Spherical Calculator

Almost all of navigation mathmatics involves just two operations:

1. Calculate the distance (Hc) and direction (Az) between two points on a sphere.

2. Calculate a Point Relative based on the distance and direction from a start point, 
	on a sphere.

These involve spherical trigonometry.  It is a case of triangle relationships, 
where given enough sides or angles of a triangle, we can find the other sides and angles.

It's interesting that both the operations mentioned above are done with the same 2 formulas.
One formula is used to solve the terrestrial triangle (Az), and the other formula is used to 
solve the astronomical triangle (Hc).  These formulas were written centuries ago, and are 
still used today in modern sight reduction tables.

Examples of the first operation to calculate the distance and direction between two points,  
include celestial sights, great circles, Lunar distances, and current vectors.

Examples of the second operation to calculate a Point Relative based on a distance and 
direction from another point, include Deduced Reckoning (DR) calculations, great circles, 
and star & planet identification.

Examples: 
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 	Solve a celestial sight. This is what Sight Reduction tables do.

On the terrestial triangle, we have our AP, GP of the body,
and the elevated pole.  Most people use the north pole as the elevated pole, 
including the folks down under, but if you want to do everything up-side-down, 
you can use the south pole as the elevated pole. 

On the astronomical triangle, we have our AP, GP of the body, and the Height Observed (Ho).

Our objective is to get the Height Calculated (Hc), and the Azimuth (Az) from the AP and GP.
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 	Great Circles:  These can also be calculated using Sight Reduction tables.

We calculate the distance and direction using the AP, destination (Dest), 
and the elevated pole. Then we project a series of points Relative to the AP to get a 
series of points to plot the great circle.  This is where you can choose the resolution 
of the points.
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	Lunar and Star Distances:  These can also be calculated using Sight Reduction tables.

In this case, all we need is the distance between the GPs' of 2 bodies, so we don't need to 
solve the terrestial triangle.  Lunar Distance calcs can be tedious because we need to 
enter the Almanac twice, to get the GP of both bodies using the same GMT.

Using the Spherical Calculator, you can quickly enter both points using the [Almanac] button.
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	Maneuvering Board: These can also be calculated using Sight Reduction tables.

This could be a series of tacks based on DR calculations.  During each tack, we estimate 
our distance and direction Relative to the last tack.  This info is then used to 
calculate a point Relative to the AP, which is then saved as the new AP.

The Spherical Calculator helps facilitate this by using Point 2 to set Point 1.
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As you can see, almost everything can be done with the Sight Reduction tables, but it would 
be very tedious.  In the days before computers, it was either the tables or a slide rule.
Most ships would carry both.

 To be continued...