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Hallo
A few questions please
1. If there are no parallel lines in spherical geometry, why are latitudes called parallels.  Would not two lines of latitude equidistant from the equator to north and south be parallel? Even if they are circles.

2. Does Spherical geometry belong to the second or third dimension?  What dimensions does euclidean geometry belong to and how do they differentiate in terms of dimensions

3. A line (first dimension) can curve, which would make it from a second dimensional plane. What dimension does the curve which introduces spherical geometry belong to, and how does this curve relate to time.

4.What would the angle be between lines of longitude (hourly meridians) at the pole
Thank you

Jeremy

Answer
Hi Jeremy,

1. If there are no parallel lines in spherical geometry, why are latitudes called parallels.  Would not two lines of latitude equidistant from the equator to north and south be parallel? Even if they are circles.

"Line" takes on a special definition in spherical geometry.  Every line in spherical geometry is part of a great circle.  (A great circle is a circle with the same diameter as the full sphere.)  So although latitudes are parallel in the sense of plane geometry, latitudes--except for the equator--are not even lines in spherical geometry.

2. Does Spherical geometry belong to the second or third dimension?  What dimensions does euclidean geometry belong to and how do they differentiate in terms of dimensions

Euclidean geometry is in 2 or 3 dimensions.  Spherical geometry is in 2 because the lines live on the surface of the sphere, a 2-dimensional object.  Note that higher-dimensional spherical geometries do exist, however.

3. A line (first dimension) can curve, which would make it from a second dimensional plane. What dimension does the curve which introduces spherical geometry belong to, and how does this curve relate to time.

Be careful.  A line is a one-dimensional object.  If the line curves, it may belong to a plane.  (It may also not; imagine tying shoelaces.)  In the case of spherical geometry, all the lines belong to a great circle, which lie in a plane.  Speaking strictly geometrically, the curves bear no relation to time.

4.What would the angle be between lines of longitude (hourly meridians) at the pole
All angles in spherical geometry are measured between great circles.  So, consider the two meridians in question and measure the angle at their intersection.  You could instead consider the planes that contain the great circles and measure the angle of their intersection.


Thanks for asking,
Azeem

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Azeem Hussain

Expertise

I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.

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4 years as a drop-in and by-appointment tutor at Champlain College. Private tutor for dozens of clients over the past 8 years.

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CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

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Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.

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