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geometrylinepointnormals

Find point along line where normal extends through another point

Given the line segment AB, how do I find the point Pn where the normal to AB passes through the point P? I also need to know if there is no normal that passes through the point (e.g. the point Q).

Line's normal through point

Solution

  • If R is any point on the normal line passing through P (different from P), then Pn is the point where AB and PR intersect.

    One way to generate point R is to rotate segment AB by 90 degrees and then translate it so that A coincides with P. The the translated B is your R:

    Rx = Px + (By - Ay)
Ry = Py - (Bx - Ax)

    Once you have your point R, it becomes a simple line-line intersection problem, which will give you your Pn (the formulas can be simplified for a specific case of perpendicular lines).

    Then you can easily check whether Pn lies between A and B or not.

    P.S. Note that solution provided in @MBo's answer is a more direct and efficient one (and if you combine and simplify/optimize the formulas required for my answer you will eventually arrive at the same thing). What I describe above might make good sense if you already have a primitive function that calculates the intersection of two lines, say, 

    find_intersection(Ax, Ay, Bx, By, Cx, Cy, Dx, Dy)
// Intersect AB and CD

    In that case finding Pn becomes a simple one-liner

    Pn = find_intersection(Ax, Ay, Bx, By, Px, Py, Px + (By - Ay), Py - (Bx - Ax))

    But if you don't have such primitive function at your disposal and/or care for making your code more efficient, then you might want to opt for a more direct dedicated sequence of calculations like the one in @MBo's answer.