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A plane curve of the third order, i.e. a set of points in the (projective, affine, Euclidean) plane the homogeneous coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c0272401.png" /> of which (in a projective, affine or Cartesian coordinate system, respectively) satisfy a homogeneous equation of the third degree:
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<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/c027/c027240/c0272402.png" /></td> </tr></table>
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A plane curve of the third order, i.e. a set of points in the (projective, affine, Euclidean) plane the homogeneous coordinates  $  x _ {0} , x _ {1} , x _ {2} $
 +
of which (in a projective, affine or Cartesian coordinate system, respectively) satisfy a homogeneous equation of the third degree:
 +
 
 +
$$
 +
F ( x)  \equiv \
 +
\sum _ {i, j, k = 0 } ^ { 2 }
 +
a _ {ijk} x _ {i} x _ {j} x _ {k}  = 0,\ \
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a _ {ijk} = a _ {jik} = a _ {ikj} .
 +
$$
  
 
The number of tangents that can be drawn to a cubic from a point outside it is known as the class of the cubic. The conic
 
The number of tangents that can be drawn to a cubic from a point outside it is known as the class of the cubic. The conic
  
<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/c027/c027240/c0272403.png" /></td> </tr></table>
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$$
 +
\sum _ {i = 0 } ^ { 2 }
 +
 
 +
\frac{\partial  F }{\partial  x _ {i} }
 +
 
 +
x _ {i}  ^  \prime  = 0
 +
$$
  
is known as the conic (or first) polar of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c0272404.png" />; the point itself is called a pole. The straight line
+
is known as the conic (or first) polar of the point $  M ^ { \prime } ( x _ {0}  ^  \prime  , x _ {1}  ^  \prime  , x _ {2}  ^  \prime  ) $;  
 +
the point itself is called a pole. The straight line
  
<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/c027/c027240/c0272405.png" /></td> </tr></table>
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$$
 +
\sum _ {i = 0 } ^ { 2 }
  
is known as the rectilinear (or second) polar of the point relative to the cubic. If the pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c0272406.png" /> is a point of the cubic, its rectilinear polar is tangent at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c0272407.png" /> to the cubic itself and to the conic polar of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c0272408.png" />. The Hessian of a cubic is the set of points whose conic polars consist of two straight lines; it is defined by the equation
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\frac{\partial  F }{\partial  x _ {i}  ^  \prime  }
  
<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/c027/c027240/c0272409.png" /></td> </tr></table>
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x _ {i}  = 0
 +
$$
  
A cubic intersects its Hessian at nine common points of inflection. The straight lines into which the conic polars of the points of the Hessian split, and also the straight lines joining pairs of corresponding points of the Hessian, form the envelope of a curve of order six and of the third class — the Cayleyan of the cubic. The set of cubics on the plane that pass through the nine points of inflection of a given cubic forms a syzygetic pencil, which contains the Hessians of all curves in the pencil and four curves, each of which splits into three straight lines and forms a syzygetic triangle. The conic polar of a point of inflection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c02724010.png" /> splits into two straight lines: The tangent to the cubic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c02724011.png" /> and the harmonic polar of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c02724012.png" /> — the set of points harmonically conjugate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c02724013.png" /> relative to the two points at which the cubic intersects the secant through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c02724014.png" />. The harmonic polars of three collinear points of inflection intersect at a single point. There exist various projective, affine and metric classifications of cubics: according to the types of the canonical equations; according to the types of the singular points of the cubic; according to the nature of the asymptotes; etc.
+
is known as the rectilinear (or second) polar of the point relative to the cubic. If the pole  $  M ^ { \prime } $
 +
is a point of the cubic, its rectilinear polar is tangent at  $  M ^ { \prime } $
 +
to the cubic itself and to the conic polar of $  M ^ { \prime } $.  
 +
The Hessian of a cubic is the set of points whose conic polars consist of two straight lines; it is defined by the equation
  
The best-known cubics on the Euclidean plane are: the folium of Descartes (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c02724015.png" />); the witch of Agnesi (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c02724016.png" />); the cubic parabola (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c02724017.png" />); the semi-cubic parabola (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c02724018.png" />); the strophoid (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c02724019.png" />); the cissoid of Diocles (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c02724020.png" />); the trisectrix (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c02724021.png" />); and the conchoid of Sluze (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027240/c02724022.png" />). In algebraic geometry, the term cubic is applied both to a [[Cubic hypersurface|cubic hypersurface]] and to a three-dimensional cubic curve.
+
$$
 +
H _ {3}  \equiv \
 +
\mathop{\rm det}  \left (
 +
 
 +
\frac{\partial  ^ {2} F }{\partial  x _ {i} \partial  x _ {j} }
 +
 
 +
\right )  =  0.
 +
$$
 +
 
 +
A cubic intersects its Hessian at nine common points of inflection. The straight lines into which the conic polars of the points of the Hessian split, and also the straight lines joining pairs of corresponding points of the Hessian, form the envelope of a curve of order six and of the third class — the Cayleyan of the cubic. The set of cubics on the plane that pass through the nine points of inflection of a given cubic forms a syzygetic pencil, which contains the Hessians of all curves in the pencil and four curves, each of which splits into three straight lines and forms a syzygetic triangle. The conic polar of a point of inflection  $  M ^ { \prime } $
 +
splits into two straight lines: The tangent to the cubic at  $  M ^ { \prime } $
 +
and the harmonic polar of  $  M ^ { \prime } $—
 +
the set of points harmonically conjugate to  $  M ^ { \prime } $
 +
relative to the two points at which the cubic intersects the secant through  $  M ^ { \prime } $.
 +
The harmonic polars of three collinear points of inflection intersect at a single point. There exist various projective, affine and metric classifications of cubics: according to the types of the canonical equations; according to the types of the singular points of the cubic; according to the nature of the asymptotes; etc.
 +
 
 +
The best-known cubics on the Euclidean plane are: the folium of Descartes ( $  x  ^ {3} + y  ^ {3} - 3axy = 0 $);  
 +
the witch of Agnesi ( $  y ( a  ^ {2} + x  ^ {2} ) = a  ^ {3} $);  
 +
the cubic parabola ( $  y = ax  ^ {3} $);  
 +
the semi-cubic parabola ( $  y  ^ {2} = ax  ^ {3} $);  
 +
the strophoid ( $  y  ^ {2} ( a - x) = x  ^ {2} ( a + x) $);  
 +
the cissoid of Diocles ( $  y  ^ {2} ( 2a - x) = x  ^ {3} $);  
 +
the trisectrix ( $  x ( x  ^ {2} + y  ^ {2} ) = a ( 3x  ^ {2} - y  ^ {2} ) $);  
 +
and the conchoid of Sluze ( $  a ( x - a) ( x  ^ {2} + y  ^ {2} ) = k  ^ {2} x  ^ {2} $).  
 +
In algebraic geometry, the term cubic is applied both to a [[Cubic hypersurface|cubic hypersurface]] and to a three-dimensional cubic curve.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.S. Smogorzhevskii, E.S. Stolova, "Handbook of the theory of planar curves of the third order" , Moscow (1961) (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.S. Smogorzhevskii, E.S. Stolova, "Handbook of the theory of planar curves of the third order" , Moscow (1961) (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Brieskorn, H. Knörrer, "Plane algebraic curves" , Birkhäuser (1986) (Translated from German) {{MR|0886476}} {{ZBL|0588.14019}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Brieskorn, H. Knörrer, "Plane algebraic curves" , Birkhäuser (1986) (Translated from German) {{MR|0886476}} {{ZBL|0588.14019}} </TD></TR></table>

Latest revision as of 17:31, 5 June 2020


A plane curve of the third order, i.e. a set of points in the (projective, affine, Euclidean) plane the homogeneous coordinates $ x _ {0} , x _ {1} , x _ {2} $ of which (in a projective, affine or Cartesian coordinate system, respectively) satisfy a homogeneous equation of the third degree:

$$ F ( x) \equiv \ \sum _ {i, j, k = 0 } ^ { 2 } a _ {ijk} x _ {i} x _ {j} x _ {k} = 0,\ \ a _ {ijk} = a _ {jik} = a _ {ikj} . $$

The number of tangents that can be drawn to a cubic from a point outside it is known as the class of the cubic. The conic

$$ \sum _ {i = 0 } ^ { 2 } \frac{\partial F }{\partial x _ {i} } x _ {i} ^ \prime = 0 $$

is known as the conic (or first) polar of the point $ M ^ { \prime } ( x _ {0} ^ \prime , x _ {1} ^ \prime , x _ {2} ^ \prime ) $; the point itself is called a pole. The straight line

$$ \sum _ {i = 0 } ^ { 2 } \frac{\partial F }{\partial x _ {i} ^ \prime } x _ {i} = 0 $$

is known as the rectilinear (or second) polar of the point relative to the cubic. If the pole $ M ^ { \prime } $ is a point of the cubic, its rectilinear polar is tangent at $ M ^ { \prime } $ to the cubic itself and to the conic polar of $ M ^ { \prime } $. The Hessian of a cubic is the set of points whose conic polars consist of two straight lines; it is defined by the equation

$$ H _ {3} \equiv \ \mathop{\rm det} \left ( \frac{\partial ^ {2} F }{\partial x _ {i} \partial x _ {j} } \right ) = 0. $$

A cubic intersects its Hessian at nine common points of inflection. The straight lines into which the conic polars of the points of the Hessian split, and also the straight lines joining pairs of corresponding points of the Hessian, form the envelope of a curve of order six and of the third class — the Cayleyan of the cubic. The set of cubics on the plane that pass through the nine points of inflection of a given cubic forms a syzygetic pencil, which contains the Hessians of all curves in the pencil and four curves, each of which splits into three straight lines and forms a syzygetic triangle. The conic polar of a point of inflection $ M ^ { \prime } $ splits into two straight lines: The tangent to the cubic at $ M ^ { \prime } $ and the harmonic polar of $ M ^ { \prime } $— the set of points harmonically conjugate to $ M ^ { \prime } $ relative to the two points at which the cubic intersects the secant through $ M ^ { \prime } $. The harmonic polars of three collinear points of inflection intersect at a single point. There exist various projective, affine and metric classifications of cubics: according to the types of the canonical equations; according to the types of the singular points of the cubic; according to the nature of the asymptotes; etc.

The best-known cubics on the Euclidean plane are: the folium of Descartes ( $ x ^ {3} + y ^ {3} - 3axy = 0 $); the witch of Agnesi ( $ y ( a ^ {2} + x ^ {2} ) = a ^ {3} $); the cubic parabola ( $ y = ax ^ {3} $); the semi-cubic parabola ( $ y ^ {2} = ax ^ {3} $); the strophoid ( $ y ^ {2} ( a - x) = x ^ {2} ( a + x) $); the cissoid of Diocles ( $ y ^ {2} ( 2a - x) = x ^ {3} $); the trisectrix ( $ x ( x ^ {2} + y ^ {2} ) = a ( 3x ^ {2} - y ^ {2} ) $); and the conchoid of Sluze ( $ a ( x - a) ( x ^ {2} + y ^ {2} ) = k ^ {2} x ^ {2} $). In algebraic geometry, the term cubic is applied both to a cubic hypersurface and to a three-dimensional cubic curve.

References

[1] A.S. Smogorzhevskii, E.S. Stolova, "Handbook of the theory of planar curves of the third order" , Moscow (1961) (In Russian)

Comments

References

[a1] E. Brieskorn, H. Knörrer, "Plane algebraic curves" , Birkhäuser (1986) (Translated from German) MR0886476 Zbl 0588.14019
How to Cite This Entry:
Cubic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cubic&oldid=46562
This article was adapted from an original article by V.S. Malakhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article