How to build an intersection in Cities Skylines


How to build the intersection of the planes

Instruction
Let the line be given by the intersection of twoplanes(see fig.), for which their general equations are given. A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0. The desired line belongs to both these planes. Accordingly, it can be concluded that all its points can be found from the solution of the system of these two equations.
Let, for example, the planes be given by the following expressions: 4x-3y4z + 2 = 0 and 3x-y-2z-1 = 0. You can solve this problem in any way convenient for you. Let z = 0, then these equations can be rewritten in the form: 4x-3y = -2 and 3x-y = 1.
Accordingly, "y" can be expressed as follows: y = 3x-1. Thus, the following expressions will occur: 4x-9x + 3 = -2; 5x = 5; x = 1; y = 3-1 = 2. The first point of the desired line is M1 (1, 2, 0).
Now assume that z = 1. From the initial equations we get: 1. 4x-3y-1 + 2 = 0 and 3x-y-2-1 = 0 or 4x-3y = -1 and 3x-y = 3. 2. y = 3x-3, then the first expression will look like 4x-9x + 9 = -1, 5x = 10, x = 2, y = 6-3 = 3. Based on this, the second point has coordinates M2 (2, 3, 1).
If you draw a straight line through M1 and M2, then the problem will be solved. Nevertheless, it is possible to present a more visual way of finding the position of the desired direct equation - the compilation of the canonical equation.
It has the form (x-x0) / m = (y-y0) / n = (z-z0) / p, here {m, n, p} = s are the coordinates of the direct vector of the line. Since in the considered example two points of the desired line were found, its guiding vector s = M2M2 = {2-1, 3-2, 1-0} = {1, 1, 1}. As M0 (x0, y0, z0), you can take any of the points (M1 or M2). Let it be M1 (1, 2, 0), then the canonical equations of the line of intersection of twoplanestakes the form: (x-1) = (y-2) = z.


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