# Solving the Great Circle Problem on a Sphere

• davidam
In summary: Since (a1, a2, a3) and (b1, b2, b3) are on that circle, you can choose the parameter values to get those two points. You will have to do some work to figure out the correct values of the parameters, given the values of a1, a2, and a3 (or b1, b2, and b3).In summary, the problem involves finding a parametric equation to describe the path from one point to another along the surface of a sphere. This can be done by finding the equation of the plane containing the points and the center of the sphere, and then using that to solve for parametric equations for the circle of intersection between the plane
davidam

## Homework Statement

I have a problem where I have a sphere with radius R centered at (0,0,0), and two given points on the surface of the sphere A=(a1,a2,a3) and B=(b1,b2,b3). I have to find/derive a parametric equation to describe the path from point A to point B along the surface of the sphere.

## Homework Equations

I need to provide x(t)=?, y(t)=?, z(t)=? in general terms to describe the path between any two points on the surface of a sphere with any radius.

## The Attempt at a Solution

When giving values to the points I can figure out how to find a plane that passes through the center of the sphere and the two points, but I don't know how to make a parametric equation to describe the circle of intersection between the sphere and the plane. If I could do that, I could set the parameters to only give me the portion of arc from A to B. I'm not exactly sure that I'm on the right track with the plane/sphere intersection idea, so any help would be much appreciated.

First, try the problem in two dimensions. You can use sin and cosine to define the circle. And then use rotations to rotate the circle to 3d. (This approach looks rather hard but not really if you can know matrices approach to do that) I think you do need to know the plane in which all three points fall.

Using above here's What I will do:

-find the plane
-find angle it makes with the z axis
-use matrices to rotate the plane to x-y plane
-solve the problem
-rotate back to the original problem using matrices again

I cannot think of any easier way at the moment.

A great circle is on a plane passing through the center of the circle and so will be the plane containing the points (0,0,0), (a1,a2,a3), and (b1,b2,b3). Find the equation of that plane: Ax+ By+ Cz= D for some numbers A,B,C,D. The equation of the sphere is x^2+ y^2+ z^2= R^2 where R^2= a1^2+ a2^2+ a3^2= b1^2+ b2^2+ b3^2 (and those last two must be equal in order that (a1, a2, a3) and (b1, b32, b3) lie on the same sphere). You have two equations in 3 variables so you can solve for two of them in terms of the third. That will give you parametric equations for the circle with that third variable as parameter.

## What is the Great Circle Problem on a Sphere?

The Great Circle Problem on a Sphere is a mathematical problem that involves finding the shortest distance between two points on a spherical surface. It is based on the concept of a great circle, which is the largest circle that can be drawn on a sphere and is equivalent to the equator on a globe.

## Why is Solving the Great Circle Problem important?

Solving the Great Circle Problem is important in various fields such as navigation, astronomy, and cartography. It allows for more accurate calculations of distances and routes on a spherical surface, which is essential for tasks such as determining flight paths and shipping routes.

## What is the formula for solving the Great Circle Problem?

The formula for solving the Great Circle Problem is the Haversine formula, which uses the latitude and longitude coordinates of two points on a sphere to calculate the central angle between them. This central angle can then be used to determine the shortest distance between the two points along the surface of the sphere.

## What are some common applications of the Great Circle Problem?

Some common applications of the Great Circle Problem include flight planning, ship navigation, and satellite orbit calculations. It is also used in geodesy, the science of measuring and mapping the Earth's surface, as well as in the study of celestial bodies.

## What challenges are involved in solving the Great Circle Problem?

One of the main challenges in solving the Great Circle Problem is accounting for the curvature of the Earth's surface. This requires the use of spherical geometry and specialized formulas, as distances and angles on a sphere are different from those on a flat surface. Additionally, the accuracy of the calculations can be affected by factors such as the precision of the input coordinates and the size of the sphere being used.

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