OidEG.m

Definitions with Explanations :

Animated Examples :   click on the " o " , then on the figure, and use the animation controls.

You can easily create variations of the Epi.... and Hypo.... examples in the MAIN worksheet.

Cycloids  vs  Trochoids  -

A circle of radius "a" rolls along the x-axis.  "P" is the point on this circle of initial contact.  As the circle rolls, the point "P" traces out a curve.  This is a cycloid .  When the point "P" is moved to a distance "b" from the center of the rolling circle the curve traced out is a trochoid .  The effects are quite different for  b < a  and  b > a.  So, a cycloid is just a trochoid with  b = a.  For simplicity,  a = 1  in the animated examples, and "P" starts at the origin.

The general parametric equations for a cycloid  are :   x = at - a sin(t) ,  y = a - a cos(t)

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The general parametric equations for a trochoid are :   x = at - b sin(t) ,  y = a - b cos(t)

Epicycloids -

A circle of radius "b" rolls on the outside of a circle of radius "a".  "P" is the point on the b-circle of initial contact with the a-circle.  As the b-circle rolls the point "P" traces out a curve in the plane.  This is an epicycloid  .  There will be "a/b" returns contacts of "P" with the a-circle as the b-circle rolls.  So, when  a/b = N  is an integer, we get a closed figure with  N  vertices (in one traversal of the a-circle).  The shape of the epicycloid is totally determined by the single number  N.

The general parametric form of an epicycloid  is :

Hypocycloids  -

A circle of radius "b" rolls on the inside of a circle of radius "a".  "P" is the point on the b-circle of initial contact with the a-circle.  As the b-circle rolls the point "P" traces out a curve in the plane.  This is a hypocycloid .  There will be "a/b" returns contacts of "P" with the a-circle as the b-circle rolls.   So, when  a/b = N  is an integer, we get a closed figure with  N  vertices (in one traversal of the a-circle).  The shape of the hypocycloid is totally determined by the single number  N.

The general parametric form of a hypocycloid  is :

Epitrochoids  -

A circle of radius "b" rotates (counter-clockwise) while its center goes around a circle of radius "a".  The b-circle spins "c" times in a full traversal of the a-circle.  A point "P" on the b-circle traces out a curve in the plane.  This is an epitrochoid .  There will be  N = c - 1  "vertices".  When  a/b = c  the curve is an epicycloid.  If  b < a/c ( = a/(N+1) ) the effect is similar to a trochoid with  b < a.  The case  b > a/c  produces loops as in a trochoid with  b > a.  The shape of the epitrochoid depends on the numbers  N  and  b/a.  Most epitrochoids have  b/a < 1,  but interesting curves can be produced with larger values.  Go back to MAIN and experiment - it's easy!

The general parametric form of an epitrochoid  is :

Hypotrochoids  -

A circle of radius "b" rotates (clockwise) while its center goes around a circle of radius "a".  The b-circle spins "c" times in a full traversal of the a-circle.  A point "P" on the b-circle traces out a curve in the plane.  This is a hypotrochoid .  There will be  N = c + 1  "vertices".  When  a/b = c  the curve is an hypocycloid.  If  b < a/c ( = a/(N-1) )  the effect is similar to a trochoid with  b < a.  The case  b > a/c  produces loops as in a trochoid with  b > a.  The shape of the hypotrochoid depends on the number  N  and  b/a.  Most hypotrochoids have  b/a < 1,  but interesting curves can be produced with larger values.  Go back to MAIN and experiment - it's easy!

The general parametric form of a hypotrochoid  is :