How to determine if a set of coordinates are vertices of a regular polygon?

Question

I have a working class that generates regular polygons given: polygon center and polygon radius and number of sides. Implementation details of the two private member functions here.

The class interface looks like this:

class RegularPolygon: public Closed_polyline{
public:
    RegularPolygon(Point c, int r, int n)
        : center(c), radius(r), sidesNumber(n) 
    { generatePoly(); }

private:
    Point center;
    int radius;
    int sidesNumber;
    void generatePoly();
    void rotateCoordinate(Point& axisOfRotation, Point& initial, 
                          double angRads, int numberOfRotations);
};

Problem:

I am asked to implement a second way of generating regular polygons by using a set of coordinates¹. The constructor needs firstly to perform a validity check of the passed coordinates:

RegularPolygon(vector<Point>& vertices)
   :center(), radius(), sideNumber()
{
   // validity check of the elements of vertices
}

My initial thought is to:

Check if each pair of coordinates produces the same side length.
Check for each lines'(generated by a pair of coordinates) relative orientation. (they should be at an angle 360/polygon sides, from each other)

Question:

~~How could I check if all lines are properly oriented, i.e. their relative orientation?~~ solved
Is there any standard algorithm that can determine if a set of coordinates are vertices of a regular polygon?

Note:

After checking [1] and all the question and answers regarding generating coordinates. I didn't found what I'm searching for.

^{1 In clockwise sequence, passed with the vector: vertices}

All the additional files for compilation could be found: here. The FLTK could be found here.

@Bartek Banachewicz you mean some difference should be smaller than `<` `epsilon` or something similar? — Ziezi, Sep 23 '15 at 14:12
Yes, precisely. IOW, you can only answer "this set of vertices forms a regular polygon with a precision of 0.00...01" — Bartek Banachewicz, Sep 23 '15 at 14:12

score 5 · Accepted Answer · edited Apr 13 '17 at 12:33

5

Your task would be a lot simpler if you could find the center of your polygon. Then you would be able to check the distance from that center to each vertex to verify the placement of the vertices on the circle, and also to check the angles from the center to each individual vertex.

Fortunately, there is an easy formula for finding the center of a polygon: all you need to do is averaging the coordinates in both dimensions. With the center coordinates in hand, verify that

The distance from the center to each vertex is the same, and
The angle between consecutive vertices is the same, and that angle is equal to 2π/N radians

These two checks are sufficient to ensure that you have a regular polygon. You do not need to check the distances between consecutive vertices.

edited Apr 13 '17 at 12:33

Community

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answered Sep 23 '15 at 14:13

Sergey Kalinichenko

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@simplicisveritatis You don't need to rotate your polygon, simply average coordinates of its vertices in each dimension. The formula works regardless of the orientation of the polygon. – Sergey Kalinichenko Sep 23 '15 at 14:19
1

_plus one_, that is brilliant! – Ziezi Sep 23 '15 at 14:20

How to determine if a set of coordinates are vertices of a regular polygon?

Problem:

Question:

Note:

1 Answers1