Geometry for Games

Distance between a point and a line

The distance betwen a point P and a line AB is the length of the segment that crosses the line, is perpendicular to it, and passes through P:

To compute it, assume that we have the locations of two points on the line: (x₁,y₁) and (x₂,y₂). In another article of this series, we saw that a vector v that is perpendicular to this line will have the following coordinates:

The distance d is the length of the projection of vector AP onto this vector v:

And this projection in turn is given by:

Here's a Java snippet that computes this distance:

  public double pointToLineDistance(Point A, Point B, Point P) {
    double normalLength = Math.sqrt((B.x-A.x)*(B.x-A.x)+(B.y-A.y)*(B.y-A.y));
    return Math.abs((P.x-A.x)*(B.y-A.y)-(P.y-A.y)*(B.x-A.x))/normalLength;
  }

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