Intersecting Lines Transformation

The intersecting straight lines pairs transformation with using (2×2) matrix is also a pair of intersecting lines. We illustrate this with an example of two lines defined by the equations:

(1.14a)

(1.14b)

In matrix notation, these equations have the form:

(1.15)

or

(1.16)

If a solution of this system exists, the lines intersect, otherwise they are parallel. The solution can be found by inversion of the matrix:

(1.17)

The inverse matrix [ M ] has the following form:

(1.18)

because [ M ] [ M ] ^-1 = [ E ], where [ E ] - the identity matrix. Therefore, the coordinates of two lines intersection point can be found as:

(1.19)

If both lines are transformed with (2×2) matrix of the total that their equations have the form

(1.20a)

(1.20b)

We can show that

(1.21)

We seek the point of lines intersection after the transformation in the same manner as in the case of the baselines:

(1.22)

Using the expressions 1.19 – 1.22, we obtain:

(1.23)

Returning now to the point of baselines intersection [ x_iy_i ] and applying the resulting transformation matrix, we have

(1.24)

Comparison of the equations of the initial and transformed lines intersection points shows that they are identical. So, the original intersection point is transformed exactly into the other intersection point.

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