View source for Quartic plane curve

{{short description|Plane algebraic curve defined by a 4th-degree polynomial}}
{{for|the univariate case|Quartic function}}

In [[algebraic geometry]], a '''quartic plane curve''' is a [[plane algebraic curve]] of the fourth [[degree of a polynomial|degree]]. It can be defined by a bivariate [[quartic equation]]:

:<math>Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0,</math>

with at least one of {{mvar|A, B, C, D, E}} not equal to zero. This equation has 15 constants. However, it can be multiplied by any non-zero constant without changing the curve; thus by the choice of an appropriate constant of multiplication, any one of the coefficients can be set to 1, leaving only 14 constants. Therefore, the space of quartic curves can be identified with the [[real projective space]] {{tmath|\mathbb{RP}^{14}.}} It also follows, from [[Cramer's theorem (algebraic curves)|Cramer's theorem on algebraic curves]], that there is exactly one quartic curve that passes through a set of 14 distinct points in [[general position]], since a quartic has 14 [[Degrees of freedom (physics and chemistry)|degrees of freedom]].

A quartic curve can have a maximum of:
* Four connected components
* Twenty-eight [[bitangent|bi-tangents]]
* Three ordinary [[double point]]s.

One may also consider quartic curves over other [[field (mathematics)|fields]] (or even [[ring (mathematics)|rings]]), for instance the [[complex numbers]]. In this way, one gets [[Riemann surface]]s, which are one-dimensional objects over {{tmath|\C,}} but are two-dimensional over {{tmath|\R.}} An example is the [[Klein quartic]]. Additionally, one can look at curves in the [[projective plane]], given by homogeneous polynomials.

==Examples==
Various combinations of coefficients in the above equation give rise to various important families of curves as listed below.

{{Col-begin}}{{Col-break}}
*[[Bicorn]]
*[[Bullet-nose curve]]
*[[Cartesian oval]]
*[[Cassini oval]]
*[[Deltoid curve]] 
*[[Devil's curve]]
*[[Hippopede]]
*[[Kampyle of Eudoxus]]
*[[Klein quartic]]
{{Col-break}}
*[[Lemniscate]]
**[[Lemniscate of Bernoulli]]
**[[Lemniscate of Gerono]]
*[[Limaçon]] 
*[[Lüroth quartic]]
*[[Spiric section]]
*[[Squircle]]
**[[Lamé's special quartic]]
*[[Toric section]]
*[[Trott curve]]
{{Col-end}}

{{Gallery
|width=130
|File:Ampersandcurve.svg
 |Ampersand curve
|File:Bean curve.svg
 |Bean curve
|File:Bicuspid curve.svg
 |Bicuspid curve
|File:Bowcurve.svg
 |Bow curve
|File:Cruciform1.png
 |Cruciform curve with parameters (b,a) being (1,1) in red; (2,2) in green; (3,3) in blue.
|File:Cruciform2.png
 |Cruciform curve with parameters (b,a) being (1,1) in red; (2,1) in green; (3,1) in blue.
|File:Spiric section.svg
 |[[Spiric section]]
|File:Three-leaved clover.svg
 |Three-leaved clover in [[Cartesian coordinates]]
|File:Three-leaved clover polar.svg
 |Three-leaved clover in [[polar coordinates]]
}}

===Ampersand curve===
The '''ampersand curve''' is a quartic plane curve given by the equation:
:<math>\  (y^2-x^2)(x-1)(2x-3)=4(x^2+y^2-2x)^2.</math>
It has [[Genus (mathematics)#Graph theory|genus]] zero, with three ordinary double points, all in the real plane. 
<ref>{{MathWorld|title=Ampersand Curve|urlname=AmpersandCurve}}</ref>

===Bean curve===
The '''bean curve''' is a quartic plane curve with the equation:
:<math>x^4+x^2y^2+y^4=x(x^2+y^2). \,</math>
The bean curve has genus zero. It has one [[Mathematical singularity|singularity]] at the origin, an ordinary triple point.
<ref>{{Citation | last1=Cundy | first1=H. Martyn | last2=Rollett | first2=A. P. | title=Mathematical models|title-link=Mathematical Models (Cundy and Rollett) | orig-year=1952 | publisher=Clarendon Press, Oxford | edition=2nd | isbn=978-0-906212-20-2  | mr=0124167 | year=1961 | page=72}}</ref><ref>{{MathWorld|title=Bean Curve|urlname=BeanCurve}}</ref>

===Bicuspid curve===
The '''bicuspid''' is a quartic plane curve with the equation
:<math>(x^2-a^2)(x-a)^2+(y^2-a^2)^2=0 \,</math>
where ''a'' determines the size of the curve.
The bicuspid has only the two cusps as singularities, and hence is a curve of genus one. 
<ref>{{MathWorld|title=Bicuspid Curve|urlname=BicuspidCurve}}</ref>

===Bow curve===
{{Redirect|Bow curve|the railway line|Bow Curve}}
The '''bow curve''' is a quartic plane curve with the equation:
:<math>x^4=x^2y-y^3. \,</math>
The bow curve has a single triple point at ''x''=0, ''y''=0, and consequently is a rational curve, with genus zero.
<ref>{{MathWorld|title=Bow|urlname=Bow}}</ref>

===Cruciform curve===
The '''cruciform curve''', or '''cross curve''' is a quartic plane curve given by the equation
:<math>x^2y^2-b^2x^2-a^2y^2=0 \,</math>
where ''a'' and ''b'' are two [[parameter]]s determining the shape of the curve.
The cruciform curve is related by a standard quadratic transformation, ''x'' ↦ 1/''x'', ''y'' ↦ 1/''y'' to the ellipse ''a''<sup>2</sup>''x''<sup>2</sup> + ''b''<sup>2</sup>''y''<sup>2</sup> = 1, and is therefore a [[algebraic curve#Rational curves|rational plane algebraic curve]] of genus zero. The cruciform curve has three double points in the [[real projective plane]], at ''x''=0 and ''y''=0, ''x''=0 and ''z''=0, and ''y''=0 and ''z''=0. 
<ref>{{MathWorld|title=Cruciform curve|urlname=Cruciform}}</ref>

Because the curve is rational, it can be parametrized by rational functions. For instance, if ''a''=1 and ''b''=2, then
:<math>x = -\frac{t^2-2t+5}{t^2-2t-3},\quad y = \frac{t^2-2t+5}{2t-2}</math>
parametrizes the points on the curve outside of the exceptional cases where a denominator is zero.

[[File:inverse_pythagorean_theorem.svg|thumb|Illustration of the inverse Pythagorean and regular Pythagorean theorems]] 
The [[inverse Pythagorean theorem]] is obtained from the above equation by substituting ''x'' with ''AC'', ''y'' with ''BC'', and each ''a'' and ''b'' with ''CD'', where ''A'', ''B'' are the endpoints of the hypotenuse of a right triangle ''ABC'', and ''D'' is the foot of a perpendicular dropped from ''C'', the vertex of the right angle, to the hypotenuse:
:<math>\begin{align}
   AC^2 BC^2 - CD^2 AC^2 - CD^2 BC^2 &= 0
\\ AC^2 BC^2                         &= CD^2 BC^2 + CD^2 AC^2
\\ \frac{1}{CD^2}                    &= \frac{BC^2}{AC^2 \cdot BC^2} + \frac{AC^2}{AC^2 \cdot BC^2}
\\ \therefore \;\; \frac{1}{CD^2}    &= \frac{  1 }{AC^2           } + \frac{  1 }{           BC^2}
\end{align}</math>

===Spiric section===
{{main|Spiric section}}
Spiric sections can be defined as [[Circular algebraic curve|bicircular]] quartic curves that are symmetric with respect to the ''x'' and ''y'' axes. Spiric sections are included in the family of [[toric section]]s and include the family of [[hippopede]]s and the family of [[Cassini oval]]s. The name is from σπειρα meaning torus in ancient Greek.

The Cartesian equation can be written as

:<math>(x^2+y^2)^2=dx^2+ey^2+f , </math>

and the equation in polar coordinates as

:<math>r^4=dr^2\cos^2\theta+er^2\sin^2\theta+f. \, </math>

===Three-leaved clover (trifolium)===
The '''three-leaved clover''' or '''trifolium'''<ref>{{MathWorld|title=Trifolium|urlname=Trifolium}}</ref> is the quartic plane curve
:<math>
x^4+2x^2y^2+y^4-x^3+3xy^2=0. \,
</math>
By solving for ''y'', the curve can be described by the following function:
:<math>
y=\pm\sqrt{\frac{-2x^2-3x\pm\sqrt{16x^3+9x^2}}{2}},
</math>
where the two appearances of ± are independent of each other, giving up to four distinct values of ''y'' for each ''x''.

The parametric equation of curve is
:<math>
x = \cos(3t) \cos t,\quad y = \cos(3t) \sin t. \,
</math><ref>
Gibson, C. G., ''Elementary Geometry of Algebraic Curves, an Undergraduate Introduction'', Cambridge University Press, Cambridge, 2001, {{isbn|978-0-521-64641-3}}. Pages 12 and 78.
</ref>

In polar coordinates (''x'' = ''r''&nbsp;cos&nbsp;φ, ''y'' = ''r''&nbsp;sin&nbsp;φ) the equation is
:<math>r = \cos(3\varphi). \,</math>
It is a special case of [[rose curve]] with ''k'' =&nbsp;3.
This curve has a triple point at the origin (0,&nbsp;0) and has three double tangents.

==See also==
*[[Ternary quartic]]
*[[Bitangents of a quartic]]

==References==
{{Reflist}}

{{Authority control}}

[[Category:Quartic curves| ]]