Circles

=Circles=

Circles on a sphere have the same basic definition as circles on a plane: **//the set of points that are a given distance from a given point, which is the center of the circle.//** On a sphere, a circle's radius would be the segment, or arc, of a great circle (i.e. a straight line on a sphere) which connects the center of the circle and a point on the circle. As an example, in the figure of a sphere below, point A is the center of circle C and line segment AB is the radius. L shows the great circle of which AB is a minor arc.

**Figure 1:** Circle on a sphere Circle C above is a **//small circle//** because it does not pass through the center of the sphere like a **//great circle//** (e.g. circle L) which does pass through the center of the sphere. As we learned when we studied straight lines on a sphere, a //**great circle**// is the intersection of a plane containing two points on a sphere and the center of the sphere. There are an infinite number of great circles on a sphere but the most familiar examples are the equator and lines of longitude on a globe. By contrast, a **//small circle//** is the intersection of the sphere and a plane which does not pass through the center of the sphere. Latitude lines on a globe are familiar examples of small circles.

__**Small Circles vs Great Circles**__
Consider the relationship between small circles and great circles on a sphere. More specifically, consider the relationship of the arc between two intersecting points on a small circle and a great circle. For example, see Figure 2a below.



The circle with center A is the great circle and the circle with center D is the small circle. The circles intersect at points B and C. In Euclidean geometry, the distance between B and C is measured at 2.78 units. (The distance from B to D is the radius of the small circle).



In Figure 2b, you can see the arc measurements of arc BC. The arc length of BC in circle D is larger than the arc length of BC in circle A. Since Circle A is a great circle, you can see that the **//shortest distance between two points is always going to be an arc on a great circle.//**

As the small circle gets bigger, the arc between B and C on the small circle gets smaller (see Figure 2c).

In fact, the only way that the arc lengths are even the same is when circle D = circle A. See Figure 2d below:

To explore these circles and the arc relationships further, use the Geogebra applet here: @http://courses.ncsu.edu/ma508/lec/001/circles.html

__Circumference and Radius__
In Euclidean geometry, the circumference of a circle = 2πr, where r is the radius of the circle. Let's see if that is true for a circle on a sphere.

__**Activity 1 - Draw a Circle on a Sphere**__
It is not feasible or practical to use a compass to draw a circle on a sphere like you can on a plane. However, a circle can be drawn on a sphere, or a ball, by fashioning a compass-like instrument using a pencil on the end of a string.
 * 1) Cut a string to the length of the desired radius.
 * 2) Attach a pencil or some other writing instrument, such as chalk, to one end of the string.
 * 3) Attach the other end of the string to a point on a sphere. That will be the center of the circle.
 * 4) Now pull the string tight and move it around until it reaches the starting point and you have drawn a circle on the sphere.

__**Activity 2 - Draw a Circle with the same radius on a Plane**__
In Activity 1, you created a circle using a string with length //r// as the radius length. Now, using that same string for the radius, draw a circle on a plane.
 * 1) Attach the end of the string to a point on a piece of paper. That will be the center of the circle.
 * 2) Pull the string tight and move it around until it reaches the starting point.
 * 3) Now you have a circle on the plane with the same radius as the one you drew on the sphere.

Compare the two circles and measure the circumferences. What do you notice about the size of the circles and their circumference?

The circumference of the circle you drew on the paper should be approximately 2πr = 2 * (3.14 * r). However, the circumference on the sphere is quite a bit smaller than that, even with the same radius. So, the relationship of π to a circle on a sphere is not the same as the relationship of π to a circle on a plane.

Try some more circles with different radii and see if you can determine if there is a consistent relationship between the radius and circumference of a circle on a sphere.

__**Questions to consider:**__

 * 1) How long can a radius be on a sphere?
 * 2) Could a radius actually go all the way around a sphere?
 * 3) Could a point be considered a circle? If so, what is the radius of the circle at the North Pole?
 * 4) Is there a consistent relationship between the radius and the circumference of a circle on a sphere like there is in circles on a plane?

Actually, a radius could go all the way around a sphere. For example, if the center of the circle was the North Pole and the radius extended all the way around the sphere, then the resulting circle would be a point at the North Pole. You could consider the radius of that circle as either the circumference of the sphere or as zero. Extending this example, if the circumference of a point as a circle is zero and the radius of the circle could the circumference of the sphere or zero, then you can see that there is no consistent relationship of radius to circumference in circles on a sphere.

**Figure 3:** North Pole as a circle with radius equal to circumference of sphere Figure 3 illustrates a circle at the North Pole with radius equal to the circumference of the whole sphere. As you can see, a radius can go all the way around a sphere in spherical geometry to form a circle with circumference of zero.