Determine Great Circles on a Sphere: Two Points or Three?

Is it three points on a sphere that determine a great circle, or is it two points? The answer to this question may seem simple, but it requires a deeper understanding of the properties of spheres and circles. This article will explore the concepts and clarify the requirements for determining a great circle on the surface of a sphere.

Defining Great Circles on a Sphere

A great circle on a sphere is the largest possible circle that can be drawn on the surface of the sphere. It is the intersection of the sphere with a plane that passes through the center of the sphere. In simpler terms, a great circle is a circle on the sphere's surface that divides the sphere into two equal hemispheres. Examples of great circles include the equator and the lines of longitude on a globe.

Exploring the Concept with Three Points

Let's start by examining the case of three points on a sphere. The first point to consider is that three points on the surface of a sphere will determine a circle on the sphere's surface, but this circle is not necessarily a great circle. In fact, a sphere can have an infinite number of circles passing through three given points, depending on the plane that intersects the sphere.

Visualizing with a Globe

To visualize this, pick a globe and place three dots on it. You will notice that the equator and the lines of longitude are great circles. However, you can place three dots on any of these lines or circles, and they will not all lie on a great circle. This demonstrates that three points on a sphere do not uniquely determine a great circle.

The Implications of Two Points

Now, let's move on to the case of two points on a sphere. Two points on a sphere will determine a straight line on the surface of the sphere. This line is the shortest path between the two points on the sphere, often referred to as the geodesic. To visualize this, use a ribbon or an elastic band to connect the two points on a sphere. The ribbon will lie along a great circle, which is the shortest path between the two points.

The Role of the Ribon Test

The "ribbon test" is a practical method to find a great circle. It involves placing a ribbon or elastic band between two points on a sphere and then stretching it to form a straight line. This line will lie along a great circle, as it is the shortest path between the two points on the surface of the sphere.

Unique Determination by Two Points

Two points on a sphere will uniquely determine a great circle. The great circle between two reference points is the set of points on the sphere that are equidistant from each point. This means that for any two points, there is exactly one great circle that passes through both of them.

Mathematically, this can be stated as follows: if you have two points on a sphere that are not antipodes (points directly opposite each other on the sphere), then these two points determine a unique plane in space. This plane intersects the sphere in a great circle. However, if the two points are antipodes, the situation changes slightly.

Special Case of Antipodal Points

When the two points are antipodes, the plane defined by these points and the center of the sphere is not unique. This is because the plane can be rotated about the line connecting the two points and the center of the sphere. Therefore, there are multiple great circles that can pass through the two antipode points.

To illustrate this, consider the Earth with the North and South Poles as the antipode points. The lines of longitude, which are the meridians, are great circles that pass through both the North and South Poles. These lines represent an infinite number of great circles through the same points.

Conclusion

In conclusion, two points on a sphere are sufficient to determine a unique great circle, unless the points are antipodes. In the case of antipodes, an infinite number of great circles can pass through these points. Understanding these concepts is crucial for applications in navigation, cartography, and various fields of mathematics and physics.