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| A '''quartic plane curve''' is a [[plane curve]] of the fourth degree. It can be defined by a quartic equation:
| | Greetings! I am Myrtle Shroyer. South Dakota is where I've always been living. He used to be unemployed but now he is a pc operator but his marketing by no means comes. Doing ceramics is what adore performing.<br><br>Also visit my web blog; [http://cs.sch.ac.kr/xe/index.php?document_srl=1203900&mid=Game13 at home std testing] |
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| :<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>
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| This equation has fifteen constants. However, it can be multiplied by any non-zero constant without changing the curve. Therefore, the space of quartic curves can be identified with the [[real projective space]] <math>\mathbb{RP}^{14}</math>. It also follows that there is exactly one quartic curve that passes through a set of fourteen distinct points in [[general position]], since a quartic has 14 [[Degrees of freedom (physics and chemistry)|degrees of freedom]].
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| A quartic curve can have a maximum of:
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| * Four connected components
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| * Twenty-eight [[bitangent|bi-tangents]]
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| * Three ordinary [[double point]]s.
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| ==Examples==
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| {{multiple image
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| |align=right
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| |image1=Ampersandcurve.svg
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| |caption1=Ampersand curve
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| |image2=Bean curve.svg
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| |caption2=Bean curve
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| |image3=Bicuspid curve.svg
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| |caption3=Bicuspid curve
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| |image4=Bowcurve.svg
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| |caption4=Bow curve
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| |image5=Cruciform1.png
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| |caption5=Cruciform curve with parameters (b,a) being (1,1) in red; (2,2) in green; (3,3) in blue.
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| |image6=Cruciform2.png
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| |caption6=Cruciform curve with parameters (b,a) being (1,1) in red; (2,1) in green; (3,1) in blue.
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| |image7=Spiric section.svg
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| |caption7=[[Spiric section]]
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| |image8=Three-leaved clover.svg
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| |caption8=Three-leaved clover in [[cartesian coordinates]]
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| |image9=Three-leaved clover polar.svg
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| |caption9=Three-leaved clover in [[polar coordinates]]
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| }}
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| Various combinations of coefficients in the above equation give rise to various important families of curves as listed below.
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| {| border="0" width="60%"
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| | [[Bicorn]] curve || [[Klein quartic]]
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| | [[Bullet-nose curve]] || [[Lemniscate of Bernoulli]]
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| | [[Cartesian oval]] || [[Lemniscate of Gerono]]
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| | [[Cassini oval]] || [[Lüroth quartic]]
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| | [[Deltoid curve]] || [[Spiric section]]
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| | [[Hippopede]] || [[Toric section]]
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| | [[Kampyle of Eudoxus]] || [[Trott curve]]
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| |}
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| ===Ampersand curve===
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| The '''ampersand curve''' is a [[quartic plane curve]] given by the equation:
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| :<math>\ (y^2-x^2)(x-1)(2x-3)=4(x^2+y^2-2x)^2.</math>
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| It is an [[algebraic curve]] of [[Genus (mathematics)#Graph theory|genus]] zero, with three ordinary double points, all in the real plane.
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| <ref>{{MathWorld|title=Ampersand Curve|urlname=AmpersandCurve}}</ref> | |
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| ===Bean curve===
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| The '''bean curve''' is a [[quartic plane curve]] with the equation:
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| :<math>x^4+x^2y^2+y^4=x(x^2+y^2) \,</math>
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| The bean curve is a [[algebraic curve|plane algebraic curve]] of [[geometric genus|genus]] zero. It has one [[Mathematical singularity|singularity]] at the origin, an ordinary triple point.
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| <ref>{{Citation | last1=Cundy | first1=H. Martyn | last2=Rollett | first2=A. P. | title=Mathematical models | origyear=1952 | publisher=Clarendon Press, Oxford | edition=2nd | isbn=978-0-906212-20-2 | mr=0124167 | year=1961 | page=72}}</ref>
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| <ref> {{MathWorld|title=Bean Curve|urlname=BeanCurve}} </ref>
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| ===Bicuspid curve===
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| The '''biscuspid''' is a [[quartic plane curve]] with the equation
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| :<math>(x^2-a^2)(x-a)^2+(y^2-a^2)^2=0 \,</math>
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| where ''a'' determines the size of the curve.
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| The bicuspid has only the two nodes as singularities, and hence is a curve of genus one.
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| <ref>{{MathWorld|title=Bicuspid Curve|urlname=BicuspidCurve}}</ref>
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| ===Bow curve===
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| The '''bow curve''' is a [[quartic plane curve]] with the equation:
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| :<math>x^4=x^2y-y^3. \,</math>
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| The bow curve has a single triple point at x=0, y=0, and consequently is a rational curve, with [[geometric genus|genus]] zero.
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| <ref>{{MathWorld|title=Bow|urlname=Bow}}</ref>
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| ===Cruciform curve===
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| The '''cruciform curve''', or '''cross curve''' is a [[quartic plane curve]] given by the equation
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| :<math>x^2y^2-b^2x^2-a^2y^2=0 \,</math>
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| where ''a'' and ''b'' are two [[parameter]]s determining the shape of the curve.
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| 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 plane algebraic curve]] of [[geometric genus|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.
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| <ref>{{MathWorld|title=Cruciform curve|urlname=Cruciform}}</ref>
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| Because the curve is rational, it can be parametrized by rational functions. For instance, if a=1 and b=2, then
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| :<math>x = -\frac{t^2-2t+5}{t^2-2t-3}, y = \frac{t^2-2t+5}{2t-2}</math>
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| parametrizes the points on the curve outside of the exceptional cases where the denominator is zero.
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| ===Spiric section===
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| {{main|spiric section}}
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| 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.
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| ===Three-leaved clover===
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| The '''three-leaved clover''' is a [[quartic plane curve]]
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| :<math>
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| x^4+2x^2y^2+y^4-x^3+3xy^2=0 \,
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| </math>
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| Alternatively, parametric equation of three-leaved clover is:
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| :<math>
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| x = \cos(3t) \cos t, y = \cos(3t) \sin t \,
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| </math><ref>
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| 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.
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| </ref>
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| Or in polar coordinates (''x'' = ''r'' cos φ, ''y'' = ''r'' sin φ):
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| :<math>r = \cos(3\varphi)\,</math>
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| It is a special case of [[rose curve]] with ''k'' = 3.
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| This curve has a triple point at the origin (0, 0) and has three double tangents.
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| ==References==
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| <references/>
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| [[Category:Algebraic curves]]
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| [[es:Curva cuártica]]
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| [[sl:Krivulja četrte stopnje]]
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Greetings! I am Myrtle Shroyer. South Dakota is where I've always been living. He used to be unemployed but now he is a pc operator but his marketing by no means comes. Doing ceramics is what adore performing.
Also visit my web blog; at home std testing