# What is a non-concave and non-convex polygon called?

I am writing a software function to plot the outer points of an n-sided polygon and I'm having trouble ensuring I use the correct terminology. The function I've written simply renders the calculated points of a predetermined polygon in two dimensional space. These polygons are neither convex nor concave and thus I believe the name I should give the function is Render Uniform Polygon; however, I believe uniform may not be the correct word and could end up in a debate over the name. The function itself performs three actions to achieve the goal:

1. Calculate the angle of each point $$\theta_n$$.
2. Calculate the new position $$P_n$$.
3. Render a 3px by 3px dot at the new position.

The math for calculating $$\theta_n$$ is:

$$\theta_n = \frac{\biggl(\bigl(\frac{360°}{p}\bigr)n + \phi\biggr)\pi}{180°}$$

Where:

• $$p$$ is the number of points in the polygon.
• $$n$$ is the current point.
• $$\phi$$ is the global rotation angle.

Once $$\theta_n$$ has been calculated with the math above, I then use it to calculate the new position with the following:

$$P_{_n{x, y}} = C_{x, y} + (\cos(\theta_n), \sin(\theta_n))r$$

Where:

• $$P_n$$ is current point's position.
• $$C_{x, y}$$ is the center of the polygon.
• $$r$$ is the radius of the polygon.

This method can be used to render any polygon with three sides or more, and since the radius is predefined there isn't a case where the polygon can be convex or concave.

What is a polygon that is neither convex nor concave called?

• All regular polygons are convex. As far as I know, a polygon that is neither convex nor concave is called "neither convex nor concave". I don't know of any special term for that. – Adrian Keister Jan 21 '19 at 18:22
• @AdrianKeister So simply Render Polygon should suffice in this case without causing any confusion? – Taco タコス Jan 21 '19 at 18:23
• Sure! A regular polygon is a polygon, but not the reverse. So if you happen to be constructing regular polygons, a construct_polygon() method is a fine name for that. – Adrian Keister Jan 21 '19 at 18:25
• For me, a concave polygon is one that has at least one angle greater than $\pi$ and a convex one has no angles greater than $\pi$. In that case there are no polygons that are neither concave nor convex. – Ross Millikan Jan 21 '19 at 18:25
• @RossMillikan: Excellent point, I agree! – Adrian Keister Jan 21 '19 at 18:26

## 1 Answer

As far as I can see your polygon is convex. It's a regular polygon whose vertices are equidistant on the circle with radius $$r$$ and center $$C$$.