Oblate Spheroidal coordinate system graphic representation of ellipse Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Finding Coordinate along Ellipse PerimeterPassing an ellipse through 3 points (where 2 two points lie on the ellipse axes)? [Updated with alternative statement of problem and new picture]Cylindrical coordinate systemEllipse representationrectangular coordinate system vs Cartesian coordinate system?Oblate Spheroidal Coordinates, Confocal Ellipsoidal Coordinates and GeodesyEllipse equation in $Bbb R^3$Scale factors for the Oblate Spheroidal Coordiante systemAffine transformation of endpoint parametrisation of an ellipse arcFrom origin walk halfway to $(8,6)$, turn $90$ degrees left, then walk twice as far.
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Oblate Spheroidal coordinate system graphic representation of ellipse
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Finding Coordinate along Ellipse PerimeterPassing an ellipse through 3 points (where 2 two points lie on the ellipse axes)? [Updated with alternative statement of problem and new picture]Cylindrical coordinate systemEllipse representationrectangular coordinate system vs Cartesian coordinate system?Oblate Spheroidal Coordinates, Confocal Ellipsoidal Coordinates and GeodesyEllipse equation in $Bbb R^3$Scale factors for the Oblate Spheroidal Coordiante systemAffine transformation of endpoint parametrisation of an ellipse arcFrom origin walk halfway to $(8,6)$, turn $90$ degrees left, then walk twice as far.
$begingroup$
I am having difficulty understanding how to interpret the coordinate system proposed by Spencer. From his handbook Field Theory Handbooks ISBN 9783540184300 proposed a system with the domains illustrated in the picture.
The domain of eta represents the curvilinear axis represented by the ellipse rotating around the minor axis. What exactly is the characteristic of the ellipse in the picture? Is 'Eta' representing a number describing the minor or mayor axis of the ellipse?
conic-sections coordinate-systems
edited Mar 26 at 6:30
J. M. is a poor mathematician
61.3k5152291
asked Mar 25 at 18:12
Jose Enrique CalderonJose Enrique Calderon
1177
$endgroup$
add a comment |
$begingroup$
I am having difficulty understanding how to interpret the coordinate system proposed by Spencer. From his handbook Field Theory Handbooks ISBN 9783540184300 proposed a system with the domains illustrated in the picture.
The domain of eta represents the curvilinear axis represented by the ellipse rotating around the minor axis. What exactly is the characteristic of the ellipse in the picture? Is 'Eta' representing a number describing the minor or mayor axis of the ellipse?
conic-sections coordinate-systems
edited Mar 26 at 6:30
J. M. is a poor mathematician
61.3k5152291
asked Mar 25 at 18:12
Jose Enrique CalderonJose Enrique Calderon
1177
$endgroup$
add a comment |
$begingroup$
I am having difficulty understanding how to interpret the coordinate system proposed by Spencer. From his handbook Field Theory Handbooks ISBN 9783540184300 proposed a system with the domains illustrated in the picture.
The domain of eta represents the curvilinear axis represented by the ellipse rotating around the minor axis. What exactly is the characteristic of the ellipse in the picture? Is 'Eta' representing a number describing the minor or mayor axis of the ellipse?
conic-sections coordinate-systems
edited Mar 26 at 6:30
J. M. is a poor mathematician
61.3k5152291
asked Mar 25 at 18:12
Jose Enrique CalderonJose Enrique Calderon
1177
$endgroup$
I am having difficulty understanding how to interpret the coordinate system proposed by Spencer. From his handbook Field Theory Handbooks ISBN 9783540184300 proposed a system with the domains illustrated in the picture.
The domain of eta represents the curvilinear axis represented by the ellipse rotating around the minor axis. What exactly is the characteristic of the ellipse in the picture? Is 'Eta' representing a number describing the minor or mayor axis of the ellipse?
conic-sections coordinate-systems
conic-sections coordinate-systems
edited Mar 26 at 6:30
J. M. is a poor mathematician
61.3k5152291
asked Mar 25 at 18:12
Jose Enrique CalderonJose Enrique Calderon
1177
edited Mar 26 at 6:30
J. M. is a poor mathematician
61.3k5152291
asked Mar 25 at 18:12
Jose Enrique CalderonJose Enrique Calderon
1177
edited Mar 26 at 6:30
J. M. is a poor mathematician
61.3k5152291
edited Mar 26 at 6:30
J. M. is a poor mathematician
61.3k5152291
edited Mar 26 at 6:30
J. M. is a poor mathematician
61.3k5152291
61.3k5152291
asked Mar 25 at 18:12
Jose Enrique CalderonJose Enrique Calderon
1177
asked Mar 25 at 18:12
Jose Enrique CalderonJose Enrique Calderon
1177
asked Mar 25 at 18:12
Jose Enrique CalderonJose Enrique Calderon
1177
1177
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Neither or both, depending on your point of view. Refer to the illustration: the surfaces with $eta$ held constant are oblate spheriods. From the equations of the transformation to Cartesian coordinates, these spheroids have the parameterization $$beginalignx &= (acosheta)sinthetacospsi \ y &= (acosheta)sinthetasinpsi \ z &= (asinheta)costheta.endalign$$ I’ve added parentheses to emphasize the constant quantity in each equation. The half-axis lengths of the spheroid are therefore $acosheta$ and $asinheta$. Similarly, the surfaces with constant $theta$ are hyperboloids of revolution with half-axis lengths $acostheta$ and $asintheta$. The angle $theta$ represents the aperture half-angle of the hyperboloid’s asymptotic cone. Surfaces with constant $psi$ are half-planes that make an angle of $psi$ with the positive $x$-$z$ half-plane.
answered Mar 25 at 18:41
amdamd
31.8k21053
$endgroup$
$begingroup$
Thanks for your clear details of the constant. I could not see this.
$endgroup$
– Jose Enrique Calderon
Mar 25 at 22:32
add a comment |
$begingroup$
For any $psi=$ constant sectioning plane containing z-axis we have confocal conic sections with the marked $F$ as common focal locus for variable $eta$ value on all ellipses formed by cutting planes$psi=$ constant.
Parametrization for horizontal ellipse sections
$$ r=sqrtx^2+y^2 = a cosh eta sin theta ;, z = a sinh eta cos theta; $$
$$ big(fracra cosh etabig)^2 + big(fracra sinh etabig)^2 =1 $$
$$ epsilon^2= 1- fracsinh^2 etacosh ^2eta= sech^2 eta $$
So have $epsilon =sech, eta$ geometrically interpreted as eccentricity of horizontal ellipses formed by cutting planes parallel to $x-y$ plane.
Circular conicoids when $eta rightarrow infty.$
answered Mar 25 at 19:18
NarasimhamNarasimham
21.2k62258
$endgroup$
$begingroup$
@ Narasimham From where is derrived the eccentricity= 𝜖=𝑠𝑒𝑐ℎ𝜂 ?
$endgroup$
– Jose Enrique Calderon
Mar 27 at 12:47
1
$begingroup$
Is the edit clear enough? $ epsilon = sech eta le 1 $
$endgroup$
– Narasimham
Mar 27 at 16:27
add a comment |
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2 Answers
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2 Answers
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$begingroup$
Neither or both, depending on your point of view. Refer to the illustration: the surfaces with $eta$ held constant are oblate spheriods. From the equations of the transformation to Cartesian coordinates, these spheroids have the parameterization $$beginalignx &= (acosheta)sinthetacospsi \ y &= (acosheta)sinthetasinpsi \ z &= (asinheta)costheta.endalign$$ I’ve added parentheses to emphasize the constant quantity in each equation. The half-axis lengths of the spheroid are therefore $acosheta$ and $asinheta$. Similarly, the surfaces with constant $theta$ are hyperboloids of revolution with half-axis lengths $acostheta$ and $asintheta$. The angle $theta$ represents the aperture half-angle of the hyperboloid’s asymptotic cone. Surfaces with constant $psi$ are half-planes that make an angle of $psi$ with the positive $x$-$z$ half-plane.
answered Mar 25 at 18:41
amdamd
31.8k21053
$endgroup$
$begingroup$
Thanks for your clear details of the constant. I could not see this.
$endgroup$
– Jose Enrique Calderon
Mar 25 at 22:32
add a comment |
$begingroup$
Neither or both, depending on your point of view. Refer to the illustration: the surfaces with $eta$ held constant are oblate spheriods. From the equations of the transformation to Cartesian coordinates, these spheroids have the parameterization $$beginalignx &= (acosheta)sinthetacospsi \ y &= (acosheta)sinthetasinpsi \ z &= (asinheta)costheta.endalign$$ I’ve added parentheses to emphasize the constant quantity in each equation. The half-axis lengths of the spheroid are therefore $acosheta$ and $asinheta$. Similarly, the surfaces with constant $theta$ are hyperboloids of revolution with half-axis lengths $acostheta$ and $asintheta$. The angle $theta$ represents the aperture half-angle of the hyperboloid’s asymptotic cone. Surfaces with constant $psi$ are half-planes that make an angle of $psi$ with the positive $x$-$z$ half-plane.
answered Mar 25 at 18:41
amdamd
31.8k21053
$endgroup$
$begingroup$
Thanks for your clear details of the constant. I could not see this.
$endgroup$
– Jose Enrique Calderon
Mar 25 at 22:32
add a comment |
$begingroup$
Neither or both, depending on your point of view. Refer to the illustration: the surfaces with $eta$ held constant are oblate spheriods. From the equations of the transformation to Cartesian coordinates, these spheroids have the parameterization $$beginalignx &= (acosheta)sinthetacospsi \ y &= (acosheta)sinthetasinpsi \ z &= (asinheta)costheta.endalign$$ I’ve added parentheses to emphasize the constant quantity in each equation. The half-axis lengths of the spheroid are therefore $acosheta$ and $asinheta$. Similarly, the surfaces with constant $theta$ are hyperboloids of revolution with half-axis lengths $acostheta$ and $asintheta$. The angle $theta$ represents the aperture half-angle of the hyperboloid’s asymptotic cone. Surfaces with constant $psi$ are half-planes that make an angle of $psi$ with the positive $x$-$z$ half-plane.
answered Mar 25 at 18:41
amdamd
31.8k21053
$endgroup$
Neither or both, depending on your point of view. Refer to the illustration: the surfaces with $eta$ held constant are oblate spheriods. From the equations of the transformation to Cartesian coordinates, these spheroids have the parameterization $$beginalignx &= (acosheta)sinthetacospsi \ y &= (acosheta)sinthetasinpsi \ z &= (asinheta)costheta.endalign$$ I’ve added parentheses to emphasize the constant quantity in each equation. The half-axis lengths of the spheroid are therefore $acosheta$ and $asinheta$. Similarly, the surfaces with constant $theta$ are hyperboloids of revolution with half-axis lengths $acostheta$ and $asintheta$. The angle $theta$ represents the aperture half-angle of the hyperboloid’s asymptotic cone. Surfaces with constant $psi$ are half-planes that make an angle of $psi$ with the positive $x$-$z$ half-plane.
answered Mar 25 at 18:41
amdamd
31.8k21053
edited Mar 26 at 4:34
answered Mar 25 at 18:41
amdamd
31.8k21053
answered Mar 25 at 18:41
amdamd
31.8k21053
answered Mar 25 at 18:41
amdamd
31.8k21053
31.8k21053
$begingroup$
Thanks for your clear details of the constant. I could not see this.
$endgroup$
– Jose Enrique Calderon
Mar 25 at 22:32
add a comment |
$begingroup$
Thanks for your clear details of the constant. I could not see this.
$endgroup$
– Jose Enrique Calderon
Mar 25 at 22:32
$begingroup$
Thanks for your clear details of the constant. I could not see this.
$endgroup$
– Jose Enrique Calderon
Mar 25 at 22:32
$begingroup$
Thanks for your clear details of the constant. I could not see this.
$endgroup$
– Jose Enrique Calderon
Mar 25 at 22:32
add a comment |
$begingroup$
For any $psi=$ constant sectioning plane containing z-axis we have confocal conic sections with the marked $F$ as common focal locus for variable $eta$ value on all ellipses formed by cutting planes$psi=$ constant.
Parametrization for horizontal ellipse sections
$$ r=sqrtx^2+y^2 = a cosh eta sin theta ;, z = a sinh eta cos theta; $$
$$ big(fracra cosh etabig)^2 + big(fracra sinh etabig)^2 =1 $$
$$ epsilon^2= 1- fracsinh^2 etacosh ^2eta= sech^2 eta $$
So have $epsilon =sech, eta$ geometrically interpreted as eccentricity of horizontal ellipses formed by cutting planes parallel to $x-y$ plane.
Circular conicoids when $eta rightarrow infty.$
answered Mar 25 at 19:18
NarasimhamNarasimham
21.2k62258
$endgroup$
$begingroup$
@ Narasimham From where is derrived the eccentricity= 𝜖=𝑠𝑒𝑐ℎ𝜂 ?
$endgroup$
– Jose Enrique Calderon
Mar 27 at 12:47
1
$begingroup$
Is the edit clear enough? $ epsilon = sech eta le 1 $
$endgroup$
– Narasimham
Mar 27 at 16:27
add a comment |
$begingroup$
For any $psi=$ constant sectioning plane containing z-axis we have confocal conic sections with the marked $F$ as common focal locus for variable $eta$ value on all ellipses formed by cutting planes$psi=$ constant.
Parametrization for horizontal ellipse sections
$$ r=sqrtx^2+y^2 = a cosh eta sin theta ;, z = a sinh eta cos theta; $$
$$ big(fracra cosh etabig)^2 + big(fracra sinh etabig)^2 =1 $$
$$ epsilon^2= 1- fracsinh^2 etacosh ^2eta= sech^2 eta $$
So have $epsilon =sech, eta$ geometrically interpreted as eccentricity of horizontal ellipses formed by cutting planes parallel to $x-y$ plane.
Circular conicoids when $eta rightarrow infty.$
answered Mar 25 at 19:18
NarasimhamNarasimham
21.2k62258
$endgroup$
$begingroup$
@ Narasimham From where is derrived the eccentricity= 𝜖=𝑠𝑒𝑐ℎ𝜂 ?
$endgroup$
– Jose Enrique Calderon
Mar 27 at 12:47
1
$begingroup$
Is the edit clear enough? $ epsilon = sech eta le 1 $
$endgroup$
– Narasimham
Mar 27 at 16:27
add a comment |
$begingroup$
For any $psi=$ constant sectioning plane containing z-axis we have confocal conic sections with the marked $F$ as common focal locus for variable $eta$ value on all ellipses formed by cutting planes$psi=$ constant.
Parametrization for horizontal ellipse sections
$$ r=sqrtx^2+y^2 = a cosh eta sin theta ;, z = a sinh eta cos theta; $$
$$ big(fracra cosh etabig)^2 + big(fracra sinh etabig)^2 =1 $$
$$ epsilon^2= 1- fracsinh^2 etacosh ^2eta= sech^2 eta $$
So have $epsilon =sech, eta$ geometrically interpreted as eccentricity of horizontal ellipses formed by cutting planes parallel to $x-y$ plane.
Circular conicoids when $eta rightarrow infty.$
answered Mar 25 at 19:18
NarasimhamNarasimham
21.2k62258
$endgroup$
For any $psi=$ constant sectioning plane containing z-axis we have confocal conic sections with the marked $F$ as common focal locus for variable $eta$ value on all ellipses formed by cutting planes$psi=$ constant.
Parametrization for horizontal ellipse sections
$$ r=sqrtx^2+y^2 = a cosh eta sin theta ;, z = a sinh eta cos theta; $$
$$ big(fracra cosh etabig)^2 + big(fracra sinh etabig)^2 =1 $$
$$ epsilon^2= 1- fracsinh^2 etacosh ^2eta= sech^2 eta $$
So have $epsilon =sech, eta$ geometrically interpreted as eccentricity of horizontal ellipses formed by cutting planes parallel to $x-y$ plane.
Circular conicoids when $eta rightarrow infty.$
answered Mar 25 at 19:18
NarasimhamNarasimham
21.2k62258
edited Mar 26 at 6:49
answered Mar 25 at 19:18
NarasimhamNarasimham
21.2k62258
answered Mar 25 at 19:18
NarasimhamNarasimham
21.2k62258
answered Mar 25 at 19:18
NarasimhamNarasimham
21.2k62258
21.2k62258
$begingroup$
@ Narasimham From where is derrived the eccentricity= 𝜖=𝑠𝑒𝑐ℎ𝜂 ?
$endgroup$
– Jose Enrique Calderon
Mar 27 at 12:47
1
$begingroup$
Is the edit clear enough? $ epsilon = sech eta le 1 $
$endgroup$
– Narasimham
Mar 27 at 16:27
add a comment |
$begingroup$
@ Narasimham From where is derrived the eccentricity= 𝜖=𝑠𝑒𝑐ℎ𝜂 ?
$endgroup$
– Jose Enrique Calderon
Mar 27 at 12:47
1
$begingroup$
Is the edit clear enough? $ epsilon = sech eta le 1 $
$endgroup$
– Narasimham
Mar 27 at 16:27
$begingroup$
@ Narasimham From where is derrived the eccentricity= 𝜖=𝑠𝑒𝑐ℎ𝜂 ?
$endgroup$
– Jose Enrique Calderon
Mar 27 at 12:47
$begingroup$
@ Narasimham From where is derrived the eccentricity= 𝜖=𝑠𝑒𝑐ℎ𝜂 ?
$endgroup$
– Jose Enrique Calderon
Mar 27 at 12:47
1
1
$begingroup$
Is the edit clear enough? $ epsilon = sech eta le 1 $
$endgroup$
– Narasimham
Mar 27 at 16:27
$begingroup$
Is the edit clear enough? $ epsilon = sech eta le 1 $
$endgroup$
– Narasimham
Mar 27 at 16:27
add a comment |
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