Ingeometry, thesubtangent and related terms are certain line segments defined using the linetangent to a curve at a given point and thecoordinate axes. The terms are somewhat archaic today but were in common use until the early part of the 20th century.

LetP = (x, y) be a point on a given curve withA = (x, 0) its projection onto thex-axis. Draw the tangent to the curve atP and letT be the point where this line intersects thex-axis. ThenTA is defined to be thesubtangent atP. Similarly, if normal to the curve atP intersects thex-axis atN thenAN is called thesubnormal. In this context, the lengthsPT andPN are called thetangent andnormal, not to be confused with thetangent line and the normal line which are also called the tangent and normal.

Letφ be the angle of inclination of the tangent with respect to thex-axis; this is also known as thetangential angle. Then

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

So the subtangent is

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

and the subnormal is

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

The normal is given by

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

and the tangent is given by

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

LetP = (r, θ) be a point on a given curve defined bypolar coordinates and letO denote the origin. Draw a line throughO which is perpendicular toOP and letT now be the point where this line intersects the tangent to the curve atP. Similarly, letN now be the point where the normal to the curve intersects the line. ThenOT andON are, respectively, called thepolar subtangent andpolar subnormal of the curve atP.

Letψ be the angle between the tangent and the rayOP; this is also known as the polar tangential angle. Then

${\displaystyle \tan \psi =\frac{r}{\frac{dr}{d\theta }}=\frac{OP}{ON}=\frac{OT}{OP}.}$

So the polar subtangent is

${\displaystyle r\tan \psi =\frac{r^2}{\frac{dr}{d\theta }},}$

and the subnormal is

${\displaystyle r\cot \psi =\frac{dr}{d\theta }.}$

• J. Edwards (1892).Differential Calculus. London: MacMillan and Co. pp. 150, 154.

• B. Williamson "Subtangent and Subnormal" and "Polar Subtangent and Polar Subnormal" inAn elementary treatise on the differential calculus (1899) p 215, 223Internet Archive

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