Editing Subtangent

{{Short description|Mathematical concept}}
[[Image:SubtangentDiagram.svg|thumb|300px|right|Subtangent and related concepts for a curve ('''black''') at a given point ''P''. The tangent and normal lines are shown in <span style="color:green;">green</span> and <span style="color:blue;">blue</span> respectively. The distances shown are the <span style="color:#800000;">'''ordinate'''</span> (''AP''), <span style="color:#008000;">'''tangent'''</span> (''TP''), <span style="color:#808000;">'''subtangent'''</span> (''TA''), <span style="color:#000080;">'''normal'''</span> (''PN''), and <span style="color:#800080;">'''subnormal'''</span> (''AN''). The angle φ is the angle of inclination of the tangent line or the tangential angle.]]
In [[geometry]], the '''subtangent''' and related terms are certain line segments defined using the line [[tangent]] to a curve at a given point and the [[Cartesian coordinate system|coordinate axes]]. The terms are somewhat archaic today but were in common use until the early part of the 20th century.

==Definitions==
Let ''P''&nbsp;=&nbsp;(''x'',&nbsp;''y'') be a point on a given curve with ''A''&nbsp;=&nbsp;(''x'',&nbsp;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|tangent line]] and the normal line which are also called the tangent and normal.

==Equations==
Let ''&phi;'' be the angle of inclination of the tangent with respect to the ''x''-axis; this is also known as the [[tangential angle]]. Then 
:<math>\tan\varphi=\frac{dy}{dx}=\frac{AP}{TA}=\frac{AN}{AP}.</math>
So the subtangent is 
:<math>y\cot\varphi=\frac{y}{\tfrac{dy}{dx}},</math>
and the subnormal is 
:<math>y\tan\varphi=y\frac{dy}{dx}.</math>
The normal is given by
:<math>y\sec\varphi=y\sqrt{1+\left(\frac{dy}{dx}\right)^2},</math>
and the tangent is given by
:<math>y\csc\varphi=\frac{y}{\tfrac{dy}{dx}}\sqrt{1+\left(\frac{dy}{dx}\right)^2}.</math>

==Polar definitions==
[[Image:PolarSubtangentDiagram.svg|thumb|250px|right|Polar subtangent and related concepts for a curve ('''black''') at a given point ''P''. The tangent and normal lines are shown in <span style="color:green;">green</span> and <span style="color:blue;">blue</span> respectively. The distances shown are the <span style="color:#800000;">'''radius'''</span> (''OP''), <span style="color:#808000;">'''polar subtangent'''</span> (''OT''), and <span style="color:#800080;">'''polar subnormal'''</span> (''ON''). The angle θ is the radial angle and the angle ψ of inclination of the tangent to the radius or the polar tangential angle.]]
Let ''P''&nbsp;=&nbsp;(''r'',&nbsp;θ) be a point on a given curve defined by [[polar coordinates]] 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 ''&psi;'' be the angle between the tangent and the ray ''OP''; this is also known as the polar tangential angle. Then
:<math>\tan\psi=\frac{r}{\tfrac{dr}{d\theta}}=\frac{OP}{ON}=\frac{OT}{OP}.</math>
So the polar subtangent is 
:<math>r\tan\psi=\frac{r^2}{\tfrac{dr}{d\theta}},</math>
and the subnormal is 
:<math>r\cot\psi=\frac{dr}{d\theta}.</math>

==References==
*{{cite book | author=J. Edwards | title=Differential Calculus
| publisher= MacMillan and Co.| location=London | pages=[https://archive.org/details/in.ernet.dli.2015.109607/page/n168 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, 223 [https://archive.org/details/anelementarytre05willgoog/page/n235 <!-- pg=215 --> Internet Archive]

[[Category:Curves]]