Namespaces
Variants
Actions

Difference between revisions of "Conic"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A second-order curve, i.e. the set of points in a (projective, affine or Euclidean) plane whose homogeneous coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249501.png" /> (with respect to some projective, affine or Cartesian coordinate system) satisfy an equation of the second degree:
+
<!--
 +
c0249501.png
 +
$#A+1 = 29 n = 0
 +
$#C+1 = 29 : ~/encyclopedia/old_files/data/C024/C.0204950 Conic
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/c024950/c0249502.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
A second-order curve, i.e. the set of points in a (projective, affine or Euclidean) plane whose homogeneous coordinates  $  x _ {0} , x _ {1} , x _ {2} $(
 +
with respect to some projective, affine or Cartesian coordinate system) satisfy an equation of the second degree:
 +
 
 +
$$
 +
F ( x)  \equiv \
 +
\sum _ {i, j = 0 } ^ { 2 }
 +
a _ {ij} x _ {i} x _ {j}  = 0,\ \
 +
a _ {ij}  = a _ {ji} .
 +
$$
  
 
The symmetric bilinear form
 
The symmetric bilinear form
  
<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/c024950/c0249503.png" /></td> </tr></table>
+
$$
 +
\Phi ( x, \widetilde{x}  )  = \
 +
\sum _ {i, j = 0 } ^ { 2 }
 +
a _ {ij} x _ {i} \widetilde{x}  _ {j}  $$
 +
 
 +
is called the polar form of  $  F ( x) $.
 +
Two points  $  M ^ { \prime } = ( x _ {0}  ^  \prime  , x _ {1}  ^  \prime  , x _ {2}  ^  \prime  ) $
 +
and  $  M ^ { \prime\prime } = ( x _ {0}  ^ {\prime\prime} , x _ {1}  ^ {\prime\prime} , x _ {2}  ^ {\prime\prime} ) $
 +
for which  $  \Phi ( x  ^  \prime  , x  ^ {\prime\prime} ) = 0 $
 +
are said to be polar conjugates with respect to the conic. If the line  $  M ^ { \prime } M ^ { \prime\prime } $
 +
intersects the conic at points  $  N _ {1} , N _ {2} $
 +
and if  $  M ^ { \prime } , M ^ { \prime\prime } $
 +
are polar conjugates with respect to the conic, then  $  N _ {2} , N _ {2} , M ^ { \prime } , M ^ { \prime\prime } $
 +
form a harmonic quadruple. The only self-conjugate points are the points of the conic itself. The pole of a given line with respect to a conic is the point that is polar conjugate with all the points of the line. The set of points in the plane that are polar conjugate with a given point  $  M ^ { \prime } $
 +
with respect to a conic is called the polar of  $  M ^ { \prime } $
 +
with respect to the conic. The polar of  $  M ^ { \prime } $
 +
is defined by the linear equation  $  \Phi ( x, x  ^  \prime  ) = 0 $
 +
in the coordinates  $  x _ {0} , x _ {1} , x _ {2} $.
 +
If  $  \Phi ( x, x  ^  \prime  ) \not\equiv 0 $,
 +
the polar of  $  M ^ { \prime } $
 +
is a straight line; if  $  \Phi ( x, x  ^  \prime  ) \equiv 0 $,
 +
the polar of  $  M ^ { \prime } $
 +
is the whole plane. In this case  $  M ^ { \prime } $
 +
lies on the conic and is called a singular point of the conic. If  $  R = \mathop{\rm rank}  ( a _ {ij} ) = 3 $,
 +
the conic has no singular points and is said to be non-degenerate or to be non-decomposing (non-splitting). In the projective plane this is a real or imaginary oval. A non-degenerate conic defines a correlation on the projective plane, i.e. a bijective mapping from the set of points onto the set of lines. A tangent to a non-degenerate conic is the polar of the point of tangency. If  $  R = 2 $,
 +
the conic is a pair of real or imaginary straight lines intersecting at a singular point. If  $  R = 1 $,
 +
every point of the conic is singular and the conic itself is a pair of coincident real straight lines (a double line). The affine properties of a conic are distinguished by the specific nature of its location and by the points and lines associated with it with respect to the distinguished line  $  x _ {0} = 0 $—
 +
the line at infinity. A conic is of hyperbolic, elliptic or parabolic type according to whether it intersects the line at infinity  $  ( \delta < 0) $,
 +
does not intersect it  $  ( \delta > 0) $
 +
or is tangent to it  $  ( \delta = 0) $.
 +
Here
  
is called the polar form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249504.png" />. Two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249505.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249506.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249507.png" /> are said to be polar conjugates with respect to the conic. If the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249508.png" /> intersects the conic at points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249509.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495010.png" /> are polar conjugates with respect to the conic, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495011.png" /> form a harmonic quadruple. The only self-conjugate points are the points of the conic itself. The pole of a given line with respect to a conic is the point that is polar conjugate with all the points of the line. The set of points in the plane that are polar conjugate with a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495012.png" /> with respect to a conic is called the polar of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495013.png" /> with respect to the conic. The polar of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495014.png" /> is defined by the linear equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495015.png" /> in the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495016.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495017.png" />, the polar of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495018.png" /> is a straight line; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495019.png" />, the polar of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495020.png" /> is the whole plane. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495021.png" /> lies on the conic and is called a singular point of the conic. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495022.png" />, the conic has no singular points and is said to be non-degenerate or to be non-decomposing (non-splitting). In the projective plane this is a real or imaginary oval. A non-degenerate conic defines a correlation on the projective plane, i.e. a bijective mapping from the set of points onto the set of lines. A tangent to a non-degenerate conic is the polar of the point of tangency. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495023.png" />, the conic is a pair of real or imaginary straight lines intersecting at a singular point. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495024.png" />, every point of the conic is singular and the conic itself is a pair of coincident real straight lines (a double line). The affine properties of a conic are distinguished by the specific nature of its location and by the points and lines associated with it with respect to the distinguished line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495025.png" /> — the line at infinity. A conic is of hyperbolic, elliptic or parabolic type according to whether it intersects the line at infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495026.png" />, does not intersect it <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495027.png" /> or is tangent to it <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495028.png" />. Here
+
$$
 +
\delta  = \
 +
\left |
  
<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/c024950/c02495029.png" /></td> </tr></table>
+
\begin{array}{ll}
 +
a _ {11}  &a _ {12}  \\
 +
a _ {21}  &a _ {22}  \\
 +
\end{array}
 +
\
 +
\right | .
 +
$$
  
 
The centre of a conic is the pole of the line at infinity, a diameter is the polar of a point at infinity, an asymptote is a tangent to the conic at a point at infinity. Two diameters are conjugate with respect to the conic if their points at infinity are polar conjugates with respect to the conic.
 
The centre of a conic is the pole of the line at infinity, a diameter is the polar of a point at infinity, an asymptote is a tangent to the conic at a point at infinity. Two diameters are conjugate with respect to the conic if their points at infinity are polar conjugates with respect to the conic.
Line 17: Line 73:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.P. Finikov,  "Analytic geometry" , Moscow  (1952)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.V. Efimov,  "A short course of analytical geometry" , Moscow  (1967)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.P. Finikov,  "Analytic geometry" , Moscow  (1952)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.V. Efimov,  "A short course of analytical geometry" , Moscow  (1967)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Salmon,  "A treatise on conic sections" , Longman  (1879)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Giering,  "Vorlesungen über höhere Geometrie" , Vieweg  (1982)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Salmon,  "A treatise on conic sections" , Longman  (1879)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Giering,  "Vorlesungen über höhere Geometrie" , Vieweg  (1982)</TD></TR></table>

Latest revision as of 17:46, 4 June 2020


A second-order curve, i.e. the set of points in a (projective, affine or Euclidean) plane whose homogeneous coordinates $ x _ {0} , x _ {1} , x _ {2} $( with respect to some projective, affine or Cartesian coordinate system) satisfy an equation of the second degree:

$$ F ( x) \equiv \ \sum _ {i, j = 0 } ^ { 2 } a _ {ij} x _ {i} x _ {j} = 0,\ \ a _ {ij} = a _ {ji} . $$

The symmetric bilinear form

$$ \Phi ( x, \widetilde{x} ) = \ \sum _ {i, j = 0 } ^ { 2 } a _ {ij} x _ {i} \widetilde{x} _ {j} $$

is called the polar form of $ F ( x) $. Two points $ M ^ { \prime } = ( x _ {0} ^ \prime , x _ {1} ^ \prime , x _ {2} ^ \prime ) $ and $ M ^ { \prime\prime } = ( x _ {0} ^ {\prime\prime} , x _ {1} ^ {\prime\prime} , x _ {2} ^ {\prime\prime} ) $ for which $ \Phi ( x ^ \prime , x ^ {\prime\prime} ) = 0 $ are said to be polar conjugates with respect to the conic. If the line $ M ^ { \prime } M ^ { \prime\prime } $ intersects the conic at points $ N _ {1} , N _ {2} $ and if $ M ^ { \prime } , M ^ { \prime\prime } $ are polar conjugates with respect to the conic, then $ N _ {2} , N _ {2} , M ^ { \prime } , M ^ { \prime\prime } $ form a harmonic quadruple. The only self-conjugate points are the points of the conic itself. The pole of a given line with respect to a conic is the point that is polar conjugate with all the points of the line. The set of points in the plane that are polar conjugate with a given point $ M ^ { \prime } $ with respect to a conic is called the polar of $ M ^ { \prime } $ with respect to the conic. The polar of $ M ^ { \prime } $ is defined by the linear equation $ \Phi ( x, x ^ \prime ) = 0 $ in the coordinates $ x _ {0} , x _ {1} , x _ {2} $. If $ \Phi ( x, x ^ \prime ) \not\equiv 0 $, the polar of $ M ^ { \prime } $ is a straight line; if $ \Phi ( x, x ^ \prime ) \equiv 0 $, the polar of $ M ^ { \prime } $ is the whole plane. In this case $ M ^ { \prime } $ lies on the conic and is called a singular point of the conic. If $ R = \mathop{\rm rank} ( a _ {ij} ) = 3 $, the conic has no singular points and is said to be non-degenerate or to be non-decomposing (non-splitting). In the projective plane this is a real or imaginary oval. A non-degenerate conic defines a correlation on the projective plane, i.e. a bijective mapping from the set of points onto the set of lines. A tangent to a non-degenerate conic is the polar of the point of tangency. If $ R = 2 $, the conic is a pair of real or imaginary straight lines intersecting at a singular point. If $ R = 1 $, every point of the conic is singular and the conic itself is a pair of coincident real straight lines (a double line). The affine properties of a conic are distinguished by the specific nature of its location and by the points and lines associated with it with respect to the distinguished line $ x _ {0} = 0 $— the line at infinity. A conic is of hyperbolic, elliptic or parabolic type according to whether it intersects the line at infinity $ ( \delta < 0) $, does not intersect it $ ( \delta > 0) $ or is tangent to it $ ( \delta = 0) $. Here

$$ \delta = \ \left | \begin{array}{ll} a _ {11} &a _ {12} \\ a _ {21} &a _ {22} \\ \end{array} \ \right | . $$

The centre of a conic is the pole of the line at infinity, a diameter is the polar of a point at infinity, an asymptote is a tangent to the conic at a point at infinity. Two diameters are conjugate with respect to the conic if their points at infinity are polar conjugates with respect to the conic.

The metric properties of a conic are determined from its affine properties by the invariance of the distance between two arbitrary points. The diameter of a conic that is orthogonal to the conjugate diameter is an axis of symmetry of the conic and is called an axis. A directrix of a conic is the polar of a focus.

References

[1] S.P. Finikov, "Analytic geometry" , Moscow (1952) (In Russian)
[2] N.V. Efimov, "A short course of analytical geometry" , Moscow (1967) (In Russian)

Comments

References

[a1] G. Salmon, "A treatise on conic sections" , Longman (1879)
[a2] O. Giering, "Vorlesungen über höhere Geometrie" , Vieweg (1982)
How to Cite This Entry:
Conic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conic&oldid=46465
This article was adapted from an original article by V.S. Malakhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article