slider detends by James McCartney

detends.tex

Slider detends 2020 by James McCartney
(https://www.desmos.com/calculator/7txykztr3m)

$q$ is the quantisation or detend spacing:

$$
q = 0.5
$$

$s \in \[0, 1\]$ is the proportion of slider throw that is in the detented zone.

$$
s = 0.2
$$


This is the detent function. 
It allows for detents spaced at intervals defined by $q$ and of size set by $s$, while maintaining an average slope of $1$. 
That is, the function passes through every detent position with $y=x.$

\begin{eqnarray}
f(x) &=\left( \frac {q} {1-s}\right) \left( a_1 - s \cdot \operatorname{floor}(a_1) -c \right)\\
a_1 &= \frac{x}{q} - \frac{s}{2}\\
a_2 &= \frac{x}{q} + \frac{s}{2}\\
b &= a_2 - \operatorname{floor}(a_2)\\
c &= \{b < s : b, 0 \}
\end{eqnarray}