Completed Proof For Incommensurate Lissajous Curves/Bowditch Curves Are Dense In The Rectangle
$begingroup$
A Lissajous curve, or a Bowditch curve, is given by the parametric equations
$x(t)=Asin(ω_{x}t + phi)$
$y(t)=Bsin(ω_{y}t+δ)$,
Now if $frac{omega_{x}}{omega_{y}}$ is irrational, and $phi$ and $delta$ are fixed, then the set $mathcal{L} = (x(t), y(t) | -infty < t < infty)$ is supposedly dense in the rectangle $R =[-A,A]$x$[-B,B]$
I have seen:
Show that a Lissajous curve has incommesurate frequencies iff it isdense in a rectangle
But a completed proof was not given. This problem is frequently mentioned in the literature, but a reference to a full proof is never given. Does anyone have such a reference, or could someone provide a proof here?
real-analysis complex-analysis analysis proof-explanation ergodic-theory
asked May 23 '17 at 19:17
The DudeThe Dude
276110
$endgroup$
add a comment |
$begingroup$
A Lissajous curve, or a Bowditch curve, is given by the parametric equations
$x(t)=Asin(ω_{x}t + phi)$
$y(t)=Bsin(ω_{y}t+δ)$,
Now if $frac{omega_{x}}{omega_{y}}$ is irrational, and $phi$ and $delta$ are fixed, then the set $mathcal{L} = (x(t), y(t) | -infty < t < infty)$ is supposedly dense in the rectangle $R =[-A,A]$x$[-B,B]$
I have seen:
Show that a Lissajous curve has incommesurate frequencies iff it isdense in a rectangle
But a completed proof was not given. This problem is frequently mentioned in the literature, but a reference to a full proof is never given. Does anyone have such a reference, or could someone provide a proof here?
real-analysis complex-analysis analysis proof-explanation ergodic-theory
asked May 23 '17 at 19:17
The DudeThe Dude
276110
$endgroup$
add a comment |
$begingroup$
A Lissajous curve, or a Bowditch curve, is given by the parametric equations
$x(t)=Asin(ω_{x}t + phi)$
$y(t)=Bsin(ω_{y}t+δ)$,
Now if $frac{omega_{x}}{omega_{y}}$ is irrational, and $phi$ and $delta$ are fixed, then the set $mathcal{L} = (x(t), y(t) | -infty < t < infty)$ is supposedly dense in the rectangle $R =[-A,A]$x$[-B,B]$
I have seen:
Show that a Lissajous curve has incommesurate frequencies iff it isdense in a rectangle
But a completed proof was not given. This problem is frequently mentioned in the literature, but a reference to a full proof is never given. Does anyone have such a reference, or could someone provide a proof here?
real-analysis complex-analysis analysis proof-explanation ergodic-theory
asked May 23 '17 at 19:17
The DudeThe Dude
276110
$endgroup$
A Lissajous curve, or a Bowditch curve, is given by the parametric equations
$x(t)=Asin(ω_{x}t + phi)$
$y(t)=Bsin(ω_{y}t+δ)$,
Now if $frac{omega_{x}}{omega_{y}}$ is irrational, and $phi$ and $delta$ are fixed, then the set $mathcal{L} = (x(t), y(t) | -infty < t < infty)$ is supposedly dense in the rectangle $R =[-A,A]$x$[-B,B]$
I have seen:
Show that a Lissajous curve has incommesurate frequencies iff it isdense in a rectangle
But a completed proof was not given. This problem is frequently mentioned in the literature, but a reference to a full proof is never given. Does anyone have such a reference, or could someone provide a proof here?
real-analysis complex-analysis analysis proof-explanation ergodic-theory
real-analysis complex-analysis analysis proof-explanation ergodic-theory
asked May 23 '17 at 19:17
The DudeThe Dude
276110
asked May 23 '17 at 19:17
The DudeThe Dude
276110
asked May 23 '17 at 19:17
The DudeThe Dude
276110
asked May 23 '17 at 19:17
The DudeThe Dude
276110
asked May 23 '17 at 19:17
The DudeThe Dude
276110
276110
add a comment |
add a comment |
1 Answer
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You can write $x$ and $y$ in complex number with the exponentials (easiest way): $mathbf rleft(tright)=left(Ae^{ω_{x}t + phi},Be^{ω_{y}t + delta}right)$. Then proof it by induction, suppose that there is a period $Tneq0$ so that $mathbf rleft(tright)=mathbf rleft(t+Tright)$.
answered Jan 18 at 2:02
David Garrido GonzálezDavid Garrido González
212
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1 Answer
1
active
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votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You can write $x$ and $y$ in complex number with the exponentials (easiest way): $mathbf rleft(tright)=left(Ae^{ω_{x}t + phi},Be^{ω_{y}t + delta}right)$. Then proof it by induction, suppose that there is a period $Tneq0$ so that $mathbf rleft(tright)=mathbf rleft(t+Tright)$.
answered Jan 18 at 2:02
David Garrido GonzálezDavid Garrido González
212
$endgroup$
add a comment |
$begingroup$
You can write $x$ and $y$ in complex number with the exponentials (easiest way): $mathbf rleft(tright)=left(Ae^{ω_{x}t + phi},Be^{ω_{y}t + delta}right)$. Then proof it by induction, suppose that there is a period $Tneq0$ so that $mathbf rleft(tright)=mathbf rleft(t+Tright)$.
answered Jan 18 at 2:02
David Garrido GonzálezDavid Garrido González
212
$endgroup$
add a comment |
$begingroup$
You can write $x$ and $y$ in complex number with the exponentials (easiest way): $mathbf rleft(tright)=left(Ae^{ω_{x}t + phi},Be^{ω_{y}t + delta}right)$. Then proof it by induction, suppose that there is a period $Tneq0$ so that $mathbf rleft(tright)=mathbf rleft(t+Tright)$.
answered Jan 18 at 2:02
David Garrido GonzálezDavid Garrido González
212
$endgroup$
You can write $x$ and $y$ in complex number with the exponentials (easiest way): $mathbf rleft(tright)=left(Ae^{ω_{x}t + phi},Be^{ω_{y}t + delta}right)$. Then proof it by induction, suppose that there is a period $Tneq0$ so that $mathbf rleft(tright)=mathbf rleft(t+Tright)$.
answered Jan 18 at 2:02
David Garrido GonzálezDavid Garrido González
212
answered Jan 18 at 2:02
David Garrido GonzálezDavid Garrido González
212
answered Jan 18 at 2:02
David Garrido GonzálezDavid Garrido González
212
answered Jan 18 at 2:02
David Garrido GonzálezDavid Garrido González
212
212
add a comment |
add a comment |
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