【今日數(shù)學(xué)】圓環(huán)面 Torus
鄭濤（Tao Steven Zheng）著
【什么是圓環(huán)面？】
圓環(huán)面（torus）是一個(gè)圓繞平面上與圓不相交的一個(gè)軸旋轉(zhuǎn)而形成的旋轉(zhuǎn)曲面（surface of revolution）。如果旋轉(zhuǎn)軸（axis of revolution）不接觸圓，該曲面稱為環(huán)形環(huán)面（ring torus）。如果旋轉(zhuǎn)軸與圓相切，該曲面稱為角環(huán)面（horn torus）。如果旋轉(zhuǎn)軸兩次穿過該圓，該曲面稱為紡錘環(huán)面（spindle torus）。
【W(wǎng)hat is a torus?】
A torus is a surface of revolution generated by revolving a circle about an axis that is coplanar with the circle. If the axis of revolution does not touch the circle, the surface is called a ring torus. If the axis of revolution is tangent to the circle, the surface is called a horn torus. If the axis of revolution passes twice through the circle, the surface is called a spindle torus.?
【圓環(huán)面的幾何學(xué)】
設(shè) 為洞的中心到圓環(huán)面“管”的中心的距離，
為圓環(huán)面“管”的半徑。那么，直角坐標(biāo)系中的關(guān)于z-軸方位角對(duì)稱的方程是
參數(shù)方程（parametric equations）為
其中 。
從這里可以推斷， 對(duì)應(yīng)于環(huán)形環(huán)面（ring torus），
對(duì)應(yīng)于角環(huán)面（horn torus），
對(duì)應(yīng)于紡錘環(huán)面（spindle torus）。
表面積
圓環(huán)面的表面積 是通過使用半徑為
的圓之周長(zhǎng)圍繞穿過圓心的圓路徑而形成的。其圓路徑的半徑為
。
體積
圓環(huán)面的內(nèi)部體積 是通過使用半徑為
的圓之面積圍繞穿過圓心的圓路徑而形成的。其圓路徑的半徑為
。
【Geometry of the Torus】
Let the distance from the center of the hole to the center of the torus "tube" be , and the radius of the torus "tube" be
. Then the equation
in Cartesian coordinates for a torus that is symmetric about the z-axis is
and the parametric equations are
where .
One can infer from here that corresponds to the ring torus,
corresponds to the horn torus, and
corresponds to the spindle torus.
Surface Area
The surface area? of the torus is generated by rotating the circumference of the circle with radius?
around a circular path that passes through the circle's center. The length of the path is the circumference of the circle with radius
.
Volume
The volume inside the torus is generated by rotating the area of the circle with radius
around a circular path that passes through the circle's center. The length of the path is the circumference of the circle with radius
.
【有趣的知識(shí)】
實(shí)心的圓環(huán)面是甜甜圈的形狀，一種標(biāo)志性的美國(guó)小吃。油炸面團(tuán)通常中心會(huì)煮不熟透、外層又過熟。這是因?yàn)槊鎴F(tuán)的中心沒有外面快熟。1847年，美國(guó)船長(zhǎng)漢森·格雷戈里（Hanson Gregory）解決了這個(gè)問題。他在油炸前在面團(tuán)的中心打了一個(gè)洞。這個(gè)洞增加了面團(tuán)的表面積，使面團(tuán)的外部更多地接觸到熱油。所以，圓環(huán)面形的甜甜圈使面團(tuán)煮得更均勻。
【Fun Fact】
The solid torus is the shape of the doughnut (more commonly spelled donut today), an iconic and popular American snack. An undercooked center and overcooked shell can often result when deep-frying dough cakes. This is because the center of the cake does not cook as fast as the outside. In 1847, Hanson Gregory (an American ship captain) resolved this problem by punching a hole in the center of dough cakes before frying. The hole increases the surface area, exposing more of the dough's exterior to the hot oil. In consequence, the torus-shaped donut makes the dough cook more evenly.
