I copied this pretty much directly from another website Define ' m to be the sum of the two circles' radii:m=a+bThe equation for the big circle of radius ' a', since it is centered at the origin, is,x^{2}+y^{2}=a^{2}The equation for the small circle is: (x-m)^{2}+y^{2}=b^{2}The point ' P' at t=0 can be represented in coordinate form by its distance from the origin:P where _{0}=(m-h,0),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 RCSince s=rø, at=bt_{1}ß=t_{1}+tSolving the equation for t_{1}t_{1}=at/bSubstituting into the equation ß=t, ß is expressed in terms of _{1}+tt:ß=at/b+tSince m=a+b
ß=mt/bConsequently, 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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