Questions About A Pelican Introduction: Our Universe: An Astronomer's Guide

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I need an expert in spherical geometry. On page 218 Jo Dunkley says, "On a surface curved like an orange, those two same parallel lines will at some point come together." I presume she is talking about lines of longitude because it is possible to draw parallel lines on a sphere which never meet: eg two circles of different radii to the side of the centre line. Longitude is not parallel except points at the equator?

Gabriel Conybeare Marsh The first two parallel lines referred to are on a flat plane. Imagine viewing the plane side on, as if you were looking at the edge of the piece of a …moreThe first two parallel lines referred to are on a flat plane. Imagine viewing the plane side on, as if you were looking at the edge of the piece of a piece of paper with the lines running front to back. Place one line on the oranges equator, then bend the plane to match the curve of the orange and finally wrap each line around the orange. The bend of the plane produces the angle which forces the lines intersect.

You can also consider the alternate scenario. Instead start with parallel lines on the orange like lines of latitude with one being the equator. View the orange in the same orientation as before, with the lines of latitude going in front of you and up and down around the orange. Let's try and get these lines onto a plane resting on top of the orange. The equator can be easily unrolled into a straight line, but the other line of latitude needs to be rotated through space to touch the plane. Once rotated, we can now unroll it, but the angle from the rotation results in a curved line mapped onto our plane. This is because the rotation brought some points closer to the plane than was the case when we unrolled the equator.

Roughly speaking this happens for the same reason you can't represent the earth properly with a flat map.(less)

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