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A hypotrochoid is a roulette traced by a point 
attached to a circle
of radius 
rolling around the inside of a fixed circle of radius
, where 
is a distance 
from the center of the interior circle. The parametric
equations for a hypotrochoid are
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(1)
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(2)
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A polar equation can be derived by computing
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(3)
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(4)
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Here, the parameter 
is not the polar angle 
but is related to it by
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(5)
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To get 
cusps in the hypotrochoid, 
, because then 
rotations of 
bring the point on the edge back to its starting position.
Special cases of the hypotrochoid are summarized in the table below.
| curve | special values |
| ellipse |  |
| hypocycloid |  |
| rose curve |  |
The arc length, curvature, and tangential angle are
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(6)
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 |  | ![(b^3-(a-b)h^2+(a-2b)bhcos((at)/b))/(|a-b|[b^2+h^2-2bhcos((at)/b)]^(3/2))](/images/equations/Hypotrochoid/Inline32.svg) |
(7)
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 |  | ![t(1-a/(2b))+cot^(-1)[(b-h)/(b+h)cot((at)/(2b))],](/images/equations/Hypotrochoid/Inline35.svg) |
(8)
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where 
is an elliptic integral of the second
kind.
