Hypotrochoid Curves are produced by a Point on a gear that rolls (orbits) around the inside of the fixed gear. There are two sets of Hypotrochoid Curves.
This topic shows Hypotrochoid Curves. The shape of the Hypotrochoid curve depends on the ratio of the teeth. With gears, the ratio of the teeth is always a rational number. Therefore, the curve will always repeat eventually even if the gear ratio is a ratio of two prime numbers, for example. 

The Revolving Gear is half the diameter of the fixed gear..The revolving gear is inside, and is half the diameter of the fixed outside gear. There are four curves One curve is given by a point on the pitchcircle  it is the Hypocycloid In this case, when the ratio of teeth is 1:2, it is a straight line. Hypotrochoid Curves In this case the three other curves are ellipses. Two curves are given by points outside the pitchcircle One curve is given by a point inside the pitchcircle 

The revolving gear inside, is 2.3 the diameter of the fixed outside gear. Hypoycloid The image shows one trace point that is on the pitchcircle It is the Hypocycloid. Hypotrochoid The image shows two curves of points outside the pitchcircle. You can see the curve is approximately a square. You can use the EulerSavary Equation to calculate the best distance from the centre of the orbiting gear to give a curve with the best 'straightline' curve.


Gear Ratio is 48:18 
Hypocycloid One curve that is given by a point on the pitchcircle  it is the Hypocycloid The Gear Ratio is 48:18. 

Straight Line Curve with Hypocycloid Gears  Internal MeshThe Video shows the TracePoint follows a perfectly straight line. You can see that the smaller gear has half the teeth of the other. This is the standard StraightLine Epicyclic GearPair configuration. 