In a hypotrochoid let r1 be the radius of the inner circle and r2 be the radius

Question

In a hypotrochoid let r1 be the radius of the inner circle and r2 be the radius of the outter circle. If r1 and r2 are whole numbers show the explicit formulas for the number of cycles and the number of petals in the shape as functions of r1 and r2. If r1 and r2 are not whole numbers - for example, if they are fractions - discuss how to find the number of cycles and petals in that case. What’s the relationship between a shape created by r1 = 1 and r2 = 3 versus a shape created by r1 = 1/5 and r2 = 3/5?

Explanation / Answer

Let r1 < r2 where both are whole numbers;

so lets define k = r1/r2; and reduce it to minimum fraction let say

r1/r2 = p/q where common divisor for p and q is 1;

here p is the number of petals and q is number of cycles inner circle will take before completing the cycle;

Suppose We have r1 = 40, r2 = 104, and therefore k = 5/13. The number of petals will be the numerator to k assuming it's been reduced as far as possible, and the number of cycles the gear makes will be its denominator. r2;

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