# Shortest Path Between 2 Points on a Sphere

- Author:
- Tim Brzezinski

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.