ssrs barcode font 19.4A Euclidean Geometry in Objective-C

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19.4A Euclidean Geometry
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Just for comparison, here are several properties of euclidean geometry: Planes are infinite in area and flat. Lines are infinite in length. The angles of triangles always sum to 180 . There are similar triangles of different sizes. The circumference of a circle with radius r is C The area of a circle with radius r is A pr 2.
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2pr 2.
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19.4B Elliptic Geometry
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The properties of elliptic geometry are Planes are shaped like spheres, finite in area. Lines are great circles, finite in length. The angles of a triangle add up to more than 180 . For an example, take two longitude circles that meet at right angles at the North and South Poles and add the equator. These will cut the globe into 8 triangles where every angle is 90 . Thus, in elliptic geometry it is possible to have a triangle with an angle sum of 270 , as shown in Fig. 19-4. Smaller triangles will have smaller angle sums, but always more than 180 .
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Fig. 19-4
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CHAPTER 19 Non-Euclidean Geometry
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The size of a triangle on a sphere is related to the sum of all its angles. For example, every triangle with an angle sum of 270 will have area equal to one eighth of the whole sphere, just like the triangles in Fig. 194. Smaller triangles will have smaller angle sums. Because of this, it is impossible to have two triangles of different sizes that have the same angle measures. Thus, there cannot be similar triangles of different sizes in elliptic space. A circle in elliptic space is defined as usual: the set of all points a given distance from a point. Because distances are measured along lines which curve around great circles on a sphere, this means that a larger radius does not always lead to a larger circle. Once the radius exceeds one quarter of a great circle s circumference, circles actually begin to shrink, as illustrated in Fig. 19-5. The distance from the North Pole N to point A is one eighth of the circumference of the sphere, from N to B is one quarter of the circumference, and from N to C is three quarters of the circumference. However, the circumference of the circle around N with radius NB (the equator) is greater than the circumference of the latitudinal circle around N with radius NC. A larger radius does not guarantee a larger circumference. In fact, the circumference of a circle with radius r in elliptic space is less than 2pr.
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Fig. 19-5
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Similarly, the area of a circle with radius r in elliptic space is A < pr2. If a circle has a very small radius, then it will be almost flat and have area close to pr2 and circumference near 2pr (though slightly less in both cases). The larger a circle s radius becomes, the further it will be from having the area and circumference of a circle in Euclidean space with the same radius. Technically, elliptic space violates the first postulate. Between the North and South Poles, any of the longitudinal great circles counts as a straight line, thus, there are many straight lines between two points. A trick to overcome this is to call the North and South Poles together as one single point. If each pair of antipodal spots on a globe is viewed as a single point, then the resulting geometry is called projective space, which has all of the above properties in common with elliptic space.
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19.4C Hyperbolic Space
Euclidean geometry is based on flat planes. A flat object is said to have zero curvature. Elliptic geometry is based on planes that are positively curved, like the surface of a sphere. Hyperbolic geometry is based on planes that are negatively curved and frilly like certain kinds of seaweed. The three kinds of curvature are illustrated in Fig. 19-6.
Fig. 19-6
Just as with elliptic space, any tiny piece of hyperbolic space is almost flat and will have properties close to those of euclidean space. As pieces get larger, however, they contain more of the warps, bends, and frills that are symptomatic of this space. The word hyperbolic, by the way, has roots meaning excessive and exaggerated.
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