How many equally spaced points are needed around a circle of radius r such that every point is d units apart from eachother?

Question

This question is a bit different than what I have seen here. I know how to calculate the positions such that n points will be equally spaced around a circle however I am trying to solve a different problem. Lets say I am trying to draw a circle with a computer program and I want to intelligently decide how many points will be needed such that the lines between the points are d units a part. This formula would allow for small circles to be drawn with few points but large circles to be drawn with many.

I have tried to solve for d using the following formula however I am not sure I solved it right. In this formula I am solving for dr or the ammount I will need to increment the angle by for each of these points. The number of points needed would then be ceil((2 * PI) / dr)

d = |(r*(cos(dr) - cos(0)))^2 + (r*(sin(dr) - sin(0)))^2|

And I solved this down to dr = (((d/r*r)-2)/-2.0f) but I do not think that that is right.

I am also told the solution may be dr = d / (2.0 * r * PI + 6) but I have no idea where this came from.

Answer 1

The figure below shows one side of the polygon that represents the circle. The radius of your circle is $g$ or $h$ and the side of the polygon is $BC$. You ask that $|BC|=d$. From the upper right triangle we have $\frac d2=h\sin \theta$. This gives that the full angle subtended by the side is $2 \theta=\arcsin \frac d{2h}$. The number of sides is $\frac \pi \theta$. If this doesn't come out even, presumably you want to round up in number of sides and down in the length of the sides.

Answer 2

If I understand your question, you can't always find a number of points $n$ that will be spaced a distance $d$ apart from each other when placed evenly around a circle. After all, suppose you have a given circle of radius $r$. You can make a table where you list various numbers of points $n$ spaced evenly around the circle, and the corresponding distance $d(n)$ between the points. These entries are discrete, rather than continuous. If you have a value of $d$ that doesn't appear in the table, then you can't find $n$ points that are spaced $d$ units apart.

That being said, you can always arrange for all but one pair of neighboring points to be distance $d$ apart, as follows.

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When $n$ points are spaced evenly around the circumference of a circle, the angle between any two pairs of points is $\theta = 2\pi/n$ (in radians). The two points form a triangle with the center of the circle being the third point; in this case, the distance $d$ between the two points is the base of the triangle, and the other two sides have length $r$, the radius of the circle. Split the triangle in half and you have a right triangle (drawing a picture can help).

By trigonometry, you can solve for $d$ in terms of $r$ (the size of the circle) and $\theta$ or $n$ (the number of points):

$$ \frac{d/2}{r} = \sin(\theta/2) = \sin\left(\pi / n\right)$$

Solving for $\theta$ or $n$:

$$\theta = 2\arcsin\left(\frac{d}{2r}\right)$$ $$n = \frac{2\pi}{\theta} = \pi/\arcsin\left(\frac{d}{2r}\right)$$

When $n$ is an integer, you'll have evenly spaced points. When $n$ isn't an integer, you'll have points spaced $\theta$ units apart still, except for the first and last points you place.