Given the parametric equations for finding the rectangular coordinates of a cycloid,
$x = r(t - \sin{t})$
$y = r(1 - \cos{t})$
where $r$ is the radius and $t$ is the angular displacement.
Using the above equations, I created a spreadsheet (MS Excel) to generate a table of $X$ and $Y$ values for a specified radius. This table is then imported into a CAD/CAM application for additional design work.
This solution needs to be modified to allow a user to specify a line segment length instead of the displacement angle ($t$). Instead of entering the incremental displacement angle we want to enter the line segment length between the plotted points. Points plotted using the angular displacement results in inconsistent segment lengths. The inconsistent segment lengths cause some challenges in our downstream process.
How can I modify my equations to accept line segment length as input instead of angular displacement?
The arc length $s$ and displacement angle $t$ of the cycloid, and radius $r$ of the generating circle, are related via $$s=4r(1-\cos\frac{t}{2})$$ From this formula you can find $t$ and then substitute to find $x$ and $y$.