Spherical Cycloids Generated by One Cone Rolling on Another

ratio of base radii:stationary cone/rolling cone

n

2.5

distance of generating point fromcenter of rolling cone's base circle

c

1.5

inclination base circle planeof rolling cone

ω

1.8

roll the cone and trace the cycloid

ϕ

5.

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In this Demonstration, we generate a spherical trochoid with a cone that rolls without slipping on another stationary cone. The generated curve is called a spherical cycloid or spherical trochoid.

Let

A

and

B

be the base circles of the stationary and rolling cones, respectively, with radii

a

and

b

. Let

c

be the distance of the generating point to the center of

B

.

A spherical cycloid is traced by a point on the edge of

B

, that is,

b=c

; a spherical trochoid is traced if

b≠c

.

A closed curve is obtained if

a/b

is rational.

Let

ω

be the angle between the planes of

A

and

B

. For a spherical hypotrochoid,

ω<π/2

, and for a spherical epitrochoid,

ω>π/2

.

In the extreme cases

ω=0

or

ω=π

, we get a planar hypotrochoid or epitrochoid, respectively.