Cycloid - Related Curves

Related Curves

Several curves are related to the cycloid.

  • Curtate cycloid: Here the point tracing out the curve is inside the circle, which rolls on a line.
  • Prolate cycloid: Here the point tracing out the curve is outside the circle, which rolls on a line.
  • Trochoid: refers to any of the cycloid, the curtate cycloid and the prolate cycloid.
  • Hypocycloid: The point is on the edge of the circle, which rolls not on a line but on the inside of another circle.
  • Epicycloid: The point is on the edge of the circle, which rolls not on a line but on the outside of another circle.
  • Hypotrochoid: As hypocycloid but the point need not be on the edge of its circle.
  • Epitrochoid: As epicycloid but the point need not be on the edge of its circle.

All these curves are roulettes with a circle rolled along a uniform curvature. The cycloid, epicycloids, and hypocycloids have the property that each is similar to its evolute. If q is the product of that curvature with the circle's radius, signed positive for epi- and negative for hypo-, then the curve:evolute similitude ratio is 1 + 2q.

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

Read more about this topic:  Cycloid

Famous quotes containing the words related and/or curves:

    “No being exists or can exist which is not related to space in some way. God is everywhere, created minds are somewhere, and body is in the space that it occupies; and whatever is neither everywhere nor anywhere does not exist. And hence it follows that space is an effect arising from the first existence of being, because when any being is postulated, space is postulated.”
    —Isaac Newton (1642–1727)

    “One way to do it might be by making the scenery penetrate the automobile. A polished black sedan was a good subject, especially if parked at the intersection of a tree-bordered street and one of those heavyish spring skies whose bloated gray clouds and amoeba-shaped blotches of blue seem more physical than the reticent elms and effusive pavement. Now break the body of the car into separate curves and panels; then put it together in terms of reflections.”
    —Vladimir Nabokov (1899–1977)