Cycloid  x = a t – b sin t y = a – b cos t where a and b are positive, a ≥ b

suggest you give Desmos a try. Try to draw some conclusions on how each parameter (or the interaction of the parameters) effects the shape of the curve. Include some example graphs to illustrate your conclusions. The amount of credit you receive depends a lot on the quality of your answer. So, if you are running short on time, it’s better to do 2 of these really well than do a sloppy job on all

3. Cycloid  x = a t – b sin t y = a – b cos t where a and b are positive, a ≥ b (hint: start with them being the same, then investigate when a is bigger) Lissajous curves (Lissajous figures)  x = sin a t y = cos b t where a and b are positive (hint: start with integers, then consider a few non-integers) Epicycloid  x = (R + r) cos t – r cos  R+r r t y = (R + r) sin t – r sin  R+r r t where R > r are positive (hint: consider where R = k*r, so R = 2 r, R = 3 r, R = 4 r, etc.)

Hypocycloid  x = (R – r) cos t + r cos  R-r r t y = (R – r) sin t – r sin  R-r r t where R > r are positive (hint: consider where R = k*r, so R = 2 r, R = 3 r, R = 4 r, etc.) Rose curves r = cos n d θ where n and d are both positive integers

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