Point coordinate in the form of the x+iy form ax+by+c=0 straight line coefficients through a gap
 The set point with coordinates (X+Yi) The set straight line Coordinates of a mirror point

Calculation of new coordinates of the set point or points which are specularly displayed concerning any set straight line is considered.

The algorithm for finding the coordinates of the mirror point is very simple

1. Determine the point of intersection of the abscissa axis and the line

2. Regarding this point, we make a rotation by an angle equal to the angular coefficient of the line.

How this is done can be found in the material turning points at an arbitrary angle

3. Using the view matrix  $\begin{pmatrix}1 \ \ \ 0 \\ 0 -1\end{pmatrix}$, we obtain a mirror image of the point relative to the abscissa axis.

4. Make a reverse turn (as in paragraph 2)

5. We get the result.

Mathematically, the solution of the problem is a product of three matrices, if the point around which the rotation is made is at the origin

$\begin{pmatrix} x1 \\ y1 \end{pmatrix}=\begin{pmatrix} x \\ y \end{pmatrix}*\begin{pmatrix} cos(\phi) & sin(\phi) \\ sin(\phi) & -cos(\phi) \end{pmatrix}*\begin{pmatrix} cos(\phi) & sin(\phi) \\ -sin(\phi) & cos(\phi) \end{pmatrix}$

Consider the implementation of this algorithm using the example shown in the figure above.

So, we have a point with coordinates C (-2,6) and the equation of the line -18.8x + 47.11-42.35 = 0

Enter the data

The coordinates of point C are written here in this form -2 + 6i

The coefficients of the line are written through the space -18.8 47.11 -42.35

 Given point with coordinates (X: Y) $-2+6i$ Direct line $-18.8*x+(47.11)*y+(-42.35)=0$ Mirror Point Coordinates $2.06150812-4.17753444i$