# Subtangent

In geometry, the subtangent is the projection of the tangent upon the axis of abscissas (i.e., the x-axis).

Tangent here specifically means a line segment which is tangential to a point P on a curve and which intersects the x-axis at point Q. The line segment PQ is the tangent, and the length of PQ is also called the "tangent".

Draw a line through P parallel to the axis of ordinates (a.k.a. y-axis). This line intersects the x-axis at P' . Then line P'Q is the "subtangent", and its length is also called the subtangent.

Let θ be the angle of inclination of the tangent with respect to the x-axis. Let the curve be described by y=f(x), let x0 be the abscissa of point P, and let θ0 be the angle of inclination of the tangent of P. Then this tangent of P is

$\displaystyle t = f(x_0) \, \csc \theta_0 \quad$

and the subtangent is

$\displaystyle t_s = t \, \cos \theta_0 = f(x_0) \cot \theta_0 \quad$

The angle of inclination θ is related to the derivative by

$\displaystyle \theta = \arctan {df \over dx}$

therefore

$\displaystyle t_s = { f(x_0) \over f'(x_0) }.$

## The subtangent in polar coordinates

In polar coordinates, the tangent to a curve can be specifically defined as a line segment, tangential to the curve, which extends from the given point P on the curve to a point T, such that line TO is perpendicular to line OP, where O is the origin. Then "tangent" specifically also means the length of PT, and the subtangent is the line TO, or -- interchangeably -- the length of line TO.

The subtangent can be found to be

$\displaystyle TO = - {\rho^2 \over \rho'}.$