Gnomonic projection: Difference between revisions
Created page with "The Gnomonic projection is projecting the surface of a sphere from the center to a tangent plane. Node that this projection can not visualize more than the surface of a hemisp..." |
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The Gnomonic projection is projecting the | The Gnomonic projection is projecting from the center of a sphere to a tangent plane. | ||
Node that this projection can not visualize more than the surface of a hemisphere. | Node that this projection can not visualize more than the surface of a hemisphere. | ||
Visualizing a hemisphere would require an infinitely large plane. | Visualizing a hemisphere would require an infinitely large plane. | ||
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==Equations== | ==Equations== | ||
===Gnomonic Projection=== | ===Gnomonic Projection=== | ||
Copied from Mathworld | Copied from [https://mathworld.wolfram.com/GnomonicProjection.html Mathworld] | ||
Inputs: | |||
<math>(\lambda, \phi)</math> Current spherical coordinate< | Inputs: | ||
<math>(\lambda_0, \phi_1)</math> Spherical coordinate of tangent plane. | * <math>(\lambda, \phi)</math> Current spherical coordinate with longitude <math>\lambda</math> and latitude <math>\phi</math> | ||
* <math>(\lambda_0, \phi_1)</math> Spherical coordinate of tangent plane. | |||
Outputs: <br> | Outputs: <br> | ||
* <math>(x,y) \in (-\infty, \infty) \times (-\infty, \infty)</math> Cartesian coordinates. | |||
<math>(x,y) \in (-\infty, \infty) \times (-\infty, \infty)</math> | |||
<math>x = \frac{\cos(\phi)\sin(\lambda - \lambda_0)}{\cos(c)}</math><br> | |||
<math>x = \frac{\cos(\ | |||
<math>y = \frac{\cos(\phi_1)\sin(\phi)-\sin(\phi_1)\cos(\phi)\cos(\lambda - \lambda_0)}{\cos(c)}</math><br> | <math>y = \frac{\cos(\phi_1)\sin(\phi)-\sin(\phi_1)\cos(\phi)\cos(\lambda - \lambda_0)}{\cos(c)}</math><br> | ||
where <math>c</math> is the angular distance of the point <math>(x,y)</math> from the center of the projection, given by<br> | where <math>c</math> is the angular distance of the point <math>(x,y)</math> from the center of the projection, given by<br> |
Latest revision as of 17:13, 16 June 2020
The Gnomonic projection is projecting from the center of a sphere to a tangent plane. Node that this projection can not visualize more than the surface of a hemisphere. Visualizing a hemisphere would require an infinitely large plane.
Equations
Gnomonic Projection
Copied from Mathworld
Inputs:
- \(\displaystyle (\lambda, \phi)\) Current spherical coordinate with longitude \(\displaystyle \lambda\) and latitude \(\displaystyle \phi\)
- \(\displaystyle (\lambda_0, \phi_1)\) Spherical coordinate of tangent plane.
Outputs:
- \(\displaystyle (x,y) \in (-\infty, \infty) \times (-\infty, \infty)\) Cartesian coordinates.
\(\displaystyle x = \frac{\cos(\phi)\sin(\lambda - \lambda_0)}{\cos(c)}\)
\(\displaystyle y = \frac{\cos(\phi_1)\sin(\phi)-\sin(\phi_1)\cos(\phi)\cos(\lambda - \lambda_0)}{\cos(c)}\)
where \(\displaystyle c\) is the angular distance of the point \(\displaystyle (x,y)\) from the center of the projection, given by
\(\displaystyle \cos(c) = \sin(\phi_1)\sin(\phi) + \cos(\phi_1)\cos(\phi)\cos(\lambda-\lambda_0)\)
Inverse Gnomonic Projection
Copied from Mathworld
\(\displaystyle \phi = \sin^{-1}\left(\cos(c)\sin(\phi_1) + \frac{y\sin(c)\cos(\phi_1)}{\rho}\right)\)
\(\displaystyle \lambda = \lambda_0 + \tan^{-1}\left(\frac{x \sin(c)}{\rho \cos(\phi_1) \cos(c) - y\sin(\phi_1)\sin(c)}\right)\)
where
\(\displaystyle \rho=\sqrt{x^2 + y^2}\)
\(\displaystyle c = \tan^{-1}(\rho)\)