Turns & Precision

Sep 6, 2025

Turns & Precision

This page formalises the angle unit used across ERG/GRG. Angles are measured in turns (fractions of a full rotation). We also define a binary fixed-point representation for hardware/software precision.

Angle unit (turns)

A full rotation is $1$ turn. Quarter-turn $= \tfrac{1}{4}$, half-turn $= \tfrac{1}{2}$, etc. Trigonometric functions are period-1 in turns: $c(u),\, s(u)$ with $c(0)=1,\ s(0)=0,\ c(\tfrac{1}{4})=0,\ s(\tfrac{1}{4})=1$.

Equal-angle partition (N sectors)

Choose the number of sectors $N$ and set the step $\Delta u = \tfrac{1}{N}$. Sampling angles: $u_k = \tfrac{k}{N}$, $k=0,\dots,N-1$.

Binary fixed-point turns (B bits)

Store angles as $u = \tfrac{k}{2^{B}}$ turns with $k \in \{0,\dots,2^{B}\!-\!1\}$. The angular resolution is $\Delta u_B = 2^{-B}$. To get exactly $N$ equal sectors, pick $N$ dividing $2^{B}$ (e.g. $N=2^{B}$). Then use indices $k = m\,j$ with $m = \tfrac{2^{B}}{N}$, $j=0,\dots,N-1$.

Example with $B = 3$ bits ⇒ $2^{3}=8$ ticks. Increasing $B$ doubles the ticks and the precision.

Why more bits ⇒ more accuracy

Circumference (chords): $ C_N \;=\; 2\,N\,r\, s\!\bigl(\tfrac{1}{2N}\bigr) $ is the inscribed $N$-gon perimeter; as $N$ doubles, the step halves and the error typically drops ≈ $\times 4$.
Area (sectors): $ A_N \;=\; \tfrac{N}{2}\,r^{2}\, s\!\bigl(\tfrac{1}{N}\bigr) $ behaves similarly (step-squared).

Quick reference

Inline: $ \Delta u = \tfrac{1}{N},\quad u_k = \tfrac{k}{N},\quad u = \tfrac{k}{2^{B}},\quad \Delta u_{B} = 2^{-B} $.
Display:

$$ C_{N} \;=\; 2\,N\,r\, s\!\Bigl(\frac{1}{2N}\Bigr), \qquad A_{N} \;=\; \frac{N}{2}\,r^{2}\, s\!\Bigl(\frac{1}{N}\Bigr). $$

Implementation notes

Keep an angle register $k$ modulo $2^{B}$ (wrap-around on $B$ bits).
For LUTs of size $2^{M}$ with $M \le B$, index by $k \gg (B-M)$.
Step-doubling check: compare $N$ vs $2N$ to gauge the remaining gap.