How can I calculate the tilt of the Earth using trigonometry?

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    Earth Model Tilt
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SUMMARY

The Earth’s axial tilt, known as obliquity, is approximately 23.439281°, influencing seasonal changes. A user has developed a preliminary MATLAB program that models the 3D Earth Equatorial Model, including rings at the equator and tilt extremes. The discussion centers on deriving a trigonometric equation to represent the 3D loop connecting the tilt maxima and minima through the origin. The user seeks assistance in formulating these equations using the parameters provided.

PREREQUISITES
  • Understanding of trigonometric functions and their applications in 3D modeling.
  • Familiarity with MATLAB programming and its graphical capabilities.
  • Knowledge of Earth's axial tilt and its impact on seasonal variations.
  • Basic concepts of spherical coordinates and their conversion to Cartesian coordinates.
NEXT STEPS
  • Research the derivation of trigonometric equations for modeling planetary tilts.
  • Learn about spherical coordinates and their application in 3D graphics.
  • Explore MATLAB functions for 3D plotting and visualization.
  • Investigate the mathematical principles behind Earth's obliquity and its effects on climate.
USEFUL FOR

Students in physics or astronomy, MATLAB programmers, and anyone interested in modeling planetary movements and understanding the Earth's tilt and its seasonal effects.

Philosophaie
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Earth has a tilt to it about 23.439281° to -23.439281° from the Tropic of Cancer to the Tropic of Capricorn. The tilt faces the sun at differrent points from 23.439281° to -23.439281° creating the seasons etc. I have written a preliminary MatLab program showing the 3D Earth Equitorial Model plus some rings at the equator and at the tilt maxima and minima. What I need is some help with the trigonomic equation of the 3D loop ring that connects the two tilt maxima and minima thru the origin. Here is an example of what I have. Center loop:

x = r * cos(theta) * sin(phi)
y = r * sin(theta) * sin(phi)
z = r * cos(phi)

where r = 6378100, phi = 90 and theta = 0 - 360
 

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I have had nothing but trouble finding a set of equations to calculate the Tilt of the Earth trigonometrically for a theta (0 to 2pi) and a given Obliquity = 23.439281° (for Earth). Please disregard the MatLab program above. Any suggestions would be much appreciated.
 
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