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'example converting ecliptic coordinates to rectangular coordinates'
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| Title of object: |
example converting ecliptic coordinates to rectangular coordinates |
| Canonical Name: |
ExampleConvertingEclipticCoordinatesToRectangularCoordinates |
| Type: |
Example |
| Created on: |
2025-03-14 00:21:36 |
| Modified on: |
2025-03-14 00:21:36 |
| Classification: |
msc:95.10.-a |
Revision comment (for changes between this and next version):
| added that article was generated by Grok |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
Consider an object with ecliptic coordinates:
\begin{itemize}
\item Longitude: \( l = 30^\circ \),
\item Latitude: \( b = 15^\circ \),
\item Distance: \( r = 1 \) AU.
\end{itemize}
The conversion to rectangular coordinates is given in ecliptic coordinate system:
\begin{align*}
x &= r \cos b \cos l, \\
y &= r \cos b \sin l, \\
z &= r \sin b.
\end{align*}
Convert to radians:
\[
l = 30^\circ = \frac{\pi}{6} \approx 0.5236, \quad b = 15^\circ = \frac{\pi}{12} \approx 0.2618.
\]
Compute:
\[
\cos b \approx 0.9659, \quad \sin b \approx 0.2588, \quad \cos l \approx 0.8660, \quad \sin l = 0.5000.
\]
Then:
\[
x = 1 \times 0.9659 \times 0.8660 \approx 0.8365,
\]
\[
y = 1 \times 0.9659 \times 0.5000 \approx 0.4830,
\]
\[
z = 1 \times 0.2588 \approx 0.2588.
\]
Thus, the rectangular coordinates are:
\[
(x, y, z) \approx (0.8365, 0.4830, 0.2588) \, \text{AU}.
\] |
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