Cassinijev oval je kriva četvrtog reda, koja se može definirati kao geometrijsko mjesto tačaka koje zadovoljavaju uslov da je konstantan proizvod njihove razdaljine od dvije fiksne tačke. Kriva je imenovana prema astronomu Giovanniju Domenicu Cassiniju , koji ju je proučavao 1680. Cassini je pogrešno smatrao da ta kriva tačnije predstavlja kretanje Zemlje .
Neka su
Q
1
{\displaystyle Q_{1}}
Q
2
{\displaystyle Q_{2}}
a
{\displaystyle a}
Q
1
{\displaystyle Q_{1}}
Q
2
{\displaystyle Q_{2}}
P
{\displaystyle P}
Q
1
{\displaystyle Q_{1}}
Q
2
{\displaystyle Q_{2}}
a
2
{\displaystyle a^{2}}
d
(
Q
1
,
P
)
×
d
(
Q
2
,
P
)
=
a
2
.
{\displaystyle \operatorname {d} (Q_{1},P)\times \operatorname {d} (Q_{2},P)=a^{2}.\,}
Neka su fokusi u
F
1
(
−
c
,
0
)
{\displaystyle F_{1}(-c,0)}
F
2
(
c
,
0
)
{\displaystyle F_{2}(c,0)}
M
(
x
;
y
)
{\displaystyle M(x;y)}
a
2
{\displaystyle a^{2}}
d
1
=
(
x
c
)
2
y
2
{\displaystyle d_{1}={\sqrt {(x c)^{2} y^{2}}}}
d
2
=
(
x
−
c
)
2
y
2
{\displaystyle d_{2}={\sqrt {(x-c)^{2} y^{2}}}}
d
1
⋅
d
2
=
a
2
{\displaystyle d_{1}\cdot d_{2}=a^{2}}
(
x
c
)
2
y
2
⋅
(
x
−
c
)
2
y
2
=
a
2
{\displaystyle {\sqrt {(x c)^{2} y^{2}}}\cdot {\sqrt {(x-c)^{2} y^{2}}}=a^{2}}
(
(
x
c
)
2
y
2
)
⋅
(
(
x
−
c
)
2
y
2
)
=
a
4
{\displaystyle {\Big (}(x c)^{2} y^{2}{\Big )}\cdot {\Big (}(x-c)^{2} y^{2}{\Big )}=a^{4}}
(
x
2
−
c
2
)
2
y
4
2
y
2
(
x
2
c
2
)
=
a
4
{\displaystyle (x^{2}-c^{2})^{2} y^{4} 2y^{2}(x^{2} c^{2})=a^{4}}
(
x
2
y
2
)
2
−
2
c
2
(
x
2
−
y
2
)
=
a
4
−
c
4
{\displaystyle (x^{2} y^{2})^{2}-2c^{2}(x^{2}-y^{2})=a^{4}-c^{4}}
Mijenja se parametar
a
{\displaystyle a}
Mijenja se parametar
c
{\displaystyle c}
Pođemo li od oblika u pravougaonim koordinatama
(
x
2
y
2
)
2
−
2
c
2
(
x
2
−
y
2
)
=
a
4
−
c
4
{\displaystyle \textstyle (x^{2} y^{2})^{2}-2c^{2}(x^{2}-y^{2})=a^{4}-c^{4}}
zamjenom
x
=
ρ
cos
φ
,
y
=
ρ
sin
φ
,
{\displaystyle x=\rho \cos \varphi ,\,y=\rho \sin \varphi ,}
(
ρ
2
cos
2
φ
ρ
2
sin
2
φ
)
2
−
2
c
2
(
ρ
2
cos
2
φ
−
ρ
2
sin
2
φ
)
=
a
4
−
c
4
{\displaystyle {\Big (}\rho ^{2}\cos ^{2}\varphi \rho ^{2}\sin ^{2}\varphi {\Big )}^{2}-2c^{2}{\Big (}\rho ^{2}\cos ^{2}\varphi -\rho ^{2}\sin ^{2}\varphi {\Big )}=a^{4}-c^{4}}
ρ
4
−
2
c
2
ρ
2
(
c
o
s
2
φ
−
sin
2
φ
)
=
a
4
−
c
4
{\displaystyle \rho ^{4}-2c^{2}\rho ^{2}(cos^{2}\varphi -\sin ^{2}\varphi )=a^{4}-c^{4}}
(
cos
2
α
−
sin
2
α
=
c
o
s
2
α
{\displaystyle \cos ^{2}\alpha -\sin ^{2}\alpha =cos2\alpha }
ρ
4
−
2
c
2
ρ
2
cos
2
φ
=
a
4
−
c
4
{\displaystyle \textstyle \rho ^{4}-2c^{2}\rho ^{2}\cos 2\varphi =a^{4}-c^{4}}
Jednačina Cassinijevog ovala ima dva nezavisna parametra;
c
{\displaystyle c}
a
{\displaystyle a}
Međusobni odnos parametara određuje oblik Cassinijevog ovala, tako da postoji više različitih oblika u zavisnosti od kvocijenta dva parametra
c
a
=
∞
{\displaystyle \textstyle {\frac {c}{a}}=\infty }
1
<
c
a
<
∞
{\displaystyle \textstyle 1<{\frac {c}{a}}<\infty }
c
a
=
1
{\displaystyle \textstyle {\frac {c}{a}}=1}
a
=
c
{\displaystyle \textstyle a=c}
Bernulijevu lemniskatu
1
2
<
c
a
<
1
{\displaystyle \textstyle {\frac {1}{\sqrt {2}}}<{\frac {c}{a}}<1}
c
<
a
<
c
2
{\displaystyle \textstyle c<a<c{\sqrt {2}}}
0
<
c
a
⩽
1
2
{\displaystyle \textstyle 0<{\frac {c}{a}}\leqslant {\frac {1}{\sqrt {2}}}}
a
⩾
c
2
{\displaystyle \textstyle a\geqslant c{\sqrt {2}}}
c
a
=
0
{\displaystyle \textstyle {\frac {c}{a}}=0}
c
=
0
{\displaystyle \textstyle c=0}
a
≠
0
{\displaystyle \textstyle a\neq 0}
Radijus zakrivljenosti u polarnim koordinatama je
R
=
a
2
ρ
ρ
2
c
2
cos
2
φ
=
2
a
2
ρ
3
c
4
−
a
4
3
ρ
4
{\displaystyle R={\frac {a^{2}\rho }{\rho ^{2} c^{2}\cos {2\varphi }}}={\frac {2a^{2}\rho ^{3}}{c^{4}-a^{4} 3\rho ^{4}}}}
