Deriving a Small Circle Equation in Spherical Coordinates: Any Pointers?

[whatta]

I need some good pointers to how could I derive arbitrary small circle equation in spherical coordinats (by hand or computer-aided), that is, LONG(t), LAT(t) or something like that.

Also, any approximate equations? How bad they are?

Please do not advice switching to cartesian, this is exactly what I would like to avoid.

Thank you for reading.

PS: if you happened to have ready equations, don't tease me, post'em.

 

[whatta]

never mind, I have found the way around it. solution would still be good to know, but no longer necessary.

 

[tacman]

Deriving equations in spherical coordinates can be challenging, but with some pointers, it can become more manageable. Here are some suggestions to help you derive the small circle equation:

1. Understand the concept: Before attempting to derive the equation, make sure you have a solid understanding of what a small circle is in spherical coordinates. A small circle is a circle on the surface of a sphere with a radius smaller than the sphere's radius. It is defined by two parameters, longitude (LONG) and latitude (LAT).

2. Use the parametric equation: The parametric equation for a small circle in spherical coordinates is LONG(t) and LAT(t), where t is a parameter that varies from 0 to 2π. This equation describes the path of a point on the small circle as t changes.

3. Use the spherical coordinate system: Familiarize yourself with the spherical coordinate system and its components, including the radial distance (r), the azimuth angle (θ), and the polar angle (φ). These components will be useful in deriving the small circle equation.

4. Use the Pythagorean theorem: The Pythagorean theorem can be used to relate the coordinates of a point on the small circle to its distance from the center of the sphere. This relationship will help you derive the equation.

5. Use trigonometric identities: Trigonometric identities such as the cosine rule and the sine rule can be helpful in deriving the small circle equation. These identities can be used to relate the coordinates of a point on the small circle to its distance from the center of the sphere.

As for approximate equations, there are several ways to approximate the small circle equation. One method is to use a series expansion, such as a Taylor series, to approximate the trigonometric functions involved in the equation. Another approach is to use a numerical method, such as the Newton-Raphson method, to solve for the coordinates of a point on the small circle.

The accuracy of these approximate equations will depend on the level of approximation used and the complexity of the small circle. However, they can still provide a good estimate for the coordinates of a point on the small circle.

In conclusion, deriving the small circle equation in spherical coordinates can be done by understanding the concept, using the parametric equation, the spherical coordinate system, the Pythagorean theorem, and trigonometric identities. There are also approximate equations available, but their accuracy may vary depending on the level of approximation used. So, choose the