Distance from a point to a line

The distance (or perpendicular distance) from a point to a line is the shortest distance from a fixed point to any point on a fixed infinite line in Euclidean geometry. It is the length of the line segment which joins the point to the line and is perpendicular to the line. The formula for calculating it can be derived and expressed in several ways. The distance (or perpendicular distance) from a point to a line is the shortest distance from a fixed point to any point on a fixed infinite line in Euclidean geometry. It is the length of the line segment which joins the point to the line and is perpendicular to the line. The formula for calculating it can be derived and expressed in several ways. Knowing the shortest distance from a point to a line can be useful in various situations—for example, finding the shortest distance to reach a road, quantifying the scatter on a graph, etc. In Deming regression, a type of linear curve fitting, if the dependent and independent variables have equal variance this results in orthogonal regression in which the degree of imperfection of the fit is measured for each data point as the perpendicular distance of the point from the regression line. In the case of a line in the plane given by the equation ax + by + c = 0, where a, b and c are real constants with a and b not both zero, the distance from the line to a point (x0,y0) is:p.14 The point on this line which is closest to (x0,y0) has coordinates: Horizontal and vertical lines In the general equation of a line, ax + by + c = 0, a and b cannot both be zero unless c is also zero, in which case the equation does not define a line. If a = 0 and b ≠ 0, the line is horizontal and has equation y = -c/b. The distance from (x0, y0) to this line is measured along a vertical line segment of length |y0 - (-c/b)| = |by0 + c| / |b| in accordance with the formula. Similarly, for vertical lines (b = 0) the distance between the same point and the line is |ax0 + c| / |a|, as measured along a horizontal line segment. If the line passes through two points P1=(x1,y1) and P2=(x2,y2) then the distance of (x0,y0) from the line is: The denominator of this expression is the distance between P1 and P2. The numerator is twice the area of the triangle with its vertices at the three points, (x0,y0), P1 and P2. See: Area of a triangle § Using coordinates. The expression is equivalent to h = 2 A b { extstyle h={frac {2A}{b}}} , which can be obtained by rearranging the standard formula for the area of a triangle: A = 1 2 b h { extstyle A={frac {1}{2}}bh} , where b is the length of a side, and h is the perpendicular height from the opposite vertex. This proof is valid only if the line is neither vertical nor horizontal, that is, we assume that neither a nor b in the equation of the line is zero.