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Parametric equation
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===Hypotrochoid=== A [[hypotrochoid]] is a curve traced by a point attached to a circle of radius {{mvar|r}} rolling around the inside of a fixed circle of radius {{mvar|R}}, where the point is at a distance {{mvar|d}} from the center of the interior circle. <gallery> Image:2-circles hypotrochoid.gif|<div class="center">A hypotrochoid for which {{math|1= ''r'' = ''d''}}</div> Image:HypotrochoidOutThreeFifths.gif|<div class="center">A hypotrochoid for which {{math|1= ''R'' = 5}}, {{math|1= ''r'' = 3}}, {{math|1= ''d'' = 5}}</div> </gallery> The parametric equations for the hypotrochoids are: <math display="block">\begin{align} x (\theta) &= (R - r)\cos\theta + d\cos\left({R - r \over r}\theta\right) \\ y (\theta) &= (R - r)\sin\theta - d\sin\left({R - r \over r}\theta\right)\,. \end{align}</math> Some examples: <gallery> Image: Param1a 6 4 1 a2.jpg|<div class="center">{{math|1= ''R'' = 6}} {{math|1= ''r'' = 4}} {{math|1= ''d'' = 1}}</div> Image: Param1a 7 4 1 a4.jpg|<div class="center">{{math|1= ''R'' = 7}} {{math|1= ''r'' = 4}} {{math|1= ''d'' = 1}}</div> Image: Param1a 8 3 2 a3.jpg|<div class="center">{{math|1= ''R'' = 8}} {{math|1= ''r'' = 3}} {{math|1= ''d'' = 2}}</div> Image: Param1a 7 4 2 a4.jpg|<div class="center">{{math|1= ''R'' = 7}} {{math|1= ''r'' = 4}} {{math|1= ''d'' = 2}}</div> Image: Param1a 15 14 1 a14.jpg|<div class="center">{{math|1= ''R'' = 15}} {{math|1= ''r'' = 14}} {{math|1= ''d'' = 1}}</div> </gallery>
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