Reparametrization of a circle
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Let $,mathbf{x}longrightarrowmathbb{R}^2,$ and $,mathbf{y}:Jlongrightarrowmathbb{R}^2;$ be two parametrizations of a circle, given by
$$mathbf{x}left(thetaright)=left(costheta,,sinthetaright)qquadmathbf{y}left(tright)=left(frac{1-t^2}{1+t^2},,frac{2t}{1+t^2}right)$$
I am asked to find a relation between both parameters $,left(right.$i.e a dipheomorphism $,f:Ilongrightarrow J,$ that verifies $,mathbf{x}=mathbf{y}circ fleft.right)$
After different attempts, I tried setting $,textrm{tg}left(frac{theta}{2}right),$ and it seems to be a right change of variables. However, it was just luck.
Question: Is there a specific method for this kind of exercise?
differential-geometry parametrization
asked Jan 18 at 19:25
CarlIOCarlIO
957
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add a comment |
$begingroup$
Let $,mathbf{x}longrightarrowmathbb{R}^2,$ and $,mathbf{y}:Jlongrightarrowmathbb{R}^2;$ be two parametrizations of a circle, given by
$$mathbf{x}left(thetaright)=left(costheta,,sinthetaright)qquadmathbf{y}left(tright)=left(frac{1-t^2}{1+t^2},,frac{2t}{1+t^2}right)$$
I am asked to find a relation between both parameters $,left(right.$i.e a dipheomorphism $,f:Ilongrightarrow J,$ that verifies $,mathbf{x}=mathbf{y}circ fleft.right)$
After different attempts, I tried setting $,textrm{tg}left(frac{theta}{2}right),$ and it seems to be a right change of variables. However, it was just luck.
Question: Is there a specific method for this kind of exercise?
differential-geometry parametrization
asked Jan 18 at 19:25
CarlIOCarlIO
957
$endgroup$
$begingroup$
Please specify the extremes of the intervals $ I $ and $ J $.
$endgroup$
– MathOverview
Jan 18 at 23:03
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@MathOverview: They are not defined. I guess I=[0,2π) and J an interval which allows to draw the whole circle.
$endgroup$
– CarlIO
Jan 21 at 10:13
add a comment |
$begingroup$
Let $,mathbf{x}longrightarrowmathbb{R}^2,$ and $,mathbf{y}:Jlongrightarrowmathbb{R}^2;$ be two parametrizations of a circle, given by
$$mathbf{x}left(thetaright)=left(costheta,,sinthetaright)qquadmathbf{y}left(tright)=left(frac{1-t^2}{1+t^2},,frac{2t}{1+t^2}right)$$
I am asked to find a relation between both parameters $,left(right.$i.e a dipheomorphism $,f:Ilongrightarrow J,$ that verifies $,mathbf{x}=mathbf{y}circ fleft.right)$
After different attempts, I tried setting $,textrm{tg}left(frac{theta}{2}right),$ and it seems to be a right change of variables. However, it was just luck.
Question: Is there a specific method for this kind of exercise?
differential-geometry parametrization
asked Jan 18 at 19:25
CarlIOCarlIO
957
$endgroup$
Let $,mathbf{x}longrightarrowmathbb{R}^2,$ and $,mathbf{y}:Jlongrightarrowmathbb{R}^2;$ be two parametrizations of a circle, given by
$$mathbf{x}left(thetaright)=left(costheta,,sinthetaright)qquadmathbf{y}left(tright)=left(frac{1-t^2}{1+t^2},,frac{2t}{1+t^2}right)$$
I am asked to find a relation between both parameters $,left(right.$i.e a dipheomorphism $,f:Ilongrightarrow J,$ that verifies $,mathbf{x}=mathbf{y}circ fleft.right)$
After different attempts, I tried setting $,textrm{tg}left(frac{theta}{2}right),$ and it seems to be a right change of variables. However, it was just luck.
Question: Is there a specific method for this kind of exercise?
differential-geometry parametrization
differential-geometry parametrization
asked Jan 18 at 19:25
CarlIOCarlIO
957
asked Jan 18 at 19:25
CarlIOCarlIO
957
asked Jan 18 at 19:25
CarlIOCarlIO
957
asked Jan 18 at 19:25
CarlIOCarlIO
957
asked Jan 18 at 19:25
CarlIOCarlIO
957
957
$begingroup$
Please specify the extremes of the intervals $ I $ and $ J $.
$endgroup$
– MathOverview
Jan 18 at 23:03
$begingroup$
@MathOverview: They are not defined. I guess I=[0,2π) and J an interval which allows to draw the whole circle.
$endgroup$
– CarlIO
Jan 21 at 10:13
add a comment |
$begingroup$
Please specify the extremes of the intervals $ I $ and $ J $.
$endgroup$
– MathOverview
Jan 18 at 23:03
$begingroup$
@MathOverview: They are not defined. I guess I=[0,2π) and J an interval which allows to draw the whole circle.
$endgroup$
– CarlIO
Jan 21 at 10:13
$begingroup$
Please specify the extremes of the intervals $ I $ and $ J $.
$endgroup$
– MathOverview
Jan 18 at 23:03
$begingroup$
Please specify the extremes of the intervals $ I $ and $ J $.
$endgroup$
– MathOverview
Jan 18 at 23:03
$begingroup$
@MathOverview: They are not defined. I guess I=[0,2π) and J an interval which allows to draw the whole circle.
$endgroup$
– CarlIO
Jan 21 at 10:13
$begingroup$
@MathOverview: They are not defined. I guess I=[0,2π) and J an interval which allows to draw the whole circle.
$endgroup$
– CarlIO
Jan 21 at 10:13
add a comment |
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$begingroup$
Please specify the extremes of the intervals $ I $ and $ J $.
$endgroup$
– MathOverview
Jan 18 at 23:03
$begingroup$
@MathOverview: They are not defined. I guess I=[0,2π) and J an interval which allows to draw the whole circle.
$endgroup$
– CarlIO
Jan 21 at 10:13