Spherical geometry
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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

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Spherical Geometry

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Spherical geometry

Spherical Geometry

TWSSP Wednesday


Welcome

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

Wednesday Agenda

  • Agenda

  • Question for today:

  • Success criteria: I can …


Taxicab circle posters

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

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

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

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

  • 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

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

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?


Distances1

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

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

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

Squares

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


Exit ticket sort of

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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