Gnomonic projection: Difference between revisions

From David's Wiki
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..."
 
 
(2 intermediate revisions by the same user not shown)
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.
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.
Line 5: Line 5:
==Equations==
==Equations==
===Gnomonic Projection===
===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>
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>
<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
\(\displaystyle \rho=\sqrt{x^2 + y^2}\)
\(\displaystyle c = \tan^{-1}(\rho)\)

References