Cartesian coordinates
are good for describing
Euclidean space (AKA flat space)---the
space
of ordinary geometry: i.e.,
Euclidean geometry.
But we do NOT usually study them mathematically, because that is hard.
Harder are 3
curved spaces because we
have NO intuitive understanding.
We do NOT notice the curvature, because it is so small on on our scale (except
near strong gravity sources
like black holes).
We are microbes
living on beach ball.
To us, the world is an
infinite flat
plane.
Features:
Airways
for aviation often follow
great circles on
Earth at least approximately since that
shortens travel distance and travel time.
This is why flightsNew York City
For more on flights
on great circles,
see the figure below
(local link /
general link: great_circle_path.html).
If the figure is absent, it can be seen at
Mathematics file:
great_circle_path.html.
Consider the sphere in
Image 2.
Note that the equator intersects two meridians both at
90°, but they are NOT
parallel elsewhere
and, in fact, meet at the
poles.
In Euclidean geometry,
of course,
geodesics
(i.e., straight lines)
that are
parallel at one place
are
parallel everywhere
and NEVER meet.
This is the same as saying that anywhere along them there is a
third geodesic that intersects
them both at 90° and has the same length.
By inspection
of Image 2, it is clear that the
sum of the
vertex
angles
for a triangle
in spherical geometry
is greater than 180°.
php require("/home/jeffery/public_html/astro/mathematics/great_circle_path.html");?>
Features:
EOF
php require("/home/jeffery/public_html/astro/relativity/curved_space_videos.html");?>
Images:
Local file: local link: space_curved.html.
Image link: Itself.
Image link: Itself.
Image link: Wikipedia:
File:Hyperbolic triangle.svg.
File: Mathematics file:
space_curved.html.