Zef Damen Regular Polygon

Regular Polygon


*How to construct an arbitrary regular N-sided polygon, inscribed in a given circle?*
	As stated in the constructions page, there are many regular polygons that cannot be constructed by the ruler-and-compass rule. One example is the heptagon. The given X- and Y-coordinates of the angular points of the heptagon are used to draw it as accurately as needed. This is a special case. In the general case, the X- and Y-coordinates are derived from the cosine and sine of the angle made by the angular point relative to the (positive) horizontal direction. If r denotes the radius of the circle, and N the total number of angular points, then the coordinates of the i-th angular point (i = 0, 1, ..., N-1) are: X = r * cos (i/N360°) Y = r sin (i/N*360°) With a scientific calculator (for instance the Windows calculator in scientific mode), these values can simply be obtained. As an example, here are the coordinates of a regular 13-sided polygon, inscribed in a circle with radius = 1.0000, and pointing to the right (calculator in the "DEG"-mode):
	i	i/N*360°	X	Y
	0	0.0000°	1.0000	0.0000
	1	27.6923°	0.8855	0.4647
	2	55.3846°	0.5681	0.8230
	3	83.0769°	0.1205	0.9927
	4	110.7692°	–0.3546	0.9350
	5	138.4615°	–0.7458	0.6631
	6	166.1538°	-0.9709	0.2393
	7	193.8462°	–0.9709	–0.2393
	8	221.5385°	-0.7458	–0.6631
	9	249.2308°	–0.3546	–0.9350
	10	276.9230°	0.1205	–0.9927
	11	304.6154°	0.5681	–0.8230
	12	332.3077°	0.8855	–0.4647

As with the heptagon, for different orientations, manipulate x- and y-values, as follows: for the polygon pointing to the left, exchange + and – of all values for the polygon pointing up, exchange x- and y-values for the polygon pointing down, do both, exchange + and –, and exchange x- and y-values If an arbitrary orientation is needed, where the starting angular point makes an angle say α with the horizontal direction, this angle must be added to the angles given above. In this case, the coordinates of the i-th angular point become: X = r * cos (i/N360° + α) Y = r sin (i/N*360° + α) The given coordinate values assume a circumscribed circle with radius 1.0000. For polygons of different sizes, multiply all x- and y-values with the desired radius of the circumscribed circle, according to the given formulas for X and Y above.