If x is a real number and [x ] denotes the greatest integer not greater than x , then
x
−
[
x
]
∈
[
0
,
1
]
.
{\displaystyle x-[x]\in [0,1].}
If
γ
∈
H
{\displaystyle \gamma \in H}
n is a positive integer, then define a curve
γ
n
{\displaystyle \gamma _{n}}
γ
n
(
t
)
=
γ
(
n
t
−
[
n
t
]
)
.
{\displaystyle \gamma _{n}(t)=\gamma (nt-[nt]).\,}
γ
n
{\displaystyle \gamma _{n}}
1 and we call it an n -curve.
Note that every curve in H is a 1-curve.
Suppose
α
,
β
∈
H
.
{\displaystyle \alpha ,\beta \in H.}
α
(
0
)
=
β
(
1
)
=
1
,
the f-product
α
⋅
β
=
β
+
α
−
e
{\displaystyle \alpha (0)=\beta (1)=1,{\mbox{ the f-product }}\alpha \cdot \beta =\beta +\alpha -e}
Example 1: Product of the astroid with the n -curve of the unit circle
edit
Let us take u , the unit circle centered at the origin and α, the astroid .
The n -curve of u is given by,
u
n
(
t
)
=
cos
(
2
π
n
t
)
+
i
sin
(
2
π
n
t
)
{\displaystyle u_{n}(t)=\cos(2\pi nt)+i\sin(2\pi nt)\,}
and the astroid is
α
(
t
)
=
cos
3
(
2
π
t
)
+
i
sin
3
(
2
π
t
)
,
0
≤
t
≤
1
{\displaystyle \alpha (t)=\cos ^{3}(2\pi t)+i\sin ^{3}(2\pi t),0\leq t\leq 1}
The parametric equations of their product
α
⋅
u
n
{\displaystyle \alpha \cdot u_{n}}
x
=
cos
3
(
2
π
t
)
+
cos
(
2
π
n
t
)
−
1
,
{\displaystyle x=\cos ^{3}(2\pi t)+\cos(2\pi nt)-1,}
y
=
sin
3
(
2
π
t
)
+
sin
(
2
π
n
t
)
{\displaystyle y=\sin ^{3}(2\pi t)+\sin(2\pi nt)}
See the figure.
Since both
α
and
u
n
{\displaystyle \alpha {\mbox{ and }}u_{n}}
n -curve with
N
=
53
{\displaystyle N=53}
Animation of n -curve for n values from 0 to 50
Example 2: Product of the unit circle and its n -curve
edit
The unit circle is
u
(
t
)
=
cos
(
2
π
t
)
+
i
sin
(
2
π
t
)
{\displaystyle u(t)=\cos(2\pi t)+i\sin(2\pi t)\,}
and its n -curve is
u
n
(
t
)
=
cos
(
2
π
n
t
)
+
i
sin
(
2
π
n
t
)
{\displaystyle u_{n}(t)=\cos(2\pi nt)+i\sin(2\pi nt)\,}
The parametric equations of their product
u
⋅
u
n
{\displaystyle u\cdot u_{n}}
are
x
=
cos
(
2
π
n
t
)
+
cos
(
2
π
t
)
−
1
,
{\displaystyle x=\cos(2\pi nt)+\cos(2\pi t)-1,}
y
=
sin
(
2
π
n
t
)
+
sin
(
2
π
t
)
{\displaystyle y=\sin(2\pi nt)+\sin(2\pi t)}
See the figure.
Example 3: n -Curve of the Rhodonea minus the Rhodonea curve
edit
Let us take the Rhodonea Curve
r
=
cos
(
3
θ
)
{\displaystyle r=\cos(3\theta )}
If
ρ
{\displaystyle \rho }
ρ
(
t
)
=
cos
(
6
π
t
)
[
cos
(
2
π
t
)
+
i
sin
(
2
π
t
)
]
,
0
≤
t
≤
1
{\displaystyle \rho (t)=\cos(6\pi t)[\cos(2\pi t)+i\sin(2\pi t)],0\leq t\leq 1}
The parametric equations of
ρ
n
−
ρ
{\displaystyle \rho _{n}-\rho }
x
=
cos
(
6
π
n
t
)
cos
(
2
π
n
t
)
−
cos
(
6
π
t
)
cos
(
2
π
t
)
,
{\displaystyle x=\cos(6\pi nt)\cos(2\pi nt)-\cos(6\pi t)\cos(2\pi t),}
y
=
cos
(
6
π
n
t
)
sin
(
2
π
n
t
)
−
cos
(
6
π
t
)
sin
(
2
π
t
)
,
0
≤
t
≤
1
{\displaystyle y=\cos(6\pi nt)\sin(2\pi nt)-\cos(6\pi t)\sin(2\pi t),0\leq t\leq 1}
n -Curving
edit
If
γ
∈
H
{\displaystyle \gamma \in H}
n -curve
γ
n
also
∈
H
{\displaystyle \gamma _{n}{\mbox{ also }}\in H}
α
→
γ
n
−
1
⋅
α
⋅
γ
n
{\displaystyle \alpha \to \gamma _{n}^{-1}\cdot \alpha \cdot \gamma _{n}}
G. We extend this map to the whole of C [0, 1], denote it by
ϕ
γ
n
,
e
{\displaystyle \phi _{\gamma _{n},e}}
n -curving with γ.
It can be verified that
ϕ
γ
n
,
e
(
α
)
=
α
+
[
α
(
1
)
−
α
(
0
)
]
(
γ
n
−
1
)
e
.
{\displaystyle \phi _{\gamma _{n},e}(\alpha )=\alpha +[\alpha (1)-\alpha (0)](\gamma _{n}-1)e.\ }
This new curve has the same initial and end points as α.
Example 1 of n -curving
edit
Let ρ denote the Rhodonea curve
r
=
cos
(
2
θ
)
{\displaystyle r=\cos(2\theta )}
x
=
cos
(
4
π
t
)
cos
(
2
π
t
)
,
{\displaystyle x=\cos(4\pi t)\cos(2\pi t),}
y
=
cos
(
4
π
t
)
sin
(
2
π
t
)
,
0
≤
t
≤
1
{\displaystyle y=\cos(4\pi t)\sin(2\pi t),0\leq t\leq 1}
With the loop ρ we shall n -curve the cosine curve
c
(
t
)
=
2
π
t
+
i
cos
(
2
π
t
)
,
0
≤
t
≤
1.
{\displaystyle c(t)=2\pi t+i\cos(2\pi t),\quad 0\leq t\leq 1.\,}
The curve
ϕ
ρ
n
,
e
(
c
)
{\displaystyle \phi _{\rho _{n},e}(c)}
x
=
2
π
[
t
−
1
+
cos
(
4
π
n
t
)
cos
(
2
π
n
t
)
]
,
y
=
cos
(
2
π
t
)
+
2
π
cos
(
4
π
n
t
)
sin
(
2
π
n
t
)
{\displaystyle x=2\pi [t-1+\cos(4\pi nt)\cos(2\pi nt)],\quad y=\cos(2\pi t)+2\pi \cos(4\pi nt)\sin(2\pi nt)}
See the figure.
It is a curve that starts at the point (0, 1) and ends at (2π, 1).
Notice how the curve starts with a cosine curve at N =0. Please note that the parametric equation was modified to center the curve at origin. Example 2 of n -curving
edit
Let χ denote the Cosine Curve
χ
(
t
)
=
2
π
t
+
i
cos
(
2
π
t
)
,
0
≤
t
≤
1
{\displaystyle \chi (t)=2\pi t+i\cos(2\pi t),0\leq t\leq 1}
With another Rhodonea Curve
ρ
=
cos
(
3
θ
)
{\displaystyle \rho =\cos(3\theta )}
we shall n -curve the cosine curve.
The rhodonea curve can also be given as
ρ
(
t
)
=
cos
(
6
π
t
)
[
cos
(
2
π
t
)
+
i
sin
(
2
π
t
)
]
,
0
≤
t
≤
1
{\displaystyle \rho (t)=\cos(6\pi t)[\cos(2\pi t)+i\sin(2\pi t)],0\leq t\leq 1}
The curve
ϕ
ρ
n
,
e
(
χ
)
{\displaystyle \phi _{\rho _{n},e}(\chi )}
x
=
2
π
t
+
2
π
[
cos
(
6
π
n
t
)
cos
(
2
π
n
t
)
−
1
]
,
{\displaystyle x=2\pi t+2\pi [\cos(6\pi nt)\cos(2\pi nt)-1],}
y
=
cos
(
2
π
t
)
+
2
π
cos
(
6
π
n
t
)
sin
(
2
π
n
t
)
,
0
≤
t
≤
1
{\displaystyle y=\cos(2\pi t)+2\pi \cos(6\pi nt)\sin(2\pi nt),0\leq t\leq 1}
See the figure for
n
=
15
{\displaystyle n=15}
Generalized n -curving
edit
In the FTA C [0, 1] of curves, instead of e we shall take an arbitrary curve
β
{\displaystyle \beta }
L
1
(
β
)
=
L
2
(
β
)
=
1
{\displaystyle L_{1}(\beta )=L_{2}(\beta )=1}
Then, for a curve γ in C [0, 1],
γ
∗
=
(
γ
(
0
)
+
γ
(
1
)
)
β
−
γ
{\displaystyle \gamma ^{*}=(\gamma (0)+\gamma (1))\beta -\gamma }
and
γ
−
1
=
γ
∗
γ
(
0
)
γ
(
1
)
.
{\displaystyle \gamma ^{-1}={\frac {\gamma ^{*}}{\gamma (0)\gamma (1)}}.}
If
α
∈
H
{\displaystyle \alpha \in H}
ϕ
α
n
,
β
{\displaystyle \phi _{\alpha _{n},\beta }}
given by
ϕ
α
n
,
β
(
γ
)
=
α
n
−
1
⋅
γ
⋅
α
n
{\displaystyle \phi _{\alpha _{n},\beta }(\gamma )=\alpha _{n}^{-1}\cdot \gamma \cdot \alpha _{n}}
is the n -curving. We get the formula
ϕ
α
n
,
β
(
γ
)
=
γ
+
[
γ
(
1
)
−
γ
(
0
)
]
(
α
n
−
β
)
.
{\displaystyle \phi _{\alpha _{n},\beta }(\gamma )=\gamma +[\gamma (1)-\gamma (0)](\alpha _{n}-\beta ).}
Thus given any two loops
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
γ
{\displaystyle \gamma }
This we shall call generalized n -curving.
Example 1
edit
Let us take
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
γ
{\displaystyle \gamma }
γ
(
t
)
=
4
π
t
+
i
cos
(
4
π
t
)
0
≤
t
≤
1
{\displaystyle \gamma (t)=4\pi t+i\cos(4\pi t)0\leq t\leq 1}
Note that
γ
(
1
)
−
γ
(
0
)
=
4
π
{\displaystyle \gamma (1)-\gamma (0)=4\pi }
For the transformed curve for
n
=
40
{\displaystyle n=40}
The transformed curve
ϕ
u
n
,
u
(
γ
)
{\displaystyle \phi _{u_{n},u}(\gamma )}
Example 2
edit
Denote the curve called Crooked Egg by
η
{\displaystyle \eta }
r
=
cos
3
θ
+
sin
3
θ
{\displaystyle r=\cos ^{3}\theta +\sin ^{3}\theta }
Its parametric equations are
x
=
cos
(
2
π
t
)
(
cos
3
2
π
t
+
sin
3
2
π
t
)
,
{\displaystyle x=\cos(2\pi t)(\cos ^{3}2\pi t+\sin ^{3}2\pi t),}
y
=
sin
(
2
π
t
)
(
cos
3
2
π
t
+
sin
3
2
π
t
)
{\displaystyle y=\sin(2\pi t)(\cos ^{3}2\pi t+\sin ^{3}2\pi t)}
Let us take
α
=
η
{\displaystyle \alpha =\eta }
β
=
u
,
{\displaystyle \beta =u,}
where
u
{\displaystyle u}
The n -curved Archimedean spiral has the parametric equations
x
=
2
π
t
cos
(
2
π
t
)
+
2
π
[
(
cos
3
2
π
n
t
+
sin
3
2
π
n
t
)
cos
(
2
π
n
t
)
−
cos
(
2
π
t
)
]
,
{\displaystyle x=2\pi t\cos(2\pi t)+2\pi [(\cos ^{3}2\pi nt+\sin ^{3}2\pi nt)\cos(2\pi nt)-\cos(2\pi t)],}
y
=
2
π
t
sin
(
2
π
t
)
+
2
π
[
(
cos
3
2
π
n
t
)
+
sin
3
2
π
n
t
)
sin
(
2
π
n
t
)
−
sin
(
2
π
t
)
]
{\displaystyle y=2\pi t\sin(2\pi t)+2\pi [(\cos ^{3}2\pi nt)+\sin ^{3}2\pi nt)\sin(2\pi nt)-\sin(2\pi t)]}
See the figures, the Crooked Egg and the transformed Spiral for
n
=
20
{\displaystyle n=20}
![{\displaystyle e(t)=1,\forall t\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fdb468cbd2be80774a0ef3b90f32a1b3f59d83e)
![{\displaystyle x-[x]\in [0,1].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb3d5e9587553bdc47512b2c5a0d18b231ab837)
![{\displaystyle \gamma _{n}(t)=\gamma (nt-[nt]).\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8989d845254012acd18f5edca5ada36a3369fe2)
![{\displaystyle \rho (t)=\cos(6\pi t)[\cos(2\pi t)+i\sin(2\pi t)],0\leq t\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40b1fa668dc7c514f60b05a7a076bef318eda8f9)
e.\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/22036a6a1276cb7921f59fca3478b3c89637ecaa)
![{\displaystyle x=2\pi [t-1+\cos(4\pi nt)\cos(2\pi nt)],\quad y=\cos(2\pi t)+2\pi \cos(4\pi nt)\sin(2\pi nt)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3e76f7752bd1b227dce89cab4762588c93d1cc)
![{\displaystyle x=2\pi t+2\pi [\cos(6\pi nt)\cos(2\pi nt)-1],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35741dfee80dbd8f01cd809a5596ed63910a720d)
.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c12e43d4880fc8c12032c52e95954ef111140e3)
![{\displaystyle x=2\pi t\cos(2\pi t)+2\pi [(\cos ^{3}2\pi nt+\sin ^{3}2\pi nt)\cos(2\pi nt)-\cos(2\pi t)],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c0ebe850dea4ae2db5db10eae286db64c9eb057)
![{\displaystyle y=2\pi t\sin(2\pi t)+2\pi [(\cos ^{3}2\pi nt)+\sin ^{3}2\pi nt)\sin(2\pi nt)-\sin(2\pi t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc20bf9e653c73d1d5fc9f4766e4ac78ca8fb84)
