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SPHERICAL CYCLOID
Curve studied by Jean Bernoulli in 1732, and later by Hachette in 1811, and by Reuleaux. |
Cartesian parametrization: Spherical curve, algebraic iff q is rational (degree = 2(numerator + denominator of q)). |
A spherical cycloid is the locus of a point on
a circle rolling without slipping on a fixed circle, the angle between
the two circles remaining constant equal to ;
here, a is the radius of the fixed circle,
that of the moving circle, and xOy the plane where lies the fixed
circle.
When Therefore, the spherical cycloid is a roulette of the motion of a sphere over a sphere. |
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Therefore, except in degenerate cases, the spherical cycloid is also the locus of a point on a cone of revolution that is rolling without slipping on a cone of revolution with the same vertex: these two cones are the cones with vertex W that contain the fixed circle and the moving circle, respectively. | ![]() ![]() |
If q is rational, then the spherical cycloids
are composed of a number of isometric arches equal to the numerator of
q.
If q is irrational, then they are composed of an infinite number
of isometric arches.
The arches meet at cuspidal points, obtained for When In the intermediate case If we change q into q/(q + 1) and
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In
this animation, the circles passing by the vertices of the two cycloids
have the same radius, but not the same altitude.
Special case q
= 1 (base and rolling circles with equal radii):
![]() Cartesian parametrization: |
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![]() The curve is the intersection between the sphere |
The superb models below show the generation of spherical cycloids by the rolling motion of a cone on another cone.
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Each point of the wheels of these bikes describes a spherical epicycloid. |
![]() The relative movements of the conical gears describe spherical epicycloids. |
See the generalization to spherical
trochoids.
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© Robert FERRÉOL, Jacques MANDONNET, Alain ESCULIER 2018