接線

関: tangent、tangential

WordNet

of or relating to or acting along or in the direction of a tangent; "tangential forces"
mark with lines; "sorrow had lined his face"
a length (straight or curved) without breadth or thickness; the trace of a moving point
a single frequency (or very narrow band) of radiation in a spectrum
reinforce with fabric; "lined books are more enduring"
a particular kind of product or merchandise; "a nice line of shoes" (同)product line, line of products, line of merchandise, business line, line of business
the road consisting of railroad track and roadbed (同)railway line, rail line
a commercial organization serving as a common carrier
acting in conformity; "in line with"; "he got out of line"; "toe the line"
something (as a cord or rope) that is long and thin and flexible; "a washing line"
a conceptual separation or distinction; "there is a narrow line between sanity and insanity" (同)dividing_line, demarcation, contrast
a formation of people or things one behind another; "the line stretched clear around the corner"; "you must wait in a long line at the checkout counter"
a formation of people or things one beside another; "the line of soldiers advanced with their bayonets fixed"; "they were arrayed in line of battle"; "the cast stood in line for the curtain call"
a fortified position (especially one marking the most forward position of troops); "they attacked the enemys line"
a mark that is long relative to its width; "He drew a line on the chart"
a spatial location defined by a real or imaginary unidimensional extent
in games or sports; a mark indicating positions or bounds of the playing area
persuasive but insincere talk that is usually intended to deceive or impress; "`let me show you my etchings is a rather worn line"; "he has a smooth line but I didnt fall for it"; "that salesman must have practiced his fast line of talk"
text consisting of a row of words written across a page or computer screen; "the letter consisted of three short lines"; "there are six lines in every stanza"
be in line with; form a line along; "trees line the riverbank" (同)run along
cover the interior of; "line the gloves"; "line a chimney"
fill plentifully; "line ones pockets"
ratio of the opposite to the adjacent side of a right-angled triangle (同)tan
a straight line or plane that touches a curve or curved surface at a point but does not intersect it at that point
providing something with a surface of a different material (同)facing
a protective covering that protects an inside surface (同)liner
the act of attaching an inside lining (to a garment or curtain etc.)
having a lining or a liner; often used in combination; "a lined skirt"; "a silk-lined jacket"
(used especially of skin) marked by lines or seams; "their lined faces were immeasurably sad"; "a seamed face" (同)seamed
bordered by a line of things; "tree lined streets"

PrepTutorEJDIC

=tangent / それた,相違した
『線』,筋,けい / (色・縫い目などの)『線』,しま;(顔・手などの)しわ(wrinkle) / (物・人の)『列』,行列(row) / 『綱』,糸,針金 / 電線,電話線,電信線 / (文章の)『行』;(詩の)『行』 / 《しばしばa~》《話》『短い手紙』(note) / 《複数形で》輪郭,外形(outline) / 境界線(boundary);限界,限度(limit) / (鉄道・バスなどの)路線;航路,空路 / 《複数形で》せりふ / 進路,道順(course);(考えなどの)筋道,方針 / 《しばしばone's~》(活動の)分野,方面;特意 / 家系,血統,血筋 / 在庫商品,仕入れ品 / 導管,パイプライン / 《しばしば複数形で》戦線,防御線 / …‘に'『線を引く』,筋(しま,しわ)をつける / …‘に'『沿って並ぶ;』(…を)…‘に'沿って並べる《+『名』+『with』+『名』》
(材料で)〈着物など〉‘に'裏を付ける《+『名』『with』+『名』》 / (…で)…‘を'満たす,(…を)…‘に'詰め込む《+『名』+『with』+『名』》
接線;タンジェント,正接(《略》tan) / 接する;(…に)接する《+to+名》 / (また tangential)接線の;(…との)接線の《+to+名》
〈U〉(服などの)裏張り,裏塗り / 〈U〉裏地,裏材

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2014/05/24 10:34:34」(JST)

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Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.

Tangent plane to a sphere

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. Informally, it is a line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f(c)) on the curve and has slope f'(c) where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.

As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.

Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space.

The word tangent comes from the Latin tangere, to touch.

1 History
2 Tangent line to a curve
- 2.1 Analytical approach
  - 2.1.1 Intuitive description
  - 2.1.2 More rigorous description
  - 2.1.3 How the method can fail
- 2.2 Equations
- 2.3 Normal line to a curve
- 2.4 Angle between curves
- 2.5 Multiple tangents at the origin
3 Tangent circles
4 Surfaces and higher-dimensional manifolds
5 See also
6 References
7 External links

History

The first definition of a tangent was "a right line which touches a curve, but which when produced, does not cut it".^[1] This old definition prevents inflection points from having any tangent. It has been dismissed and the modern definitions are equivalent to those of Leibniz. The tangent problem has given rise to differential calculus. The main ideas behind differential calculus are due to Pierre Fermat and were developed by John Wallis, Isaac Barrow, Isaac Newton and Gottfried Leibniz.

Pierre de Fermat developed a general technique for determining the tangents of a curve using his method of adequality in the 1630s.

Leibniz defined the tangent line as the line through a pair of infinitely close points on the curve.

Tangent line to a curve

A tangent, a chord, and a secant to a circle

The intuitive notion that a tangent line "touches" a curve can be made more explicit by considering the sequence of straight lines (secant lines) passing through two points, A and B, those that lie on the function curve. The tangent at A is the limit when point B approximates or tends to A. The existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as "differentiability." For example, if two circular arcs meet at a sharp point (a vertex) then there is no uniquely defined tangent at the vertex because the limit of the progression of secant lines depends on the direction in which "point B" approaches the vertex.

At most points, the tangent touches the curve without crossing it (though it may, when continued, cross the curve at other places away from the point of tangent). A point where the tangent (at this point) crosses the curve is called an inflection point. Circles, parabolas, hyperbolas and ellipses do not have any inflection point, but more complicated curves do have, like the graph of a cubic function, which has exactly one inflection point.

Conversely, it may happen that the curve lies entirely on one side of a straight line passing through a point on it, and yet this straight line is not a tangent line. This is the case, for example, for a line passing through the vertex of a triangle and not intersecting the triangle—where the tangent line does not exist for the reasons explained above. In convex geometry, such lines are called supporting lines.

At each point, the line is always tangent to the curve. Its slope is the derivative; the positivity, negativity and zeroes of the derivative are marked by green, red and black respectively.

Analytical approach

The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century. In the second book of his Geometry, René Descartes^[2] said of the problem of constructing the tangent to a curve, "And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know".^[3]

Intuitive description

Suppose that a curve is given as the graph of a function, y = f(x). To find the tangent line at the point p = (a, f(a)), consider another nearby point q = (a + h, f(a + h)) on the curve. The slope of the secant line passing through p and q is equal to the difference quotient

As the point q approaches p, which corresponds to making h smaller and smaller, the difference quotient should approach a certain limiting value k, which is the slope of the tangent line at the point p. If k is known, the equation of the tangent line can be found in the point-slope form:

More rigorous description

To make the preceding reasoning rigorous, one has to explain what is meant by the difference quotient approaching a certain limiting value k. The precise mathematical formulation was given by Cauchy in the 19th century and is based on the notion of limit. Suppose that the graph does not have a break or a sharp edge at p and it is neither plumb nor too wiggly near p. Then there is a unique value of k such that, as h approaches 0, the difference quotient gets closer and closer to k, and the distance between them becomes negligible compared with the size of h, if h is small enough. This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function f. This limit is the derivative of the function f at x = a, denoted f ′(a). Using derivatives, the equation of the tangent line can be stated as follows:

Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the power function, trigonometric functions, exponential function, logarithm, and their various combinations. Thus, equations of the tangents to graphs of all these functions, as well as many others, can be found by the methods of calculus.

How the method can fail

Calculus also demonstrates that there are functions and points on their graphs for which the limit determining the slope of the tangent line does not exist. For these points the function f is non-differentiable. There are two possible reasons for the method of finding the tangents based on the limits and derivatives to fail: either the geometric tangent exists, but it is a vertical line, which cannot be given in the point-slope form since it does not have a slope, or the graph exhibits one of three behaviors that precludes a geometric tangent.

The graph y = x^1/3 illustrates the first possibility: here the difference quotient at a = 0 is equal to h^1/3/h = h^−2/3, which becomes very large as h approaches 0. This curve has a tangent line at the origin that is vertical.

The graph y = x^2/3 illustrates another possibility: this graph has a cusp at the origin. This means that, when h approaches 0, the difference quotient at a = 0 approaches plus or minus infinity depending on the sign of x. Thus both branches of the curve are near to the half vertical line for which y=0, but none is near to the negative part of this line. Basically, there is no tangent at the origin in this case, but in some context one may consider this line as a tangent, and even, in algebraic geometry, as a double tangent.

The graph y = |x| of the absolute value function consists of two straight lines with different slopes joined at the origin. As a point q approaches the origin from the right, the secant line always has slope 1. As a point q approaches the origin from the left, the secant line always has slope −1. Therefore, there is no unique tangent to the graph at the origin. Having two different (but finite) slopes is called a corner.

Finally, since differentiability implies continuity, the contrapositive states discontinuity implies non-differentiability. Any such jump or point discontinuity will have no tangent line. This includes cases where one slope approaches positive infinity while the other approaches negative infinity, leading to an infinite jump discontinuity

Equations

When the curve is given by y = f(x) then the slope of the tangent is so by the point–slope formula the equation of the tangent line at (X, Y) is

where (x, y) are the coordinates of any point on the tangent line, and where the derivative is evaluated at .^[4]

When the curve is given by y = f(x), the tangent line's equation can also be found^[5] by using polynomial division to divide by ; if the remainder is denoted by , then the equation of the tangent line is given by

When the equation of the curve is given in the form f(x, y) = 0 then the value of the slope can be found by implicit differentiation, giving

The equation of the tangent line at a point (X,Y) such that f(X,Y) = 0 is then^[4]

This equation remains true if but (in this case the slope of the tangent is infinite). If the tangent line is not defined and the point (X,Y) is said singular.

For algebraic curves, computations may be simplified somewhat by converting to homogeneous coordinates. Specifically, let the homogeneous equation of the curve be g(x, y, z) = 0 where g is a homogeneous function of degree n. Then, if (X, Y, Z) lies on the curve, Euler's theorem implies

It follows that the homogeneous equation of the tangent line is

The equation of the tangent line in Cartesian coordinates can be found by setting z=1 in this equation.^[6]

To apply this to algebraic curves, write f(x, y) as

where each u_r is the sum of all terms of degree r. The homogeneous equation of the curve is then

Applying the equation above and setting z=1 produces

as the equation of the tangent line.^[7] The equation in this form is often simpler to use in practice since no further simplification is needed after it is applied.^[6]

If the curve is given parametrically by

then the slope of the tangent is

giving the equation for the tangent line at as^[8]

If , the tangent line is not defined. However, it may occur that the tangent line exists and may be computed from an implicit equation of the curve.

Normal line to a curve

The line perpendicular to the tangent line to a curve at the point of tangency is called the normal line to the curve at that point. The slopes of perpendicular lines have product −1, so if the equation of the curve is y = f(x) then slope of the normal line is

and it follows that the equation of the normal line at (X, Y) is

Similarly, if the equation of the curve has the form f(x, y) = 0 then the equation of the normal line is given by^[9]

If the curve is given parametrically by

then the equation of the normal line is^[8]

Angle between curves

The angle between two curves at a point where they intersect is defined as the angle between their tangent lines at that point. More specifically, two curves are said to be tangent at a point if they have the same tangent at a point, and orthogonal if their tangent lines are orthogonal.^[10]

Multiple tangents at the origin

The limaçon trisectrix: a curve with two tangents at the origin.

The formulas above fail when the point is a singular point. In this case there may be two or more branches of the curve which pass through the point, each branch having its own tangent line. When the point is the origin, the equations of these lines can be found for algebraic curves by factoring the equation formed by eliminating all but the lowest degree terms from the original equation. Since any point can be made the origin by a change of variables, this gives a method for finding the tangent lines at any singular point.

For example, the equation of the limaçon trisectrix shown to the right is

Expanding this and eliminating all but terms of degree 2 gives

which, when factored, becomes

So these are the equations of the two tangent lines through the origin.^[11]

Tangent circles

Two pairs of tangent circles. Above internally and below externally tangent

Two circles of non-equal radius, both in the same plane, are said to be tangent to each other if they meet at only one point. Equivalently, two circles, with radii of r_i and centers at (x_i, y_i), for i = 1, 2 are said to be tangent to each other if

Two circles are externally tangent if the distance between their centres is equal to the sum of their radii.

Two circles are internally tangent if the distance between their centres is equal to the difference between their radii.^[12]

Surfaces and higher-dimensional manifolds

The tangent plane to a surface at a given point p is defined in an analogous way to the tangent line in the case of curves. It is the best approximation of the surface by a plane at p, and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close to p as these points converge to p. More generally, there is a k-dimensional tangent space at each point of a k-dimensional manifold in the n-dimensional Euclidean space.

References

^ Noah Webster, American Dictionary of the English Language (New York: S. Converse, 1828), vol.2, p.733, [1]
^ Descartes, René (1954). The geometry of René Descartes. Courier Dover. p. 95. ISBN 0-486-60068-8.
^ R. E. Langer (October 1937). "Rene Descartes". American Mathematical Monthly (Mathematical Association of America) 44 (8): 495–512. doi:10.2307/2301226. JSTOR 2301226.
^ ^a ^b Edwards Art. 191
^ Strickland-Constable, Charles, "A simple method for finding tangents to polynomial graphs", Mathematical Gazette, November 2005, 466-467.
^ ^a ^b Edwards Art. 192
^ Edwards Art. 193
^ ^a ^b Edwards Art. 196
^ Edwards Art. 194
^ Edwards Art. 195
^ Edwards Art. 197
^ Circles For Leaving Certificate Honours Mathematics by Thomas O’Sullivan 1997

J. Edwards (1892). Differential Calculus. London: MacMillan and Co. pp. 143 ff.

External links

Wikimedia Commons has media related to Tangency.

Wikisource has the text of the 1921 Collier's Encyclopedia article Tangent.

Hazewinkel, Michiel, ed. (2001), "Tangent line", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Weisstein, Eric W., "Tangent Line", MathWorld.
Tangent to a circle With interactive animation
Tangent and first derivative — An interactive simulation
The Tangent Parabola by John H. Mathews

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Japanese Journal

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近年、運動器の障害により要介護リスクが高い状態を表すロコモティブシンドロームが提唱されている。そのため、簡便かつ定量的に筋肉を評価する方法が求められるが、その方法は確立されていない。本研究では、より定量的な評価を行うために腓腹筋の3次元超音波画像から羽状角を自動測定する方法を提案する。まず、上下の腱膜を曲面近似することにより決定し、それらの上下膜に挟まれた領域を筋繊維領域として抽出した。つぎに、筋 …
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… It is also shown that the graphical relation enables us to estimate the final effect after multifractionated treatment by plotting a tangent line on the curve.CONCLUSIONS:The graphical representation method is of use for improving planning in radiotherapy by determining the effective fractionation scheme. …
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Related Pictures

★リンクテーブル★

リンク元	「tangential」「接線」
関連記事	「LINE」「line」「lining」「tangent」

「tangential」

　　[★]

接線方向の、接線の

関: tangent、tangent line、tangentially

「接線」

　　[★]

英: tangent line、tangent、tangential
関: 正接、接線方向

「LINE」

　　[★]

日: 長い散在反復配列、長散在型反復配列、長い散在性反復配列、ライン、広範囲散在反復配列
英: long interspersed nuclear element, long interspersed nucleotide element, long interspersed element, long interspersed repetitive elements, LINEs, long interspersed repeat sequences
関: レトロトランスポゾン

quoted from http://en.wikipedia.org/wiki/Long_interspersed_nucleotide_elements#LINEs

Long interspersed nuclear elements[8] are a group of genetic elements that are found in large numbers in eukaryotic genomes. They are transcribed (or are the evolutionary remains of what was once transcribed) to an RNA using an RNA polymerase II promoter that resides inside the LINE. LINEs code for the enzyme reverse transcriptase, and many LINEs also code for an endonuclease (e.g. RNase H). The reverse transcriptase has a higher specificity for the LINE RNA than other RNA, and makes a DNA copy of the RNA that can be integrated into the genome at a new site.

The 5' UTR contains the promoter sequence, while the 3' UTR contains a polyadenylation signal (AATAAA) and a poly-A tail.[10] Because LINEs move by copying themselves (instead of moving, like transposons do), they enlarge the genome. The human genome, for example, contains about 20,000-40,000 LINEs, which is roughly 21% of the genome.