Download Geometry on Curved Surfaces Lab Exercise

Geometry on Curved Surfaces
Name _____________________________
Note: This worksheet is applicable to both spheres and saddles.
A surface like a sphere (or saddle) provides a place to experiment with a geometry which
is different from what you’ve seen before. Imagine you were an ant on the surface of a
ball: your experience of the “world” would be that it was a two-dimensional place; you
wouldn’t know it was curved. Today, you’ll be the ant, discovering a new geometry.
Straight Lines
1) Use string to find the shortest distance between two points. Do this for several sets of
points, some that are close together, others that are far apart. Do you notice anything
about the lines you create?
2) Use the flexible straightedge to draw a straight line between two points. You may use
the same sets of points that you used for part 1. How do your lines compare to part 1?
Can you draw more than one straight line between two points with this method?
With luck, you found that the methods for connecting the points in parts 1 and 2 gave the
same paths. We will call these paths straight lines. To you, as an ant, they appear
straight in that they don’t veer to the right or left as you crawl along the ball’s surface.
3) For the sphere, imagine that this ball is a globe of the Earth. Put two dots on the
globe at the same latitude but on different sides of the globe (about 45° works well;
imagine the Twin Cities and Urumqi in NW China). Is the line of latitude which
connects the points a straight line by our definition? (Sketch in the line of latitude
and try either method to see.) For the saddle, imagine a clock face drawn on the
surface. Mark points at 12:00 and 4:00, and draw a line which looks straight from
above to connect two points. Does this line fit our definition of a straight line?
4) How would you describe these straight lines based on your human (threedimensional) perspective? How do they fit onto the surface?
Parallel Lines
5) Draw a straight line segment on your globe.
6) Draw a short segment perpendicular to this line, sticking out from the original
segment near the center.
7) Construct another straight line perpendicular to the free end of your short segment.
8) What can you say about these lines? Are they parallel?
Triangles
9) Draw a small triangle. Find the sum of its angles.
10) Draw a big triangle. Find the sum of its angles. What is going on?
11) What are the limits to the sum of angles in a triangle? Why?
Angular Size
12) Pick a point, and make a small segment some distance away.
13) Draw straight lines from the point to each end of the segment.
14) Measure the angle the lines form at your point, and test the formula s = r.
Geodetic Rotation
15) Follow the path of the small triangle you constructed. What is the geodetic rotation?
16) Follow the path of the large triangle you constructed. What is the geodetic rotation?
Curvature
17) Find the radii of the surface and calculate the curvature.
18) Will a larger surface of this type have a larger curvature? Explain.