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Spherical Geometry

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

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Spherical Geometry

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  1. Spherical Geometry TWSSP Wednesday

  2. 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

  3. Wednesday Agenda • Agenda • Question for today: • Success criteria: I can …

  4. 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?

  5. 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?

  6. 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?

  7. 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.

  8. 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.

  9. 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?

  10. 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?

  11. 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?

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

  13. 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?

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

  15. 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?

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