Rose curve with angular frequency 2
Rotated quadrifolium Quadrifolium created with gears This article is about the geometric shape. For the plant, see
Four-leaf clover . For the symmetrical shape framework, see
Quatrefoil .
The quadrifolium (also known as four-leaved clover [ 1] rose curve with an angular frequency of 2. It has the polar equation :
r
=
a
cos
-->
(
2
θ θ -->
)
,
{\displaystyle r=a\cos(2\theta ),\,}
with corresponding algebraic equation
(
x
2
+
y
2
)
3
=
a
2
(
x
2
− − -->
y
2
)
2
.
{\displaystyle (x^{2}+y^{2})^{3}=a^{2}(x^{2}-y^{2})^{2}.\,}
Rotated counter-clockwise by 45°, this becomes
r
=
a
sin
-->
(
2
θ θ -->
)
{\displaystyle r=a\sin(2\theta )\,}
with corresponding algebraic equation
(
x
2
+
y
2
)
3
=
4
a
2
x
2
y
2
.
{\displaystyle (x^{2}+y^{2})^{3}=4a^{2}x^{2}y^{2}.\,}
In either form, it is a plane algebraic curve of genus zero.
The dual curve to the quadrifolium is
(
x
2
− − -->
y
2
)
4
+
837
(
x
2
+
y
2
)
2
+
108
x
2
y
2
=
16
(
x
2
+
7
y
2
)
(
y
2
+
7
x
2
)
(
x
2
+
y
2
)
+
729
(
x
2
+
y
2
)
.
{\displaystyle (x^{2}-y^{2})^{4}+837(x^{2}+y^{2})^{2}+108x^{2}y^{2}=16(x^{2}+7y^{2})(y^{2}+7x^{2})(x^{2}+y^{2})+729(x^{2}+y^{2}).\,}
Dual quadrifolium The area inside the quadrifolium is
1
2
π π -->
a
2
{\displaystyle {\tfrac {1}{2}}\pi a^{2}}
perimeter of the quadrifolium is
8
a
E
-->
(
3
2
)
=
4
π π -->
a
(
(
52
3
− − -->
90
)
M
′
-->
(
1
,
7
− − -->
4
3
)
M
2
-->
(
1
,
7
− − -->
4
3
)
+
7
− − -->
4
3
M
-->
(
1
,
7
− − -->
4
3
)
)
{\displaystyle 8a\operatorname {E} \left({\frac {\sqrt {3}}{2}}\right)=4\pi a\left({\frac {(52{\sqrt {3}}-90)\operatorname {M} '(1,7-4{\sqrt {3}})}{\operatorname {M} ^{2}(1,7-4{\sqrt {3}})}}+{\frac {7-4{\sqrt {3}}}{\operatorname {M} (1,7-4{\sqrt {3}})}}\right)}
where
E
-->
(
k
)
{\displaystyle \operatorname {E} (k)}
complete elliptic integral of the second kind with modulus
k
{\displaystyle k}
M
{\displaystyle \operatorname {M} }
arithmetic–geometric mean and
′
{\displaystyle '}
derivative with respect to the second variable.[ 2]
Notes
References
External links
