In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the subtangent and related terms are certain line segments defined using the line tangent

Tangent

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

 to a curve at a given point and the coordinate axes

Cartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

. The terms are somewhat archaic today but were in common use until the early part of the 20th century.

Definitions

Let P = (x, y) be a point on a given curve with A = (x, 0) its projection onto the x-axis. Draw the tangent to the curve at P and let T be the point where this line intersects the x-axis. Then TA is defined to be the subtangent at P. Similarly, if normal to the curve at P intersects the x-axis at N then AN is called the subnormal. In this context, the lengths PT and PN are called the tangent and normal, not to be confused with the tangent line

Tangent

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

 and the normal line which are also called the tangent and normal.

Equations

Let φ be the angle of inclination of the tangent with respect to the x-axis; this is also known as the tangential angle

Tangential angle

In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the x-axis. In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent...

. Then
So the subtangent is
and the subnormal is
The normal is given by
and the tangent is given by

Polar definitions

Let P = (r, θ) be a point on a given curve defined by polar coordinates and let O denote the origin. Draw a line through O which is perpendicular to OP and let T now be the point where this line intersects the tangent to the curve at P. Similarly, let N now be the point where the normal to the curve intersects the line. Then OT and ON are, respectively, called the polar subtangent and polar subnormal of the curve at P.

Polar equations

Let ψ be the angle between the tangent and the ray OP; this is also known as the polar tangential angle. Then
So the polar subtangent is
and the subnormal is