Hypotrochoid. Thomas Wood Math 50C. Definition. - A curve traced by a point P fixed to a circle with radius r rolling along the inside of a larger, stationary circle with radius R at a constant rate without slipping.
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- A curve traced by a point P fixed to a circle with radius r rolling along the inside of a larger, stationary circle with radius R at a constant rate without slipping.
-The name hypotrochoid comes from the Greek word hypo, which means under, and the Latin word trochus, which means hoop.
Wankel Rotary Engine
x-coordinate is defined as (Rcosϴ - rcosϴ) and its y-coordinate is defined as (Rsinϴ - rsinϴ). The radius of the circle created by the center point is (R-r).
The more difficult part is to find equations for a point P around the center.
Adding these equations to the equations for the center of the inner circle gives the parametric equations
for a hypotrochoid.
However, since the point P rotates about the circle traced by the center of the small circle, which has radius (R-r), ϕ is equal to (R-r)ϴ
When r=(R-1), the hypotrochoid draws R loops and has to go from 0 to 2π*r radians to complete the curve. As d increases, the size of the loop decreases. If d ≥ r, there are no longer loops, they become points.
If d=r, the point P is on the circumference of the inner circle and this is a
special case of the hypotrochoid called the hypocycloid. For a
hypocycloid, if r (which is equal to d) and R are not both even or both
odd and R is not divisible by r, the hypocycloid traces a star with R