Thank you in advance for helping.
Given a polygon with $n$ vertices, $$P = \begin{bmatrix} x_{1} & x_{2} & ... & x_{n} \\ y_{1} & y_{2} & ... & y_{n} \end{bmatrix}$$ how does one determine:
1) The axes of symmetry, if there are any
2) Radial (rotational) symmetry, if it exists
3) Any other type of symmetry (or something $close$ to a symmetry, if such a thing exists)
Methods involving linear algebra and/or group theory are more than welcome.
And if you can suggest, any thoughts for finding symmetries in three-dimensional polygons? (Not sure what they're called...) $$P = \begin{bmatrix} x_{1} & x_{2} & ... & x_{n} \\ y_{1} & y_{2} & ... & y_{n} \\ z_{1} & z_{2} & ... & z_{n} \end{bmatrix}$$
An invertible affine map is a symmetry of the polygon iff it permutes the vertices. Three non-collinear vertices will be enough to determine the map. It's convenient to use homogeneous coordinates, so the vertices are represented as $\pmatrix{x_i\cr y_i\cr 1\cr}$ and the transformation is multiplication by a $3 \times 3$ matrix with last row $[ 0\; 0\; 1]$. Then the transformation that takes a given three vertices $v_1$, $v_2$, $v_3$ to another three vertices $v'_1$, $v'_2$, $v'_3$ can be found as the matrix $V' V^{-1}$, where $V$ is the matrix formed from the columns corresponding to $v_1, v_2, v_3$, and $V'$ the matrix formed from the columns corresponding to $v'_1, v'_2, v'_3$.