#freeze
*懸垂曲線：カテナリー [#fad7623b]
カテナリー曲線 カテナリとは、電車線のことをさすが、もともとは、電車線の形づくる曲線（これを懸垂曲線という）を意味している。
In physics and geometry, the catenary is the theoretical shape a hanging chain or cable will assume when supported at its ends and acted on only by its own weight. Its surface of revolution, the catenoid, is a minimal surface and will be the shape of a soap film bounded by two circles. The curve is the graph of the hyperbolic cosine function, which has a U-like shape, similar in appearance to a parabola.

参考：[[石鹸幕が作る形>http://homepage1.nifty.com/haniu/nuas/soapfilm.html]]
*歴史的記述 [#g33e151f]
The catenary is the shape of a perfectly flexible chain suspended by its ends and acted on by gravity. Its equation was obtained by Leibniz, Huygens and Johann Bernoulli in 1691. They were responding to a challenge put out by Jacob Bernoulli to find the equation of the 'chain-curve'. 
Huygens was the first to use the term catenary in a letter to Leibniz in 1690 and David Gregory wrote a treatise on the catenary in 1690. Jungius (1669) disproved Galileo's claim that the curve of a chain hanging under gravity would be a parabola. 

The catenary is the locus of the focus of a parabola rolling along a straight line. 

The catenary is the evolute of the tractrix. It is the locus of the mid-point of the vertical line segment between the curves e^x and e^-x. 

Euler showed in 1744 that a catenary revolved about its asymptote generates the only minimal surface of revolution. 

The word catenary is derived from the Latin word catena, which means "chain". The curve is also called the "alysoid", "funicular", and "chainette". Galileo claimed that the curve of a chain hanging under gravity would be a parabola, but this was disproved by Joachim Jungius (1587-1657) and published posthumously in 1669.
In 1691, Leibniz, Christiaan Huygens, and Johann Bernoulli derived the equation in response to a challenge by Jakob Bernoulli. Huygens first used the term 'catenaria' in a letter to Leibniz in 1690, and David Gregory wrote a treatise on the catenary in 1690. However Thomas Jefferson is usually credited with the English word 'catenary'.
*アーチ：The inverted catenary arch [#p6da54e3]
The catenary is the ideal curve for an arch which supports only its own weight.
When the centerline of an arch is made to follow the curve of an up-side-down (ie. inverted) catenary, the arch endures almost pure compression, in which no significant bending moment occurs inside the material. If the arch is made of individual elements (eg., stones) whose contacting surfaces are perpendicular to the curve of the arch, no significant shear forces are present at these contacting surfaces. (Shear stress is still present inside each stone, as it resists the compressive force along the shear sliding plane.) The thrust (including the weight) of the arch at its two ends is tangent to its centerline.
The Gateway Arch in Saint Louis, Missouri, United States follows the form of an inverted catenary. It is 630 feet wide at the base and 630 feet tall.
-St. Louis Arch
#ref(catenary.JPG)

The formula for the St. Louis Arch,
 y=-127.7ft cosh(x/127.7ft);757.7
is displayed inside.

*数式：Equation [#le3f5497]
Catenaries for different values of the parameter 'a'The equation (up to translation and rotation) of a catenary in Cartesian coordinates has the form
#ref(catenary eq.png)
where cosh is the hyperbolic cosine function. The scaling factor a can be interpreted as the ratio between the horizontal component of the tension on the chain (a constant) and the weight of the chain per unit of length.
*導出 [#o194f30e]
オイラーの方程式のページに記載してある。変分法でオイラー・ラグランジェの方程式を解いて求められる
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