Spherical Geometry TWSSP Wednesday

Welcome OK, OK, I give in! You can sit wherever you want, if … You form groups of 3 or 4 You promise to assign group roles and really pay attention to them today AND you promise to stay on task, minimize your side conversations, and participate actively in our whole group discussions

Wednesday Agenda Agenda Question for today: Success criteria: I can …

Taxicab Circle Posters Go around the room and look at the posters addressing your wonders about circles. Be critical in your analysis – do you agree with the conclusions? Do you have questions about the conclusions or the justifications?

Taxicab Triangles It can be shown that taxicab geometry has many of the same properties as Euclidean geometry but does not satisfy the SAS triangle congruence postulate. Find two noncongruent right triangles with two sides and the included right angle congruent Explore taxicab equilateral triangles. What properties do they share with Euclidean equilateral triangles? How do they differ?

What do we know? Use the Think (5 min) – Go Around (5 min) – Discuss (10 min) protocol What is “straight” on the plane? How do you know if a line is straight? How can you check in a practical way if something is straight? If you want to use a tool, how do you know your tool is straight? How do you construct something straight (like laying out fence posts or constructing a straight line)? What symmetries does a straight line have? Can you write a definition of a straight line?

Straightness on the Sphere Imagine yourself to be a bug crawling around a sphere. The bug’s universe is just the surface; it never leaves it. What is “straight” for this bug? What will the bug see or experience as straight? How can you convince yourself of this? Use the properties of straightness, like the symmetries we established for Euclidean-straightness.

Great Circles Great circles are the circles which are the intersection of the sphere with a plane through the center of the sphere. Which circles on the surface of the sphere will qualify as great circles? Are great circles straight with respect to the sphere? Are any other circles on the sphere straight with respect to the sphere? The only straight lines on spheres are great circles.

Points on the sphere Given any two points on the sphere, construct a straight line between those two points. How many such straight lines can you construct? In how many points can two lines on the sphere intersect? In how many points can three lines on the sphere intersect?

Distances The Earth as a sphere in Euclidean space has a radius of 6,400 km i.e. the radius as measured from the center of the sphere to any point on the surface of Earth is 6,400 km What is Earth’s circumference? How many degrees does this represent? If two places on Earth are opposite each other, what is the distance between them in kilometers in the spherical sense? In degrees? If two places are 90o apart from each other, how far apart are they in kilometers in the spherical sense? If two places are 5026 km apart, what is their distance apart measured in degrees?

Distances Mars has a circumference of 21,321 kilometers. What does this distance represent in degrees? What is the furthest distance that two places on Mars can be apart from each other in degrees? In kilometers (in the spherical sense)? What is the minimum information we need to find the distance between two points on a sphere?

Special Lines Remind yourself of the definitions of parallel and perpendicular lines in Euclidean geometry What are parallel lines on the sphere? Perpendicular lines?

Spherical Triangles Given any three non-collinear points in the plane, how many triangles can you form between those points? Given any three non-collinear points on the sphere, how many triangles can you form between those points? What is the sum of the angles of a Euclidean-triangle? How do you know? What is the sum of the angles of a spherical-triangle? How do you know?

Squares Investigate squares on the sphere. Justify any conclusions you make.

Exit Ticket (sort of…) Compare and contrast taxicab and Euclidean circles. What do they have in common? How do they differ? Given two points on a sphere, how many possible sphere-lines (great circles) can you construct between them. Compare and contrast Euclidean and spherical triangles. What do they have in common? How do they differ?