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:$t\; =\; f(x\_0)\; ,\; csc\; heta\_0\; quad$and the subtangent is:$t\_s\; =\; t\; ,\; cos\; heta\_0\; =\; f(x\_0)\; cot\; heta\_0\; quad$The angle of inclination "θ" is related to the derivative by:$heta\; =\; arctan\; \{df\; over\; dx\}$therefore:$t\_s\; =\; \{\; f(x\_0)\; over\; f\text{'}(x\_0)\; \}.$

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

:$TO\; =\; -\; \{\; ho^2\; over\; ho\text{'}\}.$