Perpendicular distance
In geometry, the perpendicular distance is the shortest distance between a point and a line, or between two parallel lines or planes, measured along a line that is perpendicular to the given object(s). This measure represents the length of the perpendicular segment connecting the objects and is a core concept in analytic and coordinate geometry for determining spatial relationships.[1] The concept originates in Euclidean space and extends to three dimensions and higher, where it applies similarly to points and planes or parallel planes using vector projections onto normal directions.[2] In non-Euclidean spaces, analogous principles hold, though calculations are more involved.[3] Perpendicular distance is fundamental to theorems in vector geometry and methods like least-squares fitting.[4]Fundamentals
Definition
In Euclidean geometry, the perpendicular distance between a geometric object, such as a point and a line or two parallel lines, is defined as the length of the line segment that connects them and forms a right angle with both objects, thereby representing the shortest possible separation between them.[5] This measure emphasizes orthogonal separation, distinguishing it from oblique distances that would be longer paths between the same points.[6] For a point not lying on a line, the perpendicular distance is specifically the length of the segment from the point to its foot of the perpendicular on the line, where the foot is the unique point of intersection ensuring a 90-degree angle.[5] This construction provides the minimal distance, as any other line segment from the point to the line would form an acute or obtuse angle and thus be longer.[6] Geometrically, one can visualize this by imagining dropping a perpendicular from the point straight down to the line, like a plumb line in architecture, hitting the line at the closest approachable spot. Between two parallel lines, the perpendicular distance is constant and equals the length of any perpendicular segment connecting a point on one line to the other, since all such segments are equal in length due to the lines' uniform separation.[6] This uniformity arises from the parallel nature, where transversals at right angles maintain the same measure regardless of position along the lines.[5]Properties
In Euclidean geometry, the perpendicular from a given point to a given line is unique, meaning there exists exactly one such line segment that is perpendicular to the reference line and passes through the point.[7] This uniqueness follows from the properties of straight lines and the parallel postulate, ensuring no other perpendicular can be drawn under the same conditions. The defining characteristic of a perpendicular distance is its orthogonality: the line segment connecting the point to its foot on the reference line forms a right angle, measuring precisely 90 degrees, with the reference line.[8] This orthogonal relationship distinguishes perpendicular distance from other line segments, as any deviation from 90 degrees would not satisfy the perpendicular condition.[9] Perpendicular distances exhibit invariance under Euclidean isometries, such as translations and rotations, preserving their measure when the geometric configuration is rigidly transformed without scaling.[10] For parallel lines or planes, this invariance implies that the perpendicular distance remains unchanged regardless of the position along the objects, as translations and rotations do not alter relative orientations or separations in Euclidean space.[11] A key relation exists between perpendicular distance and parallelism: the perpendicular distance between two parallel lines or planes is constant throughout their extent, reflecting the uniform separation inherent in parallel configurations.[11] The perpendicular distance connects directly to the Pythagorean theorem through the right triangle it forms with any other line segment from the point to the reference line; in this triangle, the square of the hypotenuse equals the sum of the squares of the perpendicular distance and the adjacent leg, underscoring its role as the shortest path.[12]Calculation Methods
In Two Dimensions
In two-dimensional Euclidean space, the perpendicular distance from a point to a line is a fundamental computation in coordinate geometry, representing the length of the shortest path from the point to the line along a direction normal to the line. This distance is particularly useful for determining positions relative to linear boundaries in the plane. The standard formula for the perpendicular distance d from a point (x_0, y_0) to a line given by the equation ax + by + c = 0 is d = \frac{|a x_0 + b y_0 + c|}{\sqrt{a^2 + b^2}}. This formula arises from the geometry of the line's normal vector (a, b), which is perpendicular to the line and has magnitude \sqrt{a^2 + b^2}. To derive it, consider the point-to-line projection: the numerator |a x_0 + b y_0 + c| measures the signed displacement along the normal direction from the origin to the point relative to the line, normalized by the vector's length to yield the actual distance.[13] For an illustrative example, compute the perpendicular distance from the point (3, 4) to the line $2x + 3y - 6 = 0:- Substitute the coordinates into the numerator: |2(3) + 3(4) - 6| = |6 + 12 - 6| = 12.
- Compute the denominator: \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}.
- Thus, d = \frac{12}{\sqrt{13}} (approximately 3.33 units).
This step-by-step application confirms the formula's utility for specific cases.[13]