Superlesson
Project 12-1

Superlesson
Project 12-2

Geometry

Chapter 12, Astronomy and Geometric Models

Euclid's geometry postulates for planes model relatively small areas fairly well. For this reason, we can assume his postulates are true when making calculations such as those necessary to build houses and design containers. However, his postulates do not apply to many astronomical measurements. In this lesson, you will work with a non-Euclidean geometry called spherical geometry using the planet Venus as a model.

Part A, Euclidean Geometry and
Part B, Non-Euclidean Geometry

According to your text, in the spherical model of Riemannian geometry, a plane is a sphere and lines are great circles on the sphere. For this lesson, assume that Venus is a perfect sphere.

1. Go to the Venus Web site to see a list of images of Venus. For many of the images, the location is given by latitude and longitude lines like those on earth. Look at the image of the volcanoes at 20° North by 357° and the image at 35° South by 357° towards the bottom of the list. Is there a Riemannian line between these two points? If so, give the name of the line. If not, explain why not.

a. Go to the Second planet Web site and read the information about Venus. Then find the shortest distance along the planet's surface between the the two volcanoes mentioned in 1.

b. Is the distance you found in 1a shorter or longer than the Euclidean distance between these two points?

c. Is there a great circle on the surface of Venus that goes through the Golubkina crater and the volcanoes you looked at in the beginning of this problem? Is there a line of longitude that goes through these two points?

2. Does Euclid's Straight-Line Postulate hold in this spherical version of Riemannian geometry? If so, explain why. If not, give a counterexample.