geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

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

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

to a curve at a given point and the coordinate axes. 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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

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. Then :

\tan\varphi=\frac=\frac=\frac.

So the subtangent is :

y\cot\varphi=\frac,

and the subnormal is :

y\tan\varphi=y\frac.

The normal is given by :

y\sec\varphi=y\sqrt,

and the tangent is given by :

y\csc\varphi=\frac\sqrt.

Polar definitions

Let ''P'' = (''r'', θ) be a point on a given curve defined by

polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...

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 :

\tan\psi=\frac=\frac=\frac.

So the polar subtangent is :

r\tan\psi=\frac,

and the subnormal is :

r\cot\psi=\frac.

References

*{{cite book , author=J. Edwards , title=Differential Calculus , publisher= MacMillan and Co., location=London , page
150
154, year=1892 , url=https://archive.org/details/in.ernet.dli.2015.109607 * B. Williamson "Subtangent and Subnormal" and "Polar Subtangent and Polar Subnormal" in ''An elementary treatise on the differential calculus'' (1899) p 215, 22
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