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6: Elliptic Geometry

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    Elliptic geometry is the second type of non-Euclidean geometry that might describe the geometry of the universe. In this chapter, we focus our attention on two-dimensional elliptic geometry, and the sphere will be our guide. The chapter begins with a review of stereographic projection, and how this map is used to transfer information about the sphere onto the extended plane. We develop elliptic geometry in Sections 6.2 and 6.3, and then pause our story in Section 6.4 to reflect on what we have established, geometry-wise, before moving on to geometry on surfaces in Chapter 7.

    • 6.1: Antipodal Points
      Two distinct points on a sphere are called diametrically opposed points if they are on the same line through the center of the sphere. Diametrically opposed points on the sphere are also called antipodal points.
    • 6.2: Elliptic Geometry
      As was the case in hyperbolic geometry, the space in elliptic geometry is derived from C+, and the group of transformations consists of certain Möbius transformations.
    • 6.3: Measurement in Elliptic Geometry
      Rather than derive the arc-length formula here as we did for hyperbolic geometry, we state the following definition and note the single sign difference from the hyperbolic case. This sign difference is consistent with the sign difference in the algebraic descriptions of the transformations in the respective geometries.
    • 6.4: Revisiting Euclid's Postulates
      Without much fanfare, we have shown that the geometry (P^2,S) satisfies the first four of Euclid's postulates, but fails to satisfy the fifth. This is also the case with hyperbolic geometry (D,H). Moreover, the elliptic version of the fifth postulate differs from the hyperbolic version. It is the purpose of this section to provide the proper fanfare for these facts.

    This page titled 6: Elliptic Geometry is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Michael P. Hitchman via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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