In matematica una curva sestica è una curva algebrica piana di sesto grado. Può essere definita da un polinomio della forma:
![{\displaystyle A_{1}X^{6}+A_{2}Y^{6}+A_{3}X^{5}Y+A_{4}XY^{5}+A_{5}X^{4}Y^{2}+A_{6}X^{2}Y^{4}+A_{7}X^{3}Y^{3}+A_{8}X^{5}+A_{9}Y^{5}+A_{10}X^{4}Y+A_{11}XY^{4}+A_{12}X^{3}Y^{2}+A_{13}X^{2}Y^{3}+A_{14}X^{4}+A_{15}Y^{4}+A_{16}X^{3}Y+A_{17}XY^{3}+A_{18}X^{2}Y^{2}+A_{19}X^{3}+A_{20}Y^{3}+A_{21}X^{2}Y+A_{22}XY^{2}+A_{23}X^{2}+A_{24}Y^{2}+A_{25}XY+A_{26}X+A_{27}Y+A_{28}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f997d3aada352ba93eab0e2a3257c5f52d55c4)
L'equazione ha 28 coefficienti, ma la curva non cambia se li moltiplichiamo tutti per una costante non nulla. Quindi i coefficienti essenziali sono 27 e le sestiche sono ∞
. E una di esse è individuata dal suo passaggio per 27 punti generici.
Una curva sestica (
) irriducibile può avere al massimo:
componenti connesse;
punti doppi;
rette bitangenti;
punti di flesso.
![{\displaystyle 10(4x^{2}+4y^{2}-1)^{3}-500x^{2}y^{2}+3=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d7d5f8bcfb229102cfd23e8ff9bedeaf30b3923)
![{\displaystyle \left({\frac {x}{3}}\right)^{6}+\left({\frac {y}{2}}\right)^{6}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15ca43ad900267cd3bbffb1de95d2d2dd465e7d4)
![{\displaystyle x^{6}+y^{6}-x^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36139e7f6083c639ccd00b00c831b15e6b46999b)
![{\displaystyle (1-x^{2})^{3}-y^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/966b95033f3800fed1d79108b2e88b2e0852fd39)
![{\displaystyle x^{6}+y^{6}-6y^{5}-y^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57629656a19843a4d2d442826a13879a6186b919)
![{\displaystyle (x^{2}+y^{2}-1)^{3}-x^{2}y^{3}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa24eeba3ea979978aabb8a1081a4de0e2d1a5b3)
![{\displaystyle (x^{2}+y^{2})^{3}-x^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5366f0e815d3add623656484706c8c1e33a40bbc)
![{\displaystyle (x^{2}+y^{2})^{3}-x^{4}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c0e1fcb9e15b8127ffaec903c7ded601ff627d2)
![{\displaystyle x^{2}(3x^{2}-y^{2})^{2}-y^{2}(x^{2}+y^{2})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c2017f8e48ec9fd35ca9b36f5947cd41d5de3bc)
![{\displaystyle (x^{2}+y^{2})^{3}-(3x^{2}-y^{2})^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2968cb49bc6a2cb4c06facdd44465fbac062e424)
![{\displaystyle x^{6}-x^{5}+y^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e281dda4c27b51f7aa37086ab06556b043c3c300)
![{\displaystyle 4(x^{2}+y^{2}-x)^{3}-27(x^{2}+y^{2})^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45d53fe11ac21befdc530571b89de3a999c064a5)
![{\displaystyle (x^{2}+y^{2}-1)^{3}+27x^{2}y^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab3b51ba25f62fb6bda2149a4f817c6b4e8674ab)
![{\displaystyle x^{4}(x^{2}+y^{2})-(x^{2}-y^{2})^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/679d9eb091a54cda3d7e48b3b02c134ee140d0c5)
![{\displaystyle x^{6}-x^{4}+y^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b9b2b3b7a1ef5b7345ac104d90efa6b710e45a)
![{\displaystyle (x^{2}+y^{2})^{3}-x^{2}(x^{2}+20y^{2})+8y^{2}(y^{2}+2)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4fbcbf36ce36749ee3f6f658c5229276476e43d)
- Curva a quadrifoglio dritto
![{\displaystyle (x^{2}+y^{2})^{3}-(x^{2}-y^{2})^{2})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca97d1777ffeb55827beb4e9885dffcb53c32c1d)
- Curva a quadrifoglio obliquo
![{\displaystyle (x^{2}+y^{2})^{3}-4x^{2}y^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf4de9e13b6644bcf862efe14d23d90bdf0f6b7)
![{\displaystyle x^{4}(x^{2}+y^{2})-(3x^{2}-2)^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94f1bc6b288fbcf46554f1343124da65d4bedf0e)
![{\displaystyle y^{2}-(1-x^{2})(1-4x^{2})^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01e6dcdf978bbaab0793c4311e7e1182841cba0b)
![{\displaystyle (x^{2}+y^{2})^{3}-y^{2}(5y^{2}+6x^{2}-8)+3x^{4}-4=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73dc647d02c1e72714921851ef07f073f46cf75e)
![{\displaystyle (x^{2}+y^{2})(2x^{2}+2y^{2}-1)^{2}-x^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/685ce3f928890059d42008d0cc16bd950e27e4ed)
- Curva radiale dell'ellisse
![{\displaystyle (a^{2}x^{2}+b^{2}y^{2})^{3}-a^{4}b^{4}(x^{2}+y^{2})^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b369fa6006eb5b72a07a73b0aa443bfbf9fbba5)
![{\displaystyle (x^{2}+y^{2})(x^{2}+y^{2}+ax)^{2}-b^{2}(x^{2}-y^{2})^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f25ac0bfb43f7f9ab43531a8227a083c9f4a6405)
![{\displaystyle (x^{2}+y^{2})(x^{2}+y^{2}-x-y)^{2}-ax^{2}y^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ba495da1e9d1c971bd80bad054244eae465789)
![{\displaystyle (x^{2}+y^{2})(x^{2}+a^{2}-b^{2})^{2}-4a^{2}x^{4}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eef940e69f8a6db535e09cfd8e93cc81eb5edde9)
![{\displaystyle (a^{2}x^{2}+y^{2}-4)^{3}-108y^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/190ef007f63084cf451d974a8be805f9cf69cfbc)
![{\displaystyle 4x^{2}y^{2}(x^{2}+y^{2})-a^{2}(x^{2}-y^{2})^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e47bb1609cec936b2e21b5a0182c0cfb2a836e8)
![{\displaystyle (x^{2}+y^{2})(x^{2}+y^{2}-c^{2})^{2}+4a^{2}y^{2}(x^{2}+y^{2}-b^{2})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/802ca3c4782ae17df6dca9fc2edeb858d20fefe4)
![{\displaystyle a^{4}(y^{2}-3x^{2})+8a^{3}x(x^{2}+y^{2})-6a^{2}(x^{2}+y^{2})^{2}+(x^{2}+y^{2})^{3}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf749f23aa84db58ccb71d6d50aa6afa7244cc1c)
![{\displaystyle x^{2}(2x^{4}-7x^{2}+7)+3y(2y^{5}-2y^{3}-3y-1)-xy(xy^{2}+2x^{3}+7xy+7x)-3=0:}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d18b5b351af391e6267a1727eaf6b5f5a0f25ef9)
![{\displaystyle x^{2}(2x^{4}-9x^{2}+5)+9y^{2}(y^{4}+3y^{3}+2y^{2}+4y-4)+2x^{2}y(4x^{2}y-5y^{3}-6y^{2}+4y-9)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0730b57e7ddb47a94a78bae2647cc05453b0ce1e)
![{\displaystyle x^{4}(29x^{2}-45x-20)+y^{4}(14y^{2}+45y-45)+xy(35x^{4}+12y^{4}-20x^{3}+30y^{3})+5x^{2}y^{2}(6y^{2}-5x^{2}-8xy-2x-3y)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/557101b1b260bce4b9b5a2d83a6cfe6ccc66fa44)
![{\displaystyle a^{6}x^{6}+b^{6}y^{6}+3a^{4}b^{2}x^{4}y^{2}+3a^{2}b^{4}x^{2}y^{4}+(8a^{4}b^{4}-8a^{6}b^{2}-a^{8})x^{4}+(8a^{4}b^{4}-8a^{2}b^{6}-b^{8})y^{4}+(38a^{4}b^{4}-20a^{6}b^{2}-20a^{2}b^{6})x^{2}y^{2}+(8a^{6}b^{4}-32a^{4}b^{6}+16a^{2}b^{8}+8a^{8}b^{2})x^{2}+(8a^{2}b^{8}+8a^{4}b^{6}+16a^{8}b^{2}-32a^{6}b^{4})y^{2}+32a^{6}b^{6}-16a^{8}b^{4}-16a^{4}b^{8}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29f3de907548dfc1cf901cf1d38e48b573bb43fc)
![{\displaystyle [\sin {A}(x\sin {C}-y\cos {C}-d\sin {C})(x^{2}+y^{2}+b^{2}-e^{2})+y\sin {B}(x^{2}+y^{2}-2dx+d^{2}+a^{2}-f^{2})]^{2}+[\sin {A}(x\cos {C}+y\sin {C}-d\cos {C})(x^{2}+y^{2}+b^{2}-e^{2})-x\sin {B}(x^{2}+y^{2}-2dx+d^{2}+a^{2}-f^{2})]^{2}-4a^{2}\sin ^{2}{B}[\sin {C}(x^{2}-dx-y)-dy\cos {C}]^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f032b000027776db62327bc3c33a4e0fc63b14d8)
ove
sono gli angoli opposti ai lati
di un triangolo
![{\displaystyle (5x^{2}+5y^{2}-6)(3x^{2}-y^{2}-1)(9x^{2}-7y^{2}+1)-1=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0bca862c63af47e496cb955be81ac4cbb0cbb2d)
![{\displaystyle 2(5x^{2}+3y^{2}-6)(2y^{2}-4y^{2}+5)(10x^{2}-3y^{2}-7)+1=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee1e233e7d53ce0cf885feeb25fdf24b86e38047)
![{\displaystyle (x^{2}+y^{2}-1)(20x^{2}+y^{2}-2)(x^{2}+20y^{2}-2)+1=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccb71629a458e11b8ae170a807965f256655af9b)
![{\displaystyle x^{6}+5,8619y^{6}+4,0033x^{5}y-10,743xy^{5}-7,549x^{4}y^{2}+6,8793x^{2}y^{4}-2,7873x^{3}y^{3}-14,01x^{5}-14,07y^{5}+26,891x^{4}y+19,518xy^{4}+26,291x^{3}y^{2}-47,496x^{2}y^{3}-43,733x^{4}-69,449y^{4}-55,528x^{3}y+54,722xy^{3}+162,51x^{2}y^{2}+69,253x^{3}+303,43y^{3}-217,82x^{2}y-135,8xy^{2}+140,02x^{2}-419,88y^{2}+124,15xy-110,32x+204,46y+0,2014=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65e05ee119197a9f9ad5d84b08da1849944abf1b)
-
Curva a girandola
-
Superellisse
-
Curva a farfalla
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Curva a bacio
-
Curva a mongolfiera
-
Curva a cuore
-
Curva a dipolo
-
Curva a doppio uovo
-
Archi di Samotracia
-
Cicloide di Ceva
-
Curva a lacrima
-
Curva di Cayley
-
Curva astroide
-
Curva nodo a farfalla
-
Curva a manubrio
-
Curva a Croce di Malta
-
Curva a quadrifoglio dritto
-
Curva a quadrifoglio obliquo
-
Curva atriftaloide
-
Curva di Lissajous
-
Curva cornoide
-
Curva foglio di Dürer
-
Curva radiale dell'ellisse
-
Curva a scarabeo dritto
-
Curva a scarabeo obliquo
-
Curva a biella-manovella
-
Curva nefroide
-
Curva a mulino a vento
-
Curva di Watt
-
Curva nefroide di Freeth
-
Curva a vaso
-
Curva a telefono
-
Curva a tomahawk
-
Curva di Talbot
-
Curva a tre barre
-
Curva con 20 flessi
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Curva decaconnessa
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Curva a 144 bitangenti
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Curva ennanodata