# Finding vertices of regular polygon

I am trying to find the vertices of a regular polygon using just the number of sides and 2 vertices. After the second vertex, I will make left turns to find each subsequent vertex that follows.

For example, If I have 4 sides, and 2 points, (0,0) & (0,10), how would I go about find the next to point of the square? I know the points would be (10,10) and then (10,0), but I can't think of a formula to accomplish this.

Thank you for any help, and I hope I provided enough information.

• What's your level of mathematical expertise? Do you know trig? Vectors and linear algebra? Complex numbers? – Blue May 5 '15 at 5:17
• Complex number would be best, rotation theorem, real smooth. – Mann May 5 '15 at 5:18
• I have taken trig and linear algebra. – user2587878 May 5 '15 at 5:19
• Use the fact that angle subtended at centre by each side is $(2\pi/n)$ , draw a circumscribing and inscribing circle, see the relations. – Mann May 5 '15 at 5:21
• how do I do that without knowing the radius or center? Is there a way to find the center coordinates with just 2 points and side length? – user2587878 May 5 '15 at 5:34

If two consecutive vertices are given then the angle they make at origin can be found. Call it $\theta=2\pi/n$. Now all points are obtainable from any single point by rotating repeatedly by this angle (centred at origin). As rotation is a linear transformation, this can be achieved by matrix multiplication $$R_\theta=\pmatrix{\cos\theta & -\sin\theta\cr \sin\theta &\cos\theta\cr}$$ Write the co-ordinates of a vertex of the polygon as a column vector $v_1=\displaystyle {x_1\choose y_1}$. Matrix multiplication $v_2=R_\theta v_1, v_3=R_\theta v_2$ etc will give the co-ordinates of all the vertices.
• If the centre is the point (vector) $v_0$ subtract it from given two points (vectors), and appeal to the known case and find the other vertices. Now add the vertex $v_0$ to all the vertices you calculated. – P Vanchinathan May 5 '15 at 6:03