Bézier & Cycloid Curves #1

ID: bez1 (R = 30) C: 188.5 circumference of circle = length of path K: 240 length of cycloid arc (8 * R) CP1: 178.09,399.19 (blue dot) CP2: 321.91,399.19 (blue dot) ID: bez2 (R = 40)');

The top illustration (bez1) shows a cubic Bézier curve (blue) representing a cycloid curve, generated by a point on a circle with radius 30 as it rotates along a straight line without slipping.

This was produced using a JavaScript formula.

After generating several dozen cycloid curves & and their 'best fit' Bézier curves to gather some data, we have possibly come up with a solution to the problem of determining where to place the 2 Control Points for the Bézier curve.

We have determined the following (with cycloid ABOVE the baseline):

In red we show a Bézier curve produced using the values from our 'best fit' research.

Our research has revealed key points:

  1. Control Point 'x' positions are symmetrical. They are the same distance from the Midpoint of the path length
  2. Control Point 'y' positions must be > radius of the circle above the baseline

Two values have been determined for Control Points as follows:

  1. x value = 0.417185960529024
    x coordinate: divide R by this value
  2. y value = 0.673383067889061
    y coordinate: multiply R by this value

These values were derived after dozens of visual tests, and are used in a particular way to determine both the x and y coordinates of the cycloid so that they can be use in a path command.