geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, 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, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

to a curve at a given point and the

coordinate axes In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...

. 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, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

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 specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...

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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