Subtangent

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

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

${\displaystyle \tan \varphi ={\frac {dy}{dx}}={\frac {AP}{TA}}={\frac {AN}{AP}}.}$

So the subtangent is

${\displaystyle y\cot \varphi ={\frac {y}{\tfrac {dy}{dx}}},}$

and the subnormal is

${\displaystyle y\tan \varphi =y{\frac {dy}{dx}}.}$

The normal is given by

${\displaystyle y\sec \varphi =y{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}},}$

and the tangent is given by

${\displaystyle y\csc \varphi ={\frac {y}{\tfrac {dy}{dx}}}{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}.}$