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Sir Col
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Parallel Lines
« on: Oct 20th, 2003, 4:06pm » |
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This is a genuine question with regards parallel lines.
In the first instance, how do we define parallel lines on a uniform flat surface? Same direction, same distance apart, never meeting in a finite distance?
Secondly, how do we define parallel lines on the surface of a sphere?
Imagine two people, 50 m apart, and both standing on the equator. They both travel north, which is perpendicular to the equator, yet they meet at the pole. So either, travelling in the same direction is not parallel, or parallel lines meet in a finite distance; which is it?
Instead, imagine one person travelling from the equator, through the North pole, continuing south, through the South pole, and back to their starting point on the equator. If the second person, starting 50 m away on the equator, travels in such a way that the distance between them and the first person is maintained at 50 m, what path do they follow? What direction do they travel in relation to the equator?
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Icarus
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Re: Parallel Lines
« Reply #1 on: Oct 20th, 2003, 5:15pm » |
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In Euclidean geometry, parallel lines are defined to be lines in the same plane which never meet. They also have the property that the distance between them never changes.
In Hyperbolic geometry, there are infinitely many lines in the same plane as a given line and passing through a given point which do not intersect the given line. Defining all of them to be parallel to the given line is not a useful definition. Instead two rays are defined to be parallel if (1) they never meet, and (2) there is no ray that lies entirely between them (any ray in the middle must eventually intersect one or the other). If parallel rays are extended in the opposite direction, the opposite rays are not parallel. Lines are said to be parallel if they are parallel in one direction.
In spherical geometry, or projective geometry (spherical geometry in which each point is identified with its antipode), there is no such thing as parallel lines. All lines intersect.
In your second example, the first person is following the spherical equivalent of a line - a great circle on the sphere. The second person is traversing another circle concentric with the first, but not a great circle.
More generally, you cannot define a concept of parallel geodesics (lines) on manifolds (other geometries). You can define a concept of "parallel translation" along curves. That is, given a vector at one end of a curve, you can define vector valued function on the curve which represents the parallel translation of the initial vector along it. How this translation is defined determines the geometric properties of your manifold. Usually, if you parallel translate a vector around a closed curve, the resulting vector will be different from the original. The limit of this change as the curve shrinks to nothing determines the curvature of the manifold.
Try it on the sphere: Parallel translation along a great circle maintains a constant angle between the vector being translated and the tangent vector to the sphere (both vectors always being tangent to the sphere). Calculate the deviation created by parallel translating a vector around a spherical triangle. What happens if you let the size of the triangle go to zero?
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