Modulo operator to short-cut counter-clockwise degrees I am trying to short-cut the following with modulo, but it is not working for 90 and 270 degrees. What is the correct short-cut?
\begin{equation}
\delta'=
    \begin{cases}
        \delta & \text{if } 0 \le \delta \le 90 \\\\
        180 - \delta & \text{if } 90 \lt \delta \le 180 \\\\
        \delta - 180 & \text{if } 180 \lt \delta \le 270 \\\\
        360 - \delta & \text{if } 270 \lt \delta \le 360 \\\\
    \end{cases}
\end{equation}
Trying: \delta % 90




delta
expected delta-prime
calculated
equal?




0
0
0
TRUE


30
30
30
TRUE


45
45
45
TRUE


60
60
60
TRUE


90
90
0
FALSE


105
75
75
TRUE


130
50
50
TRUE


150
30
30
TRUE


180
0
0
TRUE


200
20
20
TRUE


220
40
40
TRUE


230
50
50
TRUE


270
90
0
FALSE


280
80
80
TRUE


300
60
60
TRUE


320
40
40
TRUE


330
30
30
TRUE


350
10
10
TRUE


360
0
0
TRUE



 A: Short answer:
$$
\delta' = \bigl\lvert \operatorname{mod}(x - 90, 180) - 90 \bigr\rvert.
$$
The graph of $\delta'$ as a function of $\delta$ is a triangle wave. (I'm assuming that this function is periodic every $360$ degrees, but you can restrict the domain to just $[0, 360]$ if you desire.)

A: It appears that you are looking for a way to specify the reference angle for a given angle $\delta$ expressed in degrees.
Perhaps you should consider$\mod 180$ rather than $\mod 90$.
The reference angle is the positive angle between $\delta$ and the nearest whole multiple of $180$.
There is a unique integer $n$ such that $180n\le\delta<180(n+1)$. That is, such that $n\le\frac{\delta}{180}<n+1$. Using the floor function, we have that $n=\lfloor\frac{\delta}{180}\rfloor$.
So we have $180\lfloor\frac{\delta}{180}\rfloor\le\delta<180\left(\lfloor\frac{\delta}{180}\rfloor+1\right)$
The reference angle $\delta^\prime$ is the smaller of the two positive distances between delta and the two extremes of the inequality.
That is, $\delta^\prime$ is the smaller of $\delta-180\lfloor\frac{\delta}{180}\rfloor$ and $180\left(\lfloor\frac{\delta}{180}\rfloor+1\right)-\delta$.
This gives the definition of the reference angle as follows:
$$ \delta^\prime=\min\left\{\delta-180\left\lfloor\frac{\delta}{180}\right\rfloor,180-\left(\delta-180\left\lfloor\frac{\delta}{180}\right\rfloor\right)\right\} $$
Check and you will find that for angles such as $\delta=90^\circ$ and $\delta=270^\circ$, the value of $\delta^\prime$ will be $90^\circ$.
