Spherical Geometry Ideas

Steve Phelps, Latoya Brewer, Sep 21, 2020

This is a GeoGebraBook of some basics in spherical geometry.

Table of Contents

  1. Spherical Geometry Basics
    1. Spherical Lines: Great Circles and Poles
    2. Spherical Lines: Angles Formed by Great Circles
    3. Spherical Lines: Great Circles
    4. Spherical Lines: Angles Formed by Great Circles 2
    5. A Regular Quadrilateral on a Sphere
    6. Constructing a Spherical Segment
  2. Triangle Basics
    1. Spherical Triangle
    2. Spherical Triangle: Polar Triangles
    3. The Pythagorean Theorem for a Spherical Right Triangle
    4. Spherical Triangle: A Quadrant Triangle
    5. Spherical Triangle: A Birectanglular Triangle
    6. Spherical Triangle: Napier's Rules
    7. Spherical Triangle: Napier's Rules II
    8. Spherical Triangle: Isosceles Right Triangle
    9. Spherical Triangle: Another Special Right Triangle
  3. Points of Concurrency
    1. The Orthocenter
    2. The Spherical Circumcenter and Circumcircle.
    3. The Spherical Incenter and Incircle.
    4. The Spherical Nine Point Circle
    5. Spherical Triangle: The Medians and Centroid
  4. Conic Constructions
    1. The Parabola Construction on a Sphere
  5. Tessellations
    1. Tessellating the Sphere: Trirectangular Spherical Triangle
    2. A Spherical Cuboctahedron
    3. A spiral on a sphere

1.

Spherical Geometry Basics

  • 1. Spherical Lines: Great Circles and Poles
  • 2. Spherical Lines: Angles Formed by Great Circles
  • 3. Spherical Lines: Great Circles
  • 4. Spherical Lines: Angles Formed by Great Circles 2
  • 5. A Regular Quadrilateral on a Sphere
  • 6. Constructing a Spherical Segment

1.1.

Spherical Lines: Great Circles and Poles

Every point on the sphere can be associated with a great circle. From point A, construct a line through the center of the sphere. Then, construct a plane perpendicular to this line through the center of the sphere. The intersection is the great circle associated with the point A. Point A is called the pole of the great circle. Every great circle has two poles (do you see why?)
Spherical Lines: Great Circles and Poles
– Steve Phelps

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1.3.

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1.4.

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1.5.

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1.6.

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2.

Triangle Basics

  • 1. Spherical Triangle
  • 2. Spherical Triangle: Polar Triangles
  • 3. The Pythagorean Theorem for a Spherical Right Triangle
  • 4. Spherical Triangle: A Quadrant Triangle
  • 5. Spherical Triangle: A Birectanglular Triangle
  • 6. Spherical Triangle: Napier's Rules
  • 7. Spherical Triangle: Napier's Rules II
  • 8. Spherical Triangle: Isosceles Right Triangle
  • 9. Spherical Triangle: Another Special Right Triangle

2.1.

Spherical Triangle

This is an example of a spherical triangle. Points A, B, and C are dragable!
Spherical Triangle
– Steve Phelps

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2.7.

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3.

Points of Concurrency

  • 1. The Orthocenter
  • 2. The Spherical Circumcenter and Circumcircle.
  • 3. The Spherical Incenter and Incircle.
  • 4. The Spherical Nine Point Circle
  • 5. Spherical Triangle: The Medians and Centroid

3.1.

The Orthocenter

In Euclidean geometry, the ORTHOCENTER is the intersection of the three altitudes of a triangle. In other words, the altitudes are concurrent at the orthocenter. In the sketch below, the [color=#FF0000]RED[/color] great circles are the altitudes of the [color=#0000FF]BLUE[/color] triangle, and the [color=#FF0000]RED[/color] point is the point of concurrency of the altitudes; the spherical orthocenter.[br][br]Drag the blue points around, but do so slowly! Things change quickly on a sphere!
The Orthocenter
– Steve Phelps

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4.

Conic Constructions

  • 1. The Parabola Construction on a Sphere

4.1.

The Parabola Construction on a Sphere

Most geometry teachers know how to construct a parabola when given a focus and a directrix. Below we see the same parabola construction performed on a sphere. The [color=#006600]GREEN[/color] point is the focus. The [color=#0000FF]BLUE[/color] great circle is the directrix. The small [color=#FF0000]RED[/color] point is the point that traces the parabola.
The Parabola Construction on a Sphere
Looks like an ellipse?
– Steve Phelps

5.

Tessellations

  • 1. Tessellating the Sphere: Trirectangular Spherical Triangle
  • 2. A Spherical Cuboctahedron
  • 3. A spiral on a sphere

5.1.

Tessellating the Sphere: Trirectangular Spherical Triangle

Just like there are certain figures that can be used to tessellate a Euclidean plane, there are certain figures that can tessellate a sphere. In the figure below, the sphere is tessellated with 8 trirectangular spherical triangles ([url]http://books.google.com/books?id=OwmF0wxXfIUC&pg=PA460&lpg=PA460&dq=robbin%27s+plane+and+solid+geometry+trirectangular&source=bl&ots=qwMZkfgS-1&sig=PyFrEde55MeFpEsUMLcx7uhMtBc&hl=en&sa=X&ei=YbW0U-TNOaG_8QHF-oHIBw&ved=0CB0Q6AEwAA#v=onepage&q=robbin's%20plane%20and%20solid%20geometry%20trirectangular&f=false[/url]
Tessellating the Sphere: Trirectangular Spherical Triangle
– Steve Phelps

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