Parallel Lines Transformation

Straight line can be determined by the two vectors defining the coordinates of its endpoints. The location and direction of the line connecting these two points may vary depending on the vectors position.

Result of the conversion of two parallel lines with the (2×2)-matrix will be again two parallel lines. This can be seen by considering the line between the points [ A ] = [ x₁ y₁ ], [ B ] = [ x₂ y₂ ] and parallel to the line running between the points E and F. We must show that by any transformation these lines preserves parallelism. Since AB and EF are parallel, the inclination of the lines AB and EF is defined from

(1.10)

Transform the endpoints of AB, using a matrix transformation (2×2):

(1.11)

The inclination of the A*B* is defined as:

(1.12)

or

(1.13)

Since the inclination m* is independent of x₁, x₂, y₁, y₂, and m, a, b, c and d are the same for EF and AB, then m* is the same for E*F* and A*B*. Thus, parallel lines remain parallel after transformation. This means that the transformation (2×2) parallelogram is converted to another parallelogram.

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