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Think of the Earth’s circles of latitude. They are all parallel, but never cross. The circles of longitude, on the other hand, each cross all the other circles of longitude exactly twice: once at each pole.
There are no parallel lines on earth’s surface, because all lines cross. All lines are geodesics. ‘Lines’ of latitude except the equator are not actually lines, because they are not the shortest distance between two points.
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It ignores skewed lines and lines that are the same.
Is it correct to say that if your parallel lines are on a sphere they meet twice?
No. If they are parallel, they never meet.
Think of the Earth’s circles of latitude. They are all parallel, but never cross. The circles of longitude, on the other hand, each cross all the other circles of longitude exactly twice: once at each pole.
There are no parallel lines on earth’s surface, because all lines cross. All lines are geodesics. ‘Lines’ of latitude except the equator are not actually lines, because they are not the shortest distance between two points.
All lines, *on earth, are geodesics.
Please credit Cowbirds in Love for this image.
This is why I choose a projective or spherical geometry.