Finding transformation in $mathbb{R}^2$
I have to find the transformation $T: mathbb{R}^2 to mathbb{R}^2$ that goes from the 2D annulus to a square, and I am unsure if my procedure is right.
More precisely, the problem is as follows:
Find the transformation from the domain $A$ to $B$, where $A$ is described according to $(xi, eta) in A = [-1,1]^2$ and $(r,theta)in B subseteq mathbb{R}^2$ where $r in [r_0, r_0 + Delta r]$ and $theta in [-frac{Delta theta}{2}, frac{Delta theta}{2}]$
So I found from http://www-users.math.umn.edu/~olver/ln_/cml.pdf (page 29) that this is equivalent to the conformal mapping
$$(x, y) = (e^xicos eta, e^xi sin eta)$$
But how do I constructively come up with this conclusion?
complex-analysis transformation conformal-geometry
asked Nov 28 at 4:38
The Bosco
536212
add a comment |
I have to find the transformation $T: mathbb{R}^2 to mathbb{R}^2$ that goes from the 2D annulus to a square, and I am unsure if my procedure is right.
More precisely, the problem is as follows:
Find the transformation from the domain $A$ to $B$, where $A$ is described according to $(xi, eta) in A = [-1,1]^2$ and $(r,theta)in B subseteq mathbb{R}^2$ where $r in [r_0, r_0 + Delta r]$ and $theta in [-frac{Delta theta}{2}, frac{Delta theta}{2}]$
So I found from http://www-users.math.umn.edu/~olver/ln_/cml.pdf (page 29) that this is equivalent to the conformal mapping
$$(x, y) = (e^xicos eta, e^xi sin eta)$$
But how do I constructively come up with this conclusion?
complex-analysis transformation conformal-geometry
asked Nov 28 at 4:38
The Bosco
536212
You are using the transformation $T(bf{x})=Mbf{x}$. What makes you believe that your transformation will be linear? The edges of the annular region are not straight lines.
– Anurag A
Nov 28 at 4:54
You're right. How can I find it if it is non-linear?
– The Bosco
Nov 28 at 5:12
add a comment |
I have to find the transformation $T: mathbb{R}^2 to mathbb{R}^2$ that goes from the 2D annulus to a square, and I am unsure if my procedure is right.
More precisely, the problem is as follows:
Find the transformation from the domain $A$ to $B$, where $A$ is described according to $(xi, eta) in A = [-1,1]^2$ and $(r,theta)in B subseteq mathbb{R}^2$ where $r in [r_0, r_0 + Delta r]$ and $theta in [-frac{Delta theta}{2}, frac{Delta theta}{2}]$
So I found from http://www-users.math.umn.edu/~olver/ln_/cml.pdf (page 29) that this is equivalent to the conformal mapping
$$(x, y) = (e^xicos eta, e^xi sin eta)$$
But how do I constructively come up with this conclusion?
complex-analysis transformation conformal-geometry
asked Nov 28 at 4:38
The Bosco
536212
I have to find the transformation $T: mathbb{R}^2 to mathbb{R}^2$ that goes from the 2D annulus to a square, and I am unsure if my procedure is right.
More precisely, the problem is as follows:
Find the transformation from the domain $A$ to $B$, where $A$ is described according to $(xi, eta) in A = [-1,1]^2$ and $(r,theta)in B subseteq mathbb{R}^2$ where $r in [r_0, r_0 + Delta r]$ and $theta in [-frac{Delta theta}{2}, frac{Delta theta}{2}]$
So I found from http://www-users.math.umn.edu/~olver/ln_/cml.pdf (page 29) that this is equivalent to the conformal mapping
$$(x, y) = (e^xicos eta, e^xi sin eta)$$
But how do I constructively come up with this conclusion?
complex-analysis transformation conformal-geometry
complex-analysis transformation conformal-geometry
asked Nov 28 at 4:38
The Bosco
536212
asked Nov 28 at 4:38
The Bosco
536212
edited Nov 28 at 6:37
asked Nov 28 at 4:38
The Bosco
536212
asked Nov 28 at 4:38
The Bosco
536212
asked Nov 28 at 4:38
The Bosco
536212
536212
You are using the transformation $T(bf{x})=Mbf{x}$. What makes you believe that your transformation will be linear? The edges of the annular region are not straight lines.
– Anurag A
Nov 28 at 4:54
You're right. How can I find it if it is non-linear?
– The Bosco
Nov 28 at 5:12
add a comment |
You are using the transformation $T(bf{x})=Mbf{x}$. What makes you believe that your transformation will be linear? The edges of the annular region are not straight lines.
– Anurag A
Nov 28 at 4:54
You're right. How can I find it if it is non-linear?
– The Bosco
Nov 28 at 5:12
You are using the transformation $T(bf{x})=Mbf{x}$. What makes you believe that your transformation will be linear? The edges of the annular region are not straight lines.
– Anurag A
Nov 28 at 4:54
You are using the transformation $T(bf{x})=Mbf{x}$. What makes you believe that your transformation will be linear? The edges of the annular region are not straight lines.
– Anurag A
Nov 28 at 4:54
You're right. How can I find it if it is non-linear?
– The Bosco
Nov 28 at 5:12
You're right. How can I find it if it is non-linear?
– The Bosco
Nov 28 at 5:12
add a comment |
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You are using the transformation $T(bf{x})=Mbf{x}$. What makes you believe that your transformation will be linear? The edges of the annular region are not straight lines.
– Anurag A
Nov 28 at 4:54
You're right. How can I find it if it is non-linear?
– The Bosco
Nov 28 at 5:12