Shortest Path Between 2 Points on a Sphere

In the context of a SPHERE, A GREAT CIRCLE is defined to be a CIRCLE that lies on the SURFACE OF THE SPHERE and LIES ON A PLANE that PASSES THROUGH THE CIRCLE's CENTER. In essence, the center of a GREAT CIRCLE and the center of the sphere are the same. Consequently, a GREAT CIRCLE also the largest possible circle one can draw on a sphere. In the applet below, the pink arc and blue arc make up a GREAT CIRCLE. To explore this resource in Augmented Reality, see the directions below the applet.

1.

Note that the black arc and yellow arc (put together) DO NOT make a great circle. Why is this?

See below this applet for directions.

Directions: Move the 2 WHITE POINTS anywhere you'd like on the sphere. The PINK ARC is part of a GREAT CIRCLE of this SPHERE. You can move the YELLOW POINT anywhere you'd like as well. Again, note that the YELLOW ARC is NOT PART of a great circle. Slide the slider slowly and carefully observe what happens.

2.

How would you describe the SHORTEST DISTANCE between 2 POINTS along a SPHERE? Explain.

TO EXPLORE IN AUGMENTED REALITY:

1) Open GeoGebra 3D app on your device. 2) Press the 3 horizontal bars (upper left corner). Select OPEN. 3) In the SEARCH TAB that appears, type Gh58sVPx Note this string of characters can be found in the URL here. Be sure to either copy & paste this code or type it just the way you see it here. 4) The slider named j controls the entire animation. Slider k_1 adjusts the opacity (shading) of the green sphere. Slider a adjusts the radius of the sphere.