O curbă algebrică
C
n
{\displaystyle C_{n}\!}
n care trece prin punctele ciclice al planului se numeşte curbă circulară .
Dacă
C
n
{\displaystyle C_{n}\!}
m (
2
m
≤
n
{\displaystyle 2m\leq n\!}
m -circulară.
Pentru a determina ecuaţia unei astfel de curbe luăm:
φ
k
(
x
,
y
)
=
a
k
0
x
k
+
a
k
−
1
1
x
k
−
1
y
+
⋯
+
a
0
y
k
k
=
0
,
1
,
⋯
,
n
{\displaystyle \varphi _{k}(x,y)=a_{k0}x^{k}+a_{k-1\;1}x^{k-1}y+\cdots +a_{0}y^{k}\;\;k=0,1,\cdots ,n\!}
(1) Avem:
C
n
:
F
(
x
,
y
)
=
φ
n
(
x
,
y
)
+
φ
n
−
1
(
x
,
y
)
+
⋯
+
φ
1
(
x
,
y
)
+
φ
0
=
0.
{\displaystyle C_{n}\ :F(x,y)=\varphi _{n}(x,y)+\varphi _{n-1}(x,y)+\cdots +\varphi _{1}(x,y)+\varphi _{0}=0.\!}
(2) Dacă
C
n
{\displaystyle C_{n}\!}
I
(
1
,
i
,
0
)
∈
C
n
{\displaystyle I(1,i,0)\in C_{n}\!}
J
(
1
,
−
i
,
0
)
∈
C
n
,
{\displaystyle J(1,-i,0)\in C_{n},\!}
φ
n
(
1
,
i
)
=
0
,
φ
n
(
1
,
−
i
)
=
0
;
{\displaystyle \varphi _{n}(1,i)=0,\;\varphi _{n}(1,-i)=0;\!}
C
n
{\displaystyle C_{n}\!}
φ
n
(
x
,
y
)
=
(
x
2
+
y
2
)
ζ
n
−
2
(
x
,
y
)
.
{\displaystyle \varphi _{n}(x,y)=(x^{2}+y^{2})\zeta _{n-2}(x,y).\!}
(3)
C
n
{\displaystyle C_{n}\!}
m -circulară, atunci:
φ
n
=
(
x
2
+
y
2
)
m
ζ
n
−
2
m
,
φ
n
−
1
=
{\displaystyle \varphi _{n}=(x^{2}+y^{2})^{m}\zeta _{n-2m},\;\;\varphi _{n-1}=\!}
=
(
x
2
+
y
2
)
m
−
1
ζ
n
−
2
m
+
1
,
⋯
,
φ
n
−
m
+
1
=
(
x
2
+
y
2
)
ζ
n
−
m
−
1
{\displaystyle =(x^{2}+y^{2})^{m-1}\zeta _{n-2m+1},\cdots ,\varphi _{n-m+1}=(x^{2}+y^{2})\zeta _{n-m-1}\!}
Căci, dacă
C
n
{\displaystyle C_{n}\!}
I şi J sunt puncte duble (puncte multiple de ordinul doi) pentru curbă.
Deci, în (3) avem:
ζ
n
−
2
(
x
,
y
)
=
(
x
2
+
y
2
)
ζ
n
−
4
(
x
,
y
)
.
{\displaystyle \zeta _{n-2}(x,y)=(x^{2}+y^{2})\zeta _{n-4}(x,y).\!}
Pe de altă parte, dreapta
y
=
i
x
+
λ
{\displaystyle y=ix+\lambda \!}
y
=
−
i
x
+
λ
{\displaystyle y=-ix+\lambda \!}
I , (respectiv J ), taie pe
C
n
{\displaystyle C_{n}\!}
I (J ) şi în alte n-2 puncte la distanţă finită; abscisele acestor puncte sunt rădăcinile ecuaţiei:
F
(
x
,
i
x
+
λ
)
=
φ
n
−
1
(
1
,
i
)
x
n
−
1
+
{\displaystyle F(x,ix+\lambda )=\varphi _{n-1}(1,i)x^{n-1}+\!}
+
[
−
4
λ
2
ζ
n
−
4
(
1
,
i
)
+
∂
φ
n
−
1
(
1
,
i
)
∂
y
+
φ
n
−
2
(
1
,
i
)
]
x
n
−
2
+
⋯
+
a
00
=
0.
{\displaystyle +\left[-4\lambda ^{2}\zeta _{n-4}(1,i)+{\frac {\partial \varphi _{n-1}(1,i)}{\partial y}}+\varphi _{n-2}(1,i)\right]x^{n-2}+\cdots +a_{00}=0.\!}
Deci I e punct dublu dacă
φ
n
−
1
(
1
,
i
)
=
0
;
{\displaystyle \varphi _{n-1}(1,i)=0;\!}
φ
n
−
1
=
(
x
2
+
y
2
)
ζ
n
−
3
.
{\displaystyle \varphi _{n-1}=(x^{2}+y^{2})\zeta _{n-3}.\!}
Aşadar, ecuaţia unei curbe bicirculare
C
n
{\displaystyle C_{n}\!}
(
x
2
+
y
2
)
2
ζ
n
−
4
+
(
x
2
+
y
2
)
ζ
n
−
3
+
φ
n
−
2
+
⋯
+
φ
1
+
φ
0
=
0.
{\displaystyle (x^{2}+y^{2})^{2}\zeta _{n-4}+(x^{2}+y^{2})\zeta _{n-3}+\varphi _{n-2}+\cdots +\varphi _{1}+\varphi _{0}=0.\!}
![{\displaystyle +\left[-4\lambda ^{2}\zeta _{n-4}(1,i)+{\frac {\partial \varphi _{n-1}(1,i)}{\partial y}}+\varphi _{n-2}(1,i)\right]x^{n-2}+\cdots +a_{00}=0.\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/2a044da8f5631baf1db5bfc4e4961ded148b8adc)
