7: Geometry on Surfaces
In hyperbolic geometry (\(\mathbb{D}, \mathcal{H}\)) and elliptic geometry (\(\mathbb{P}^2, \mathcal{S}\)), the area of a triangle is determined by the sum of its angles. This is a significant difference from Euclidean geometry, in which a triangle with three given angles can be built to have any desired area. Does this mean that if a bug lives in a world adhering to elliptic geometry, it can never stumble upon a triangle with three right angles having area \(3π\)? Yes and no. In the elliptic geometry as defined in Chapter 6, no such triangle exists because a triangle with \(3\) right angles must have area \((\dfrac{π}{2} + \dfrac{π}{2} + \dfrac{π}{2}) − π = \dfrac{π}{2}\). So the answer appears to be yes. However, the elliptic geometry (\(\mathbb{P}^2, \mathcal{S}\)) models the geometry of the unit sphere, and this choice of sphere radius is somewhat arbitrary. What if the radius of the sphere changes? Imagine a triangle with three right angles having one vertex at the north pole and two vertices on the equator. If the sphere uniformly expands, the angles of the triangle will stay the same, but the area of the triangle will increase. So, if a bug is convinced she lives in a world with elliptic geometry, but is also convinced she has found a triangle with three right angles and area \(3π\), the bug might be drawn to conclude she lives in a world modeled on a larger sphere than the unit \(2\)-sphere.
The key geometrical property of a space dictating the relationship between the angles of a triangle and its area is called curvature. Curvature also dictates the relationship between the circumference of a circle and its radius.
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- 7.1: Curvature
- The curvature of the curve at a point is a measure of how drastically the curve bends away from its tangent line, and this curvature is often studied in a multivariable calculus course. The radius of curvature at a point corresponds to the radius of the circle that best approximates the curve at this point.
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- 7.3: Hyperbolic Geometry with Curvature k < 0
- We may do the same gentle scaling of the Poincaré model of hyperbolic geometry as we did in the previous section to the disk model of elliptic geometry. In particular, for each negative number k<0 we construct a model for hyperbolic geometry with curvature k.
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- 7.4: The Family of Geometries (Xₖ, Gₖ)
- A strong connection exists between the family (Pk^2, Sk) of elliptic geometries with curvature k>0 and the family (Dk, Hk) of hyperbolic geometries with curvature k<0. The two families sport identical descriptions of the transformation group, identical descriptions of straight lines, identical arc-length and area formulas, as well as identical formulas for the area of a triangle.
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- 7.6: Geometry of Surfaces
- If you've got a surface in your hand, you can find a homeomorphic version of the surface on which to construct hyperbolic geometry, elliptic geometry, or Euclidean geometry. And the choice of geometry is unique: No surface admits more than one of these geometries. As we shall see, of the infinitely many surfaces, all but four admit hyperbolic geometry (two admit Euclidean geometry and two admit elliptic geometry).
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- 7.7: Quotient Spaces
- A relation on a set S is a subset R of S x S. In other words, a relation R consists of a set of ordered pairs of the form (a,b) where a and b are in S. A partition of a set 𝐴 consists of a collection of non-empty subsets of A that are mutually disjoint and have union equal to A. An equivalence relation on a set A serves to partition A by the equivalence classes.