[Math] Calculate angle between two lines

geometry

We have four points: a, b, c and d. We only know length of line cd and line ab. We also know that points c and d have same x coordinate, also points a and b have same x coordinate. Lines cd and ab are parallel. How can I find angle (marked as green) between lines ca and line db? Please provide example of calculations.

enter image description here

Best Answer

Given the points with their coordinates $a_x, a_y ... d_x, d_y$, you can compute the angle of each line to the x-axis (using the arcustangens function), and then just compute the difference between these angles.

So you can compute the angles

$\alpha_0 = atan2(c_y-a_y, c_x-a_x)$

and

$\alpha_1 = atan2(d_y-b_y, d_x-b_x)$

(see the wikipedia link for the definition of the $atan2$ function).

The angle between the lines is then simply $\alpha_1-\alpha_0$

EDIT: The example:

  • $a_x = 100$, $a_y = 5$
  • $b_x = 100$, $b_y = 0$
  • $c_x = 0$, $c_y = 8$
  • $d_x = 0$, $d_y = 0$

Now compute

$\alpha_0 = atan2(c_y-a_y, c_x-a_x) = atan2(3, -100)$ $\alpha_1 = atan2(d_y-b_y, d_x-b_x) = atan2(0, -100)$

According to the cases described in this image from wikipedia:

enter image description here

We have

$\alpha_0 = atan(3/-100) + \pi \approx 3.1116016487329152$

$\alpha_1 = atan(0/-100) + \pi \approx 3.141592653589793$

And the angle is then the difference

$\alpha = \alpha_1 - \alpha_0 \approx 0.02999100485687789 \approx 1.718°$

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