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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク conical surface ： ウィキペディア英語版 conical surface In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the ''apex'' or ''vertex'' — and any point of some fixed space curve — the ''directrix'' — that does not contain the apex. Each of those lines is called a ''generatrix'' of the surface. Every conic surface is ruled and developable. In general, a conical surface consists of two congruent unbounded halves joined by the apex. Each half is called a nappe, and is the union of all the rays that start at the apex and pass through a point of some fixed space curve. (In some cases, however, the two nappes may intersect, or even coincide with the full surface.) Sometimes the term "conical surface" is used to mean just one nappe. If the directrix is a circle $C$, and the apex is located on the circle's ''axis'' (the line that contains the center of $C$ and is perpendicular to its plane), one obtains the ''right circular conical surface''. This special case is often called a ''cone'', because it is one of the two distinct surfaces that bound the geometric solid of that name. This geometric object can also be described as the set of all points swept by a line that intercepts the axis and rotates around it; or the union of all lines that intersect the axis at a fixed point $p$ and at a fixed angle $\backslash theta$. The ''aperture'' of the cone is the angle $2\; \backslash theta$. More generally, when the directrix $C$ is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of $C$, one obtains a ''conical quadric'', which is a special case of a quadric surface. A cylindrical surface can be viewed as a limiting case of a conical surface whose apex is moved off to infinity in a particular direction. Indeed, in projective geometry a cylindrical surface is just a special case of a conical surface. ==Equations== A conical surface $S$ can be described parametrically as :$S(t,u)\; =\; v\; +\; u\; q(t)$, where $v$ is the apex and $q$ is the directrix. A right circular conical surface of aperture $2\backslash theta$, whose axis is the $z$ coordinate axis, and whose apex is the origin, it is described parametrically as :$S(t,u)\; =\; (u\; \backslash cos\backslash theta\; \backslash cos\; t,\; u\; \backslash cos\backslash theta\; \backslash sin\; t,\; u\; \backslash sin\backslash theta)$ where $t$ and $u$ range over $[0,2\backslash pi)$ and $(-\backslash infty,+\backslash infty)$, respectively. In implicit form, the same surface is described by $S(x,y,z)\; =\; 0$ where :$S(x,y,z)\; =\; (x^2\; +\; y^2)(\backslash cos\backslash theta)^2\; -\; z^2\; (\backslash sin\; \backslash theta)^2.$ More generally, a right circular conical surface with apex at the origin, axis parallel to the vector $\backslash mathbf$, and aperture $2\backslash theta$, is given by the implicit vector equation $S(\backslash mathbf)\; =\; 0$ where :$S(\backslash mathbf)\; =\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf)^2\; -\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf)\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf)\; (\backslash cos\; \backslash theta)^2$ or :$S(\backslash mathbf)\; =\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; -\; |\backslash mathbf|\; |\backslash mathbf|\; \backslash cos\; \backslash theta$ where $\backslash mathbf=(x,y,z)$, and $\backslash mathbf\; \backslash cdot\; \backslash mathbf$ denotes the dot product. In three coordinates, x, y and z, the general equation for a cone with apex at origin is a homogeneous equation of degree 2 given by :$S(x,\; y,\; z)\; =\; ax^2+by^2+cz^2+2uxy+2vyz+2wzx=0$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「conical surface」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.