- Subtangent
In

geometry , the**subtangent**is theprojection of thetangent upon the axis ofabscissa s (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 "x

_{0}" be theabscissa of point "P", and let "θ_{0}" 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

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

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### Look at other dictionaries:

**Subtangent**— Sub*tan gent, n. (Geom.) The part of the axis contained between the ordinate and tangent drawn to the same point in a curve. [1913 Webster] … The Collaborative International Dictionary of English**subtangent**— [sub tan′jənt] n. Geom. the segment of the x axis included between the ordinate of a given point on a curve and the tangent at that point … English World dictionary**subtangent**— /sub tan jeuhnt/, n. Geom. the part of the x axis cut off between the ordinate of a given point of a curve and the tangent at that point. [1705 15; SUB + TANGENT] * * * … Universalium**subtangent**— sub·tangent … English syllables**subtangent**— |səb+ noun Etymology: sub + tangent : the projection on the x axis of the portion of the tangent to a curve between the x axis and the point of tangency … Useful english dictionary**Tangent**— For the tangent function see trigonometric functions. For other uses, see tangent (disambiguation). In geometry, the tangent line (or simply the tangent) to a curve at a given point is the straight line that just touches the curve at that point… … Wikipedia**Isaac Barrow**— Infobox Scientist box width = 200px name = Isaac Barrow image size = 300px caption = Isaac Barrow (1630 1677) birth date = October 1630 birth place = London, England nationality = United Kingdom death date = death date and… … Wikipedia**List of curve topics**— This is a list of curve topics in mathematics. See also curve, list of curves, and list of differential geometry topics.*acnode *algebraic curve *arc *asymptote *asymptotic curve *Barbier s theorem *barycentric… … Wikipedia**Differential geometry of curves**— This article considers only curves in Euclidean space. Most of the notions presented here have analogues for curves in Riemannian and pseudo Riemannian manifolds. For a discussion of curves in an arbitrary topological space, see the main article… … Wikipedia**List of mathematics articles (S)**— NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… … Wikipedia