1) A "small circle" is a circle on a sphere that is NOT a "great circle".
What is the locus of the centers of ALL small circles of a sphere having radius 1/2 that of the sphere?
I think the answer is a plane?
But I'm not sure.
Please help me!
To determine the locus of the centers of all small circles on a sphere with radius 1/2 of that sphere, let's break down the problem.
First, it's important to understand the difference between a small circle and a great circle on a sphere. A great circle is a circle whose diameter is equal to the diameter of the sphere. Examples of great circles are the equator or any meridian on Earth. A small circle, on the other hand, has a diameter smaller than that of the sphere.
Next, let's consider the centers of the small circles. Since the small circles have a radius 1/2 that of the sphere, it means that each small circle is located halfway between the sphere's center and its surface.
The locus of the centers of these small circles is a concentric sphere, coinciding with the original sphere but with half the radius. This is because all points on this smaller sphere are equidistant from both the center of the original sphere and its surface. Therefore, this locus forms a perfectly symmetrical sphere with a radius equal to 1/2 of the original sphere's radius.
In conclusion, the locus of the centers of all small circles on a sphere with a radius 1/2 that of the sphere is a sphere with half the radius of the original sphere.