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Difference between revisions of "Conical surface"

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''cone''
 
''cone''
  
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A second-order cone is one which has the form of a [[Surface of the second order|surface of the second order]]. The canonical equation of a real second-order conical surface is
 
A second-order cone is one which has the form of a [[Surface of the second order|surface of the second order]]. The canonical equation of a real second-order conical surface is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024980/c0249801.png" /></td> </tr></table>
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$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0;$$
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024980/c0249802.png" />, the surface is said to be circular or to be a conical surface of rotation; the canonical equation of an imaginary second-order canonical surface is
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if $a=b$, the surface is said to be circular or to be a conical surface of rotation; the canonical equation of an imaginary second-order canonical surface is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024980/c0249803.png" /></td> </tr></table>
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$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=0;$$
  
the only real point of an imaginary conical surface is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024980/c0249804.png" />.
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the only real point of an imaginary conical surface is $(0,0,0)$.
  
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024980/c0249806.png" />-th order cone is an algebraic surface given in affine coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024980/c0249807.png" /> by the equation
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An $n$-th order cone is an algebraic surface given in affine coordinates $x,y,z$ by the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024980/c0249808.png" /></td> </tr></table>
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$$f(x,y,z)=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024980/c0249809.png" /> is a homogeneous polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024980/c02498010.png" /> (a form of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024980/c02498011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024980/c02498012.png" />). If the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024980/c02498013.png" /> lies on a cone, then the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024980/c02498014.png" /> also lies on the cone (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024980/c02498015.png" /> is the coordinate origin). The converse is also true: Every algebraic surface consisting of lines passing through a single point is a conical surface.
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where $f(x,y,z)=0$ is a homogeneous polynomial of degree $n$ (a form of degree $n$ in $x,y,z$). If the point $M(x_0,y_0,z_0)$ lies on a cone, then the line $OM$ also lies on the cone ($O$ is the coordinate origin). The converse is also true: Every algebraic surface consisting of lines passing through a single point is a conical surface.

Latest revision as of 17:29, 11 April 2014

cone

The surface formed by the movement of a straight line (the generator) through a given point (the vertex) intersecting a given curve (the directrix). A conical surface consists of two concave pieces positioned symmetrically about the vertex.

A second-order cone is one which has the form of a surface of the second order. The canonical equation of a real second-order conical surface is

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0;$$

if $a=b$, the surface is said to be circular or to be a conical surface of rotation; the canonical equation of an imaginary second-order canonical surface is

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=0;$$

the only real point of an imaginary conical surface is $(0,0,0)$.

An $n$-th order cone is an algebraic surface given in affine coordinates $x,y,z$ by the equation

$$f(x,y,z)=0,$$

where $f(x,y,z)=0$ is a homogeneous polynomial of degree $n$ (a form of degree $n$ in $x,y,z$). If the point $M(x_0,y_0,z_0)$ lies on a cone, then the line $OM$ also lies on the cone ($O$ is the coordinate origin). The converse is also true: Every algebraic surface consisting of lines passing through a single point is a conical surface.

How to Cite This Entry:
Conical surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conical_surface&oldid=31530
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article