“In spherical geometry, there are no parallel lines”
Read that in a book that’s not about navigation. That might be a good subject for a question on a CG license exam. First I thought that’s right, then no, it’s not…etc.
1 Like
I think you will learn more about how mathematical reasoning works by studying Non-Euclidean Geometry than any other topic. I’m talking about how a change in axioms propagates through lemmas and theorems. I was lucky enough to have a math prof who believed this so much he had a textbook reprinted to teach it. And of course there’s "I have a friend in Minsk … who has a friend in Pinsk … "
Cheers,
Earl
5 Likes
“In spherical geometry, there are no parallel lines”
If two different great circles are drawn on a sphere they are going to cross somewhere. Lines of longitude for example all cross at the poles.
What about the parallels or parallels of latitude? They are parallel but don’t meet the definition of a line as they are not the shortest path on the surface With the exception of the equator the parallels are circles, not lines.
The shortest path on the surface of a sphere is called a geodesic. which is a segment of a great circle. Couldn’t remember at first where I’d heard that term before, just now recalled Bucky’s geodesic domes.
1 Like
About 20 years ago I programmed a high-precision tool for calculating distances on the Earth’s surface (millimeter range).
Knowing that I probably wouldn’t remember the basics of these calculations a few years later, I wrote an explanation of the code >>>
5 Likes
As I progressed in my education and career it became apparent my life would have been much easier if my first required geometry course in the 9th grade was Non-Euclidean Geometry instead of ‘plane geometry’.
1 Like
Depends on which definition you use.
“a continuous extent of length, straight or curved, without breadth or thickness.”
-or-
“a circle of the terrestrial or celestial sphere”
Regardless you can have parallel lesser circles (lines of latitude or really any full circle that doesn’t pass through the center of the sphere) that are “lines.”
Back in the day of the Magnavox Transit satellite navigator it only calculated Great Circle routes between way points.
To facilitate voyage planning for a voyage from New Zealand to Panama or West Coast USA I used a HP programmable calculator to calculate the intermediate points to ensure we missed the many shoals and islands in the way.
This is true. The text used in this case the definition is the shortest path, a line on a plane, a geodesic on a sphere.
Isn’t a line of position (LOP) in celestial navigation actually a segment of a small circle? The center of the circle being the geographical position (GP) of the body?
The meaning of terms often varies with context. In technical matters, mathematics for example, precise definitions are often provided in the textbook.
Here’s the Wolfram’s entry on spherical geometry.
The study of figures on the surface of a sphere (such as the spherical triangle and spherical polygon), as opposed to the type of geometry studied in plane geometry or solid geometry. In spherical geometry, straight lines are great circles, so any two lines meet in two points. There are also no parallel lines. The angle between two lines in spherical geometry is the angle between the planes of the corresponding great circles, and a spherical triangle is defined by its three angles. There is no concept of similar triangles in spherical geometry.
1 Like
Correct
1 Like
It becomes very apparent when taking a meridian altitude when declination and latitude are very close. An observation 4 minutes before and after, plotting the GP of the sun and using the zenith distance as a radius a position can be obtained with more accuracy than the actual observation.
1 Like
That’s an interesting observation. Plane geometry seems based upon every day common sense, only to have the foundation overturned in the next chapter of the text book,
I had no idea when I started this thread.
Has to do with Euclid’s 5th postulate.
From here: Gödel, Escher, Bach: An Eternal Golden Braid
This is Farkas Bolyai telling his son János Bolyai not to go down the parallel line rabbit hole. Written in 1820.
You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of parallels alone….I thought I would sacrifice myself for the sake of the truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction. For here it is true that si paullum a summo discessit, vergit ad imum [if it’s failed to make the grade, even by a smidgeon, it might as well be the worst]. I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind…I have traveled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin nof my disposition and my fall date back to this time. I thoughtlessly risked my life and happiness—aut Caesar aut nihil [either Caesar or nothing].
3 Likes
I do not understand 
More neat sphere stuff:
Cheers,
Earl
What if you drill two parallel holes through the middle of the sphere?
Seems impossible, that only one can go through the middle, the other goes infinitesimally close to, but not through the middle.
1 Like
You are right, I misspoke when I said middle. Two parallel tunnels through the interior of the sphere.
It looks like that police cruiser is blocking the lane of traffic, I guess it depends on perspective.