Spherical geometry

Description

in plane geometry, the basic concept is that points and lines. On a spherical surface, and define the concept of points remain unchanged, but the line is no longer a "straight line", but the shortest distance between two points, called geodesic. On a spherical surface, most of the great circle arc is short, so that the planar line is substituted in the great circle of the spherical geometry. Similarly, the angle of the spherical geometry is defined between two great circles. The result is the usual spherical trigonometry and trigonometry has many differences. For example: spherical triangular interior angle greater than 180 °.

In contrast to a point by at least two parallel lines, or even an infinite hyperboloid geometry plurality of parallel lines, parallel lines do not pass a particular point of spherical geometry is elliptical geometry easiest model.

ants found

is contemplated to have a flat surface ants living in the two-dimensional, two-dimensional because the biological, the third dimension is no feeling. If the ants living in the big plane, on the creation of Euclidean geometry from practice. If it lives in a sphere, triangle will be the creation of a greater than 180 degrees, is less than the circumference of the spherical geometry 3.14. However, if the ants living in a big sphere, when its "science" was not fully developed, the scope of activities is not big enough, it is not sufficient to find curved sphere, which is similar to the life of small spherical plane, so it first founded Euclidean geometry. When it's "science and technology" developed, it will find the triangle and greater than 180 degrees, the ratio of the circumference of less than 3.14 and other "experimental facts." If ants smart enough, it will conclude that their universe is a curved two-dimensional space, when it is his own "universe" measurement times, come to the conclusion that the universe is closed (perimeter also We will return to the place), limited, and because the degree of bending "space" (surface) (curvature) is the same everywhere, they will own the universe and the universe round up the analogy, that the universe is "round." Since there is no third dimension feeling, so it can not imagine how their universe is curved into a ball, but they can not imagine this "boundless" there is a spherical universe is finite area in a three-dimensional flat space. They are difficult to answer "What is outside the universe," such problems. Because of their limited universe is boundless closed two-dimensional space, it is difficult to form the "outside" of the concept.

For the ants have to rely on the fact that the abstract can be found in the "advanced technology", a bee but can be easily described with the visual image of it. Because bees are biological three-dimensional space, for embedded in three-dimensional space of the two-dimensional surface is "clear", the concept is also very easy to form a sphere. Ants with their own "science and technology" to get to the same conclusion, but which is not the image of the strictly mathematical.

Thus, not only biological high-dimensional space can be found in low-dimensional space, clever ants can be found bent sphere, and ultimately establish a perfect spherical geometry, its depth of knowledge not much worse than a bee.

critical

spherical geometry is important in shaping the critical real projection plane, the opposite is obtained antipodes on the sphere by identifying (opposed points on both lateral sides of disaggregated ). On the ground, the projection plane has all the properties of spherical geometry, but with a different overall characteristics, in particular, he is no orientation.

use

spherical geometry in marine science and astronomy are practical and important uses.

Development

spherical trigonometry is part of a spherical geometry, mainly in the processing of finding and interpreting a polygonal (especially triangular) spherical surface angle contact and associated with the edges. On the importance of astronomy for celestial navigation in calculating celestial orbit and the Earth's surface and space travel.