Gnomonic projection: Difference between revisions

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The Gnomonic projection is projecting the ~~surface~~ of a sphere ~~from the center~~ to a tangent plane.

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.

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==Equations==

===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:

~~Cartesian coordinates.~~

* <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>

<math>x = \frac{\cos(\~~theta~~)\sin(\lambda - \lambda_0)}{\cos(c)}</math>

<math>y = \frac{\cos(\phi_1)\sin(\phi)-\sin(\phi_1)\cos(\phi)\cos(\lambda - \lambda_0)}{\cos(c)}</math>

where <math>c</math> is the angular distance of the point <math>(x,y)</math> from the center of the projection, given by

@@ Line 1: / Line 1: @@
-The Gnomonic projection is projecting the surface of a sphere from the center to a tangent plane.
+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.
@@ Line 5: / Line 5: @@
 ==Equations==
 ===Gnomonic Projection===
-Copied from Mathworld<br>
+Copied from [https://mathworld.wolfram.com/GnomonicProjection.html Mathworld]
-Inputs: <br>
-<math>(\lambda, \phi)</math> Current spherical coordinate<br>
+Inputs:
-<math>(\lambda_0, \phi_1)</math> Spherical coordinate of tangent plane.<br>
+* <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>
-Cartesian coordinates.
+* <math>(x,y) \in (-\infty, \infty) \times (-\infty, \infty)</math> Cartesian coordinates.
-<math>(x,y) \in (-\infty, \infty) \times (-\infty, \infty)</math><br>
-<br>
+<math>x = \frac{\cos(\phi)\sin(\lambda - \lambda_0)}{\cos(c)}</math><br>
-<math>x = \frac{\cos(\theta)\sin(\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>