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### 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?

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