressions.

Noun

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