Question

For each property listed from plane Euclidean geometry, write a corresponding statement for non-Euclidean spherical geometry.
A straight line is infinite.

A great circle is finite.
A great circle is infinite.
A straight line is finite.
A straight line is a line segment.

Answer 1

nice question. . this was not mention by our prof. .

Answer 2

http://www.cs.unm.edu/~joel/NonEuclid/noneuclidean.html

Answer 3

there is no such term like line in non-euclidean geometry. .

therefor, i suspect, figures present here are curves. .

Answer 4

hehe Im so lost :[

Answer 5

you're not alone. . .lol

Answer 6

haha This is not even one of thoughs Brain busters i post later at night lol

Answer 7

Im just asking all the problems i had a hard time with to better understand it :|

Answer 8

given that the figures in non-euclidean geometry are curves or perhaps circles, and this figures are infinite in this field, therefor i could speculate that the second option is correct. .

just an assumption,. (i'm good at dreaming and assuming) lol. .

Answer 9

there is quote of an anonymous mathematician ; just wanted to share. .

"if in euclidean geometry, parallel lines has no way to intersect each other, in non-euclidean geometry there exist" . . .

Answer 10

A great circle in spherical Geometry is the counterpart of a line in Euclidean Geometry. While a line is infinite in extent, a great circle is not. Yet, the great circle contains infinitely many points. The tricky part is the change of definition of the terms.Jerwyn and I will check the properties of "great circles" in spherical Geometry and select what we think is the correct answer. At the moment, I'm thinking that the answer is "A great circle is finite" but that is not my final answer. Hold on a sec while we think.

Answer 11

Here's some background information on spherical Geometry.

Riemannian Geometry (also called elliptic geometry or spherical geometry): A non-Euclidean geometry using as its parallel postulate any statement equivalent to the following:

If l is any line and P is any point not on l , then there are no lines through P
that are parallel to l .


Riemannian Geometry is named for the German mathematician, Bernhard Riemann, who in 1889 rediscovered the work of Girolamo Saccheri (Italian) showing certain flaws in Euclidean Geometry.



Riemannian Geometry is the study of curved surfaces. Consider what would happen if instead of working on the Euclidean flat piece of paper, you work on a curved surface, such as a sphere. The study of Riemannian Geometry has a direct connection to our daily existence since we live on a curved surface called planet Earth.



What effect does working on a sphere, or a curved space, have on what we think of as geometrical truths?

In curved space, the sum of the angles of any triangle is now always greater than 180°.

On a sphere, there are no straight lines. As soon as you start to draw a straight line, it curves on the sphere.


In curved space, the shortest distance between any two points (called a geodesic) is not unique. For example, there are many geodesics between the north and south poles of the Earth (lines of longitude) that are not parallel since they intersect at the poles.


In curved space, the concept of perpendicular to a line can be illustrated as seen in the picture at the right.

http://regentsprep.org/Regents/math/geometry/GG1/Euclidean.htm

Answer 12

lol, this kind of problem will result a subjective answer,. .

i hope a great mathematician is around. .

Answer 13

this is a theoretical question. .

Answer 14

if someone could precisely answer this,i condemn his/her is the greatest mathematician of this generation. . lol

Answer 15

of course with proofs. .

Answer 16

1.3 Spherical Geometry:
Spherical geometry is a plane geometry on the surface of a sphere. In a plane geometry, the basic concepts are points and lines. In spherical geometry, points are defined in the usual way, but lines are defined such that the shortest distance between two points lies along them. Therefore, lines in spherical geometry are great circles. A great circle is the largest circle that can be drawn on a sphere. The longitude lines and the equator are great circles of the Earth. Latitude lines, except for the equator, are not great circles. Great circles are lines that divide a sphere into two equal hemispheres.

Spherical geometry is used by pilots and ship captains as they navigate around the globe. Working in spherical geometry has some non-intuitive results. For example, did you know that the shortest flying distance from Florida to the Philippine Islands is a path across Alaska? The Philippines are south of Florida - why is flying north to Alaska a short-cut? The answer is that Florida, Alaska, and the Philippines are collinear locations in spherical geometry (they lie on a great circle). Another odd property of spherical geometry is that the sum of the angles of a triangle is always greater then 180°. Small triangles, like those drawn on a football field, have very, very close to 180°. Big triangles, however, (like the triangle with veracities: New York, L.A. and Tampa) have significantly more than 180°.

http://www.cs.unm.edu/~joel/NonEuclid/NonEuclid.html

Answer 17

lol oh noz @Hero we need you :P

Answer 18

Just listen to directrix

Answer 19

But directrix, use your own intuition to come up with results rather than consult outside sources.

Answer 20

@hero, he just wanted to make his statements reliable. .

Answer 21

But their not "his". If he posts them from other sources, he's supposed to cite those sources. Otherwise it can be viewed as plaigarism :P

Sorry, just got done taking a writing course so....

Answer 22

they're*

Answer 23

http://www.cs.unm.edu/~joel/NonEuclid/NonEuclid.html

curves are present in this subject not line. even triangle uses curves,.
read my link. .

Answer 24

lol, parallel lines in no-euclidean geometry intersects. .

Answer 25

@Directrix , what do you think is the answer base on the options?

Answer 26

There's no definition of "finite" that I see in spherical Geometry. Because distances in spherical Geometry are measured along a great circle of the sphere, and the circle is, of course, confined to the sphere, I will go with A) for my final answer.

Answer 27

B) is my second choice but I considered that though a circle is comprised of infinitely many points, it's "length" as in circumference is finite.

Answer 28

I agree with that

Answer 29

uhhm,. i wanted to go with B, but my assumption above has no enough proof.


the given states that a line is infinite, which is true considering that a line has two arrows which signifies infinity,.

for sphere, it is a bounded figure, therefor we can't say it's infinite, although points are infinite, but there is no proof that it's totally infinite, so i suspect it's points are still countable. .

Answer 30

wow This is Interesting