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Math Help - Distance of a point from a triangular surface....

  1. #1
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    Distance of a point from a triangular surface....

    Can someone help me finding a distance of a point (x1,y1,z1) from a triangular surface
    where the distances from three corners to the point are known.
    and three sides of the triange are known.
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  2. #2
    Member Greengoblin's Avatar
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    For a triangle ABC, with sides of length a,b,c respectively, you might be able to solve by letting A lie at the origin of the plane, and B lie in the x-axis. Then the distance of each corner from the point p=(x_1,y_1,z_1) will represent the radii, r_a, r_b, and r_c, of three spheres with centres at A, B, and C respectively.

    The intersection of all three spheres will be the point, p, which might be obtained from finding p=(x_1,y_1,z_1), which satisfies the following three equations for spheres simultaneously:

    x^2+y^2+z^2=r_a^2

    (x-c)^2+y^2+z^2=r_b^2

    (x-c_x)^2+(y-c_y)^2+z^2=r_c^2

    You can calculate the point C=(c_x,c_y,0), knowing that A is at the origin and also knowing the lengths a, b, and c.

    The intersection of the first two spheres should give you the equation for a circle parrallel to the z-y plane. The x value that satisfies both this, and the equation of your third sphere will be your x ordinate.

    Assuming the distance from the triangle to the point is not zero (the point does not lie in the plane) there should be two solutions - one for the point above the plane, and one for the point below. The modulus of either of these (z-ordinates) is the distance.
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  3. #3
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    thanks!

    thank you for the solution.
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