maths - find vertices when 1 vertex and center point is given in polygon

Question

I want to know if there is any general formula to find out vertices (co-ordinates) of a polygon (3 or more equal sides) when following is given:

Co-ordinates of one of the vertices 
Center point (distance between each vertex and center point is equal)

For example, what would be A and B vertices of following equilateral triangle:

(For equilateral triangle i assume that center point = centroid)

     A
    /\
   /  \                 Center point = (2, 5)
  /____\ 
 B      C (4, 6)

Thanks for help.

Regards

Answer 1

If you have a polygon with equal sides and equal distance from center to all vertices it seems to be a regular convex polygon

EDIT I've found much easier way.

Assume that center of the polygon has coordinates (x_0,y_0) and known vertice has coordinates (x_n,y_n). Also assume that we are considering n-sided polygon.

Coordinates of i-th vertce (0<i<n) can be calculated using this formulae

x_i = x_0+R*cos(a+2*Pi*i/n)
y_i = y_0+R*sin(a+2*Pi*i/n)

where

     _______________________
R = v(x_n-x_0)^2+(y_n-y_0)^2
a = acos((x_n-x_0)/R)

According to your example computations using formula above shows that

A=(4, 6)
B=(0.1339745962155614, 6.2320508075688776)
C=(1.8660254037844377, 2.7679491924311228)

You can check (e.g. using this calculator) that distances between A and B, B and C, C and A are the same and equal to 3.8729833462074166. Also yu can calculate distance between center and each vertice and see that they all will be the same.

==================

That means you can find length of the side of such polygon using this formula a=2Rsin(Pi/n), where R is a distance between center c of your poly and its known vertice p.

   _____________________________
R=v(c.x - p.x)^2 + (c.y - p.y)^2

So you will have a triangle based on center c of your poly and its first p and second s vertices. Since you know coordinates of c and p and length of all sides of this triangle (distance between c and p is R, distance between p and s is a and distance between s and c is again R) you can determine coordinates of s.

Answer 2

The coordinate gives you the distance from each point to the center:

            _______________________________
distance = v (c.x - p.x)^2 + (c.y - p.y)^2

Now that you know the distance, you can construct the polygon. I'm not sure what you mean by "find out vertices", so I'll just assume you want to find the coordinate of an arbitrary vertex (hexagon in this example).

   _____. v
  /   r/\
 /    /  \
 \       /
  \_____/

Just assume it's centered at the origin and subdivide the unit circle into that many "slices", starting from 0:

v(n, k, c, p) = (cos(pi * (n/k)), sin(pi * (n/k))) Unit circle
        * sqrt((c.x - p.x)^2 + (c.y - p.y)^2)  Orig. length
        + (c.x, c.y)                           Move center

This spits out the coordinate of the kth vertex in a n sided polygon centered at c with a point p. So in a very verbose OOP notation:

def v(n, k, c, p):
  z = Point(0, 0)
  d = sqrt((c.x - p.x)^2 + (c.y - p.y)^2) # Distance from
                                          # vertex to center

  z.x = cos(2pi * (n/k))  # Coordinates of point
  z.y = sin(2pi * (n/k))  # on unit circle.

  z.x *= d # Unit circle has a radius of 1.
  z.y *= d # You need a radius of d.

  z.x += c.x # Center the point at c (the center of
  z.y += c.y # the original polygon, as it's at the origin.

  return z

No guarantees that this works ;)