Deriving the Parametric Equations I copied this pretty much directly from another website Define 'm to be the sum of the two circles' radii: m=a+b The equation for the big circle of radius 'a', since it is centered at the origin, is, x2+y2=a2 The equation for the small circle is: (x-m)2+y2=b2 The point 'P' at t=0 can be represented in coordinate form by its distance from the origin: P0=(m-h,0),   where h is the distance from the small circle's center to the point 'P' As the small circle with radius 'b' revolves counterclockwise around the big circle with radius 'a', the coordinates of the point 'P' can be described with the equation: P=m[cos(t), sin(t)]-h[cos(ß), sin(ß)]. Now ß needs to be expressed in terms of t so the parametric equation only has one variable. As the small circle rolls on the big one, it 'unwinds' the same arc length as the big one: arcBC=arc RC Since s=rø, at=bt1 ß=t1+t Solving the equation for t1 t1=at/b Substituting into the equation ß=t1+t, ß is expressed in terms of t: ß=at/b+t Since m=a+b ß=mt/b Consequently, the parametric equations for the epitrochoid are: x=m cos(t)-h cos(mt/b) y=m sin(t) -h sin(mt/b)

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