Does $\frac{\sin(tx)}{\sin(x)}$ have a name?

Question

Does the following function have a name?

$$\operatorname{boxySine}(t,x) = \begin{cases} \frac{\sin(tx)}{\sin(x)} & x \neq 0 \\ t & \text{otherwise} \end{cases} $$

It appears in describing the position of a point on a circular segment in terms of a position on the chord. For a circular segment of central angle $\theta$ and chord length $c$ the arc (or line segment if $\theta = 0$) can be described in terms of $\operatorname{boxySine}$ as $$ t \in [-1,1] $$ $$ \alpha = \frac{\theta}{2} $$ $$ x = \frac{c}{2}\cdot\operatorname{boxySine}\left(t,\alpha\right) $$ $$ y= -c\cdot\sin\left(\alpha\right)\left(\left(\operatorname{boxySine}\left(\frac{t}{2},\alpha\right)\cdot\cos\left(\frac{\alpha}{2}\right)\right)^{2}-\left(\cos\left(\frac{\alpha t}{2}\right)\cdot \operatorname{boxySine}\left(\frac{1}{2},\alpha\right)\right)^{2}\right) $$

An alternative way to parameterize a circular arc or line segment without using $\operatorname{boxySine}$ would also be an appropriate answer, as it would explain why it might not have a name.

Desmos graph demonstrating that parameterization of an arc

I called it $\operatorname{boxySine}$ because for small values of $t$ the graph of it looks boxy.

Answer 1

To the best of my knowledge, the function boxySine(t,x) does not have a commonly used name. It appears to be a specific function created to parameterize a circular segment in a certain way, rather than a function with a general use or application. However, it's not uncommon for functions to be named after their creators or the context in which they were developed, so it's possible that boxySine has been named informally by the person who created it or others in the field who use it.

That being said, there are other ways to parameterize circular arcs or line segments, such as using trigonometric functions like sine and cosine directly, or using parametric equations with angles and radii. These alternative parameterizations may be more commonly used and have established names, but they may not necessarily be better suited for the specific application or context in which boxySine was developed.