hypotrochoids

6

Vintage depiction of the patterns made by Spirograph, geometric drawing toy that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids.

2

I used to have the original Spirograph set.  And I don’t know where that went.  Imagine my joy years later when I found the Hypotrochoid Art Set at Domus NYC.  It was something like $9.99 there and it gave me hours of joyous nostalgia.   

Plus, I always like visiting Domus, which is at 413 West 44 Street at 9th Ave.   The ladies who run the place, Louisa and Nikki, are just the kindest, sweetest souls ever.   They’re well-traveled and whimsical, with an excellent aesthetic sense.    More on Domus in a future post, but I’ve found so many gifts for loved ones there and they’ve always been very well-received.  :-)

4

Hypotrochoid

Images1-2-3-4: a:b:c = 4:2:½;  a:b:c = 4:1:2;  a:b:c = 4:1:3;  a:b:c = 4:1:4

(Gif: Hypotrochoid by Tokioka)

And I illustrated some pictures to explain for the equation above (The equation of the trace point P) :

When the rolling curve is a circle and the fixed curve is either a line or a circle, A specific subset of a roulette (the trace point P) Will be found. In order to develop the parametric equations for this curve, we need to be able to pinpoint the position of our trace point (P) at any given time.

  • We have OP = OO’ + O’P ,  ( i, j, OP, OO’, O’P are vectors)
  • OO’ = (a-b).Sinθ.(i) + (a-b).Cosθ.(j) – (1)

          O’P = - PO’ = - [c.Sin(π-β).(i) + c.Cos(π-β).(j)]

            = - c.[Sin(β).(i) - Cos(β).r(j) ] = c.[Cos(β).(i) - Sin(β).(j)]

  • O’P = Cos[(a-b)/b]*θ.(i) - Sin[(a-b)/b]*θ.(j)] –(2)

     (1) & (2) => OP = [(a-b).Sinθ + c.Cos[(a-b)/b]*θ].(i)

                                 + [(a-b).Cosθ - c.Sin[(a-b)/b]*θ].(j) = X.i + Y.j.

This leads us to the parametric equations:

X = (a-b).Sinθ + c.Cos[(a-b)/b]*θ

Y = (a-b).Cosθ - c.Sin[(a-b)/b]*θ

Finding the relationship between arc length, angle, and radius by the same length arcs: β.b = θ.a - θ.b