curate cycloid

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curate cycloid

A small wheel rolls along a flat surface, with a point inside it tracing a curate cycloid.

Definition
  1. Noun:
    • A type of cycloid: A "curate cycloid" is a specific curve in mathematics. It is generated by tracing the path of a point that is located inside a circle as that circle rolls along a straight line without slipping.
Usage
  • The term "curate cycloid" is used exclusively in technical contexts, primarily in geometry, mathematics, and physics, to describe this specific type of roulette curve.
  • It is often discussed in contrast to a "common cycloid" (point on the circle's rim) and a "prolate cycloid" (point outside the circle).
Examples
  • Noun:
    • The path traced by a point on a spoke of a rolling wheel is a curate cycloid.
    • In the diagram, the inner curve represents a curate cycloid.
Advanced Usage
  • Mathematical Parameterization: The curate cycloid can be defined by parametric equations. If the rolling circle has radius and the generating point is at a distance from the center (where ), its coordinates are given by: and , where is the angle of rotation.
  • Physical Applications: The curate cycloid can model the motion of certain points in gear systems or planetary motion models.
Variants and Related Words
  • Cycloid (n): The general family of curves traced by a point on a circle rolling along a line. A "curate cycloid" is a subtype.
  • Common cycloid (n) or Standard cycloid (n): The cycloid generated by a point on the of the rolling circle.
  • Prolate cycloid (n): The cycloid generated by a point the rolling circle.
  • Hypocycloid (n): A curve generated by a point on a circle rolling another fixed circle.
  • Trochoid (n): A more general term for roulette curves, which includes cycloids (curate, common, and prolate).
Synonyms
  • Short cycloid: An alternative, though less common, name for a curate cycloid, referring to the fact that its loops are shorter than those of a prolate cycloid.
  • Contracted cycloid: Another historical synonym.
Related Terms and Concepts
  • Roulette (n): In geometry, a curve traced by a point attached to one curve as it rolls upon another fixed curve. A cycloid is a roulette where the fixed curve is a straight line.
  • Cusp (n): A sharp point on a curve. A common cycloid has cusps, but a curate cycloid does not; it is a smooth, wavy curve.
curate cycloid

A small wheel rolls along a flat surface, with a point inside it tracing a curate cycloid.

Noun
  1. a cycloid generated by a point inside the rolling circle