# Is the small circle the straight line on the sphere?

What are straight lines? On the plane, the straight lines are the locus of the point where the direction of motion does not change.

On the sphere, we can regard any given circle as the circle of latitude (the equator is a special circle of latitude). The point along the circle of latitude movement, is the east-west direction of movement, that is, the movement does not change direction. So circles on the sphere are straight lines . Great circles are straight lines, and small are straight lines.

On the definition of the direction on the sphere:East-West is well defined, and the vertical to the South-North is the East-West.

direction

The direction is determined at three points in the plane, and the direction is determined on the surface of the sphere. As shown in the figure, The ab line is the tangent of the surface of the sphere., tangent point is a point, ab only a point on surface of the sphere, all other points are away from the surface of sphere, not on the surface of sphere. The direction on the surface of the sphere must be on the surface of the sphere, so that ab can not represent the direction of the sphere. As a matter of fact, ab is tangent to the red, yellow and blue circles. In theory, ab is tangent to an infinite number of circles, so, of course, ab can't represent the direction of the sphere. In fact, each circle represents a direction on the surface of the sphere. So, the surface of the sphere is directed by the circle. And the circle is determined by three. So, three points on the surface of the sphere determine a direction.

Spherical Angle:

A spherical angle is a particular dihedral angle; it is the angle between two intersecting arcs of circles on a sphere. It measured by the angle between the planes containing the arcs .

Spherical coordinate system:

As shown in the figure below, the horizontal lines are the latitude lines, the different latitude lines are parallel to each other, the vertical lines are the longitude lines, and the different longitude lines are parallel to each other. The longitude lines and the latitude lines are perpendicular to each other. The direction is defined in this spherical coordinate system. In this coordinate system, we can define what direction to move. For example, we can move from any given point on the latitude line and move in any direction.

We will find that the so-called straight lines on the sphere are the circles on the sphere, Because the directions of the circles are constant.

Tangent on a sphere

The tangent of the circle on the plane is a straight line on the plane, and the tangent of the circle on the surface of the sphere is the circle itself. For example, A is a circle on the surface of a sphere, then the tangent of the A circle is the circle of A itself.The tangent of the big circle is the big circle itself, the tangent of the small circle is the small circle itself. Transitivity

On the sphere, if the two lines are parallel to the third lines, the two lines are parallel to each other. Figure, the yellow line, the red line is parallel to the blue line, so the yellow line and red line parallel to each other. Angles of a transversal

On the sphere, if the two lines are parallel, consecutive interior angles are supplementary, corresponding angles are equal, and alternate angles are equal.

Angles of a transversal diagram

Rhumb line

The great circle is a straight line, and the direction is constant. The angle of the intersection of a great circle and a longitude line is obviously different. The angle between the longitude line and the Rhumb line is constant, so Rhumb line is not a straight line. The direction of Rhumb line is not constant. It also shows that in the sphere, a direction is determined from three points, and two points on the sphere can not determine the direction.

distance

The distance between the sphere and the plane is different. If we want to know the distance between two points on the sphere, we must know what direction the distance is, the distance from the direction of the great circle, or the distance from the direction of the small circle. In fact, there are infinitely many distances between two points on a sphere.

yellow line represents South to north

The direction in the sphere is determined by three points, not at two points. So the blue line represents the direction of the East and West, and the other lines do not represent the East - west direction. yellow line represents South to north, and the other lines are not South - North.

yellow line is South to north diagram

the parallel postulate

In a sphere, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.

The distance between two points

There are a lot of distances between the given two points on the sphere. This is different from the plane. For example, the distance between two points of AB can be the distance along the red line and the distance along the blue line, or the distance along the direction of the great circle, and so on. There are countless distances between two points on the sphere. Suppose B is the north pole, because in fact, the compass points to the yellow line from A to B, and the yellow line is the common chord of all the arcs from A to B. So even if we use the compass, we can also follow the red straight line from A to B, or along the straight blue straight line from A to B, or from other straight lines from A to B, not necessarily along the great circle from A to B.

The distance between two points diagram

Theorem:

On the plane, two points determine a straight line, and on the sphere, the three point determines a straight line.

What direction does the blue line represent?

Suppose that the blue line is tangent to the sphere, and the tangent point is a point. The yellow circle (horizontal circle) intersects with the red circle (vertical circle) at a point. If the yellow circle represents east to west, the red line represents South North. What direction does the blue line represent?

diagram of the direction of the blue line

So we have to be careful to avoid a simple understanding of the concepts on the sphere. Nor can we simply copy the concept on the plane.

• Comments are not for extended discussion; this conversation has been moved to chat. – Alexander Gruber Nov 26 '17 at 9:54
• @fleablood There are important supplements to the problem. – enbin zheng Nov 27 '17 at 5:37
• @MvG There are important supplements to the problem. – enbin zheng Nov 27 '17 at 5:38
• As was pointed out five days ago, a curve maintaining a constant cardinal direction is i) inconsistant and ii) insignificant and iii) does not contain any of the characteristics generally implied by the term "straight line". Maintaining a constant cardinal direction is not a relevant nor important characteristic of a curve, whereas spherical geodesics are. Hence it is universally agreed upon that the great circles best maintain the relevent characteristics of a line. (symetric and universal, and optimal geodesics [which small circles fail on all accounts]). – fleablood Nov 27 '17 at 6:57
• The only thing you have done has been to declare that you've decided to change the definition of "straight" and "line" from the conventional and useful usage of geodesic great circles, to the capricious and useless one of small circles. As I said in the very first post, we could define "straight lines" to be "panda bears" and feed our straight lies bamboo. The fact that some, but not most, small circles keep a constant but arbitrary value of direction and some, but not most, of the curves that do so are small circles (most are loxodromes) is just not significant, useful nor interesting. – fleablood Nov 28 '17 at 16:46

While my original answer should still serve to answer your original question, the updated question gives more insight into where you went wrong, and thus allows me to write an answer to correct this.

The point along the circle of latitude movement, is the east-west direction of movement, that is, the movement does not change direction.

Suppose you are standing at a latitude of 45° north, and facing due east. Now you take a step. Try to picture this process of taking a step as follows: at the point where you are standing, there is a tangent plane. It touches the sphere at your initial position only. On that plane you have a clear north-south axis with north pointing towards the north pole and south pointing towards the south pole. Orthogonal to that you have an east-west axis. So you can pick a point one step east on that tangent plane.

But you don't want to keep walking on the tangent plane, you want to walk on the sphere. So you lower your leg further, through the tangent plane and down to the surface of the sphere. But as the tangent plane makes a 45° angle with the axis of the earth, moving your foot through the plane down towards the center of the earth means you put it down a bit further south than where you started. Which also means you end up facing a tiny bit to the south. If you continue taking steps in this fashion, without correcting for the change in heading, you eventually end up in the point antipodal to where you started, after completing half a great circle.

If you want to stay at constant latitude, you need to move your second foot a bit further left, or rotate a bit left after each (infinitesimally small) step. So in effect you're walking a curve.

And keep in mind that constant latitude is just a special case of constant compass heading, which I'd consider a more precise term for what you call direction. For constant compass heading due north or south you get a meridian, a great circle. For due east or west you get a parallel, a small circle except at the equator. But for other headings, you don't get a circle at all, but instead a rhumb line or loxodrome. Would you call this a “straight line” as well?

I just found https://www.mapthematics.com/Essentials.php and I think the sections from Great circle through Direction should be useful to you. In particular, the section on Direction covers the misconception you seem to have here as well.

As the term “direction” is kind of central to many of the statements in the various comments here, let's have a closer look at that. One precise mathematical concept that can be assicuated with that term is the concept of a tangent vector. It points in a given direction from the current position. In the case of the sphere, it's an element from the tangent plane. It can be used to take an infinitely small step into that direction. After taking a step, the tangent plane changes, and thus the original vector is not a valid tangent vector any more. Most everyday uses of a “direction” allow using the same name to define a direction in different locations. So you get a “north” direction in almost any place of the earth. In this sense, a direction would be best described as a map from locations to tangent vectors. Many directions are undefined in some singular locations, e.g. at the poles. So it's in fact a partial map. In this sense, valid directions are “towards New York”, “parallel to the nearest coastline, in clockwise direction” or “orthogonal to a line from here to San Francisco”. This concept of direction has no obvious connection to straight lines. If you want to demonstrate that a certain definition of direction leads to straight lines, more work is needed.

You can also think about spherical geometry in a different way: instead of the earth, think of a perfect metal ball: smooth and round, and very symmetric. It has no poles. So suppose you are an ant standing on such a sphere, and facing a given direction. Start walking. What curve do you make? If you believe that the curve is fully specified by the initial position and orientation, then it has to be a great circle for the sake of symmetry. The moment you claim it's forming a small circle, it would separate the surface of the sphere into a larger and a smaller part, which would break the symmetry of the initial setup. Or you could believe that the initial position and orientation are insufficient to uniquely define the curve. There might be several different ways of walking “straight ahead”. But if that were the case, then the concept of walking straight is of little value.

• "But for other headings, you don't get a circle at all, but instead a loxodrome or rhumb line. Would you call those “straight lines” as well?" Dang. I totally forgot about rhumb lines. And I'm a sailor. – fleablood Nov 22 '17 at 0:24
• @fleablood: This confusion between great circles and rhumb lines when it comes to directions reminds me of an answer I posted a few years ago. I doubt I could convince the OP back then, but perhaps I could help other visitors. – MvG Nov 22 '17 at 10:52
• @MvG Because the East West is always perpendicular to the south north, so the east west direction of the second step has changed, the tangent plane is different, so your analysis is wrong, and I'm right. – enbin zheng Nov 23 '17 at 13:37
• @enbinzheng: I find a statement of “your analysis is wrong, and I'm right” to be rude and an indication of unwillingness to learn. I don't follow the argument that led you to that statement, but seeing how several people are spending considerable effort teaching you, a refusal to learn is rather ill received, on my part. If you were to say ”I don't understand that argument, the way I see it things look different” and explain that, I might continue the discussion. After that comment just now, I feel any more discussion will only be a waste of my time. Please think about your behavior. – MvG Nov 23 '17 at 17:19
• Everyone has his own point of view — ...and some of them are wrong. – JeffE Dec 5 '17 at 19:32

If you move on a sphere and always go straight ahead, following the curvature but never veering left or right, then the course you follow is a great circle. For better illustration: if you drive in a car on that sphere, you hold the steering wheel in center position at all times and you end up with a great circle.

In order to follow a small circle, you'd need to diverge from the straight ahead course, and keep that deviation constant. In the car analogy, you'd be holding the steering wheel at the same position at all times, but that position would not be the center position.

So to use the vocabulary from your question, I'd say that the great circles on the sphere correspond to straight lines, while all other circles correspond to circles in the plane.

• Comments are not for extended discussion; this conversation has been moved to chat. – Alexander Gruber Nov 26 '17 at 9:55
• @fleablood You've been asking yourself, "how did you determine the east west direction on earth?" Why don't you mistake the east west direction on the earth? Why do you fly around the earth without making a mistake in the east west direction? Why is that? – enbin zheng Nov 26 '17 at 10:46
• @MvG My question adds to the direction and other aspects. – enbin zheng Nov 28 '17 at 15:33

Your comment "Don't turn around in small circles" is interesting, and I believe there is a way to explain geodesics by taking that comment into account.

Since your question is formulated in an intuitive fashion, I will try to write an intuitive description of how to understand geodesics on surfaces. But before I start, I'll say that this intuitive discussion can be made rigorous, in the field of mathematics known as differential geometry. If you are interested in knowing the more rigorous mathematical concepts underlying this answer, that's what you should study.

Consider a curve on a smooth surface --- perhaps a sphere, or a plane, or a surface with many varying undulations. Suppose you want to test whether that curve is a geodesic. Here's how that test goes, from an intuitive point of view:

1. Imagine walking upright along the curve, always forcing yourself to face in the direction tangent to the curve.
2. Ask yourself: As you do this, do you have to gradually turn for some portion of your walk, in order to keep facing in the direction tangent to the curve?

If the answer is "Yes" then that curve is not a geodesic. For instance, if you walk counterclockwise around a small circle on the sphere, and if you force yourself to face in the direction tangent to that circle, then at all times you must constantly turn left left left left. If you don't make that effort to turn left, then you'll start facing in a direction that is not tangent to that small circle. So, that curve is not a geodesic. Dancers understand this: a dancer who walks in a small circle, facing tangent to that circle, has to constantly twist their feet, forcing themselves to turn in order to keep facing tangent to that circle. Although the dancer could walk around the circle without twisting their feet, that would violate the requirement that they always face in the direction tangent to the circle: sometimes they walk forward facing in the direction of the circle; sometimes they walk diagonally facing at an angle away from the circle; sometimes they walk sideways facing at a right angle to the circle.

If on the other hand the answer is "No. Without any turning effort whatsoever, I will always continue facing in the direction tangent to the curve", then that curve is a geodesic.

• Thank you for explaining that in detail. I want to ask you a question: The straight lines on the planes are the lines where the direction does not change. Do you recognize the fact? – enbin zheng Nov 18 '17 at 15:31
• Your statement "The straight lines on the planes are the lines where the direction does not change" is vague. Without a more rigorous formulation, your question is unanswerable. Using differential geometry, one formulation might be this: "If $f : [a,b] \to \mathbb{R}^2$ is a geodesic parameterizing a line segment, if $p=f(a)$ and $q=f(b)$, and if $df/dt(a) \in T_p\mathbb{R}^2$ and $df/dt(b) \in T_q\mathbb{R}^2$ are the tangent vectors, then parallel translation along $f$ from $a$ to $b$ takes $df/dt(a)$ to $df/dt(b)$." If that is what you mean, then yes, I recognize this fact. – Lee Mosher Nov 18 '17 at 17:16
• Actually, it's very simple. On the plane, the derivative of a straight line is constant, so the direction of the straight line is constant. Do you agree? – enbin zheng Nov 20 '17 at 13:03
• "From this consensus, we can deduce the direction on the sphere" no we can not. There are two ways of defining the direction of a line in a plane. Either through an internal orientation of the points to its immediate neighbors. Or as a global orientation to an universal frame of reference. On the plane these two "directions" are consistent and compatible. On a sphere, they are not. – fleablood Nov 23 '17 at 18:46
• "So how do people move in the direction of the sphere?" I can't tell if you are being deliberately disingenuous for the sake of being argumentative or if you are truly genuinely stupid. No, there is no and can not be any global universal direction on a spere. What is left or right or up or down or North or west will be the exact opposite for a person on the opposite side of a globe. When people refer to direction the mean a local direction in relation to their current position. Paths and curves based on them, such as a latitude or longitude line or a rhumb line are not straight. – fleablood Nov 26 '17 at 3:35

The direction is defined in this spherical coordinate system. In this coordinate system, we can define what direction to move.

No, you can not. Notice that all four of the colored circles below all have direction $0$, West, by your definition of direction. What is the purple direction? Or the direction of the other tangent circles? How can a person on your globe tell the difference between any of these cotangent directions as the will and read the exact same ($90^{\circ}$) the compass?

How can you tell the difference between directions when at any point there are an infinite number of tangent circles. • Comments are not for extended discussion; this conversation has been moved to chat. – Daniel Fischer Dec 1 '17 at 11:36
• A special description of the distance between two points is added at the bottom of the top floor. – enbin zheng Dec 1 '17 at 14:12
• i never asked about distance. I asked about direction. All of thosecircle (except the purple one) are tangent with the 40th latitude. So if you had a conventional compass (your definition of direction is not conventionl) it would read WEST for all of the circles. You claim the yellow one is North-South (but don't explain why), that the blue latitude line is East-West (but don't explain why), That the green and red are not East-west (but don't answer what directions they are) and you don't have any direction for the upper blue and green circles. – fleablood Dec 1 '17 at 16:48
• And you still haven't explain how the purple and yellow likes can both be the same direction at the same point. And if you parrot the nonsensical "three points determine a direction on a sphere" I'll scream. But then I'll point out that your ENTIRE premise we that a circle is a path of A point where the direction does not change. You've just develope a system where a single point (the intersection of yellow purple circle) can not have a single SOUTH direction, both yellow and purple are south. If the point need another point to have direction thn you can't trace dir of a single pt. – fleablood Dec 1 '17 at 16:54
• 87% of what you write is incomprehensible; 5% is inconsistant, 3% is wrong, and the remaining 2% is the trivial statement. Three points determine a unique circle on a sphere. – fleablood Dec 1 '17 at 16:56

We need definitions:

On a plane we have: an x-axis. On a sphere we have an equator.

On a plane we have: a y-axis. On the sphere we have a prime meridian.

At a point we have an angular direction: On a plane it is measured by a line through the point that is parralell to x-axis, and measuring an angle relative to it. Note there are an infinite number of paths with this direction. Only one of them is a line. The line will be so that every point will have the same direction. The line is the only path that is called straight. That line would be straight regardless as to how we set the x and y axes (which are ultimately arbitrary) and it would keep the direction constant regardless as to how we set the x and y axes.

On a sphere it is measured by a circle through the point that is parallel to the equator, If the point is not on the equator this is a small circle of latitude, and measuring an angle relative to the point. Note there are an infinite number of paths with this direction. Only one of them is a great circle. Only one of them the paths will keep the direction constant.

BUT THE PATH THAT KEEPS THE DIRECTION CONSTANT AND THE GREAT CIRCLE ARE NOT THE SAME; THE PATH THAT KEEPS THE DIRECTION CONSTANT IS ARBITRARILY DECIDED BY THE CHOICE OF EQUATOR; AND NATURE OF THE PATH WILL VARY DRASTICALLY BASED UPON THE CHOICE OF DIRECTION-- SOMETIMES IT WILL BE A GREAT CIRCLE, SOME TIMES A SMALL CIRCLE BUT MOST OF THE TIME IT WILL BE A LONG SPIRALLING LOXODROME.

The one that will keep the direction constant will be the great circle only if the direction is $90$ or $270$ or if point is on the equator and the direction is $0$ or $180$. Other wise, if the direction is $0$ or $180$ the path that will keep direction constant is the small circle of latitude. If the direction is not $0,90,180,270$ and the point is not on the equator than the path that keeps the direction constant would not be a small nor a great circle but a rhumb line. This is sort of a long helix.

NONE of the paths that keep the direction constant, will keep the direction constant with different chooses of equator (which is ultimately arbitrary.)

So.. You CAN define a "straight line" to be "keep the direction straight" but this will be entirely dependent upon the choices of equator and will not have any bearing on the nature of the path which could be great circle, or small circle or rhumb line.

And this will have nothing to do with the shortest distance on a geodesic.

This is all in all a very poor, or at best, minimally useful definition of "straight".

At the heart is the problem with the direction of "angular direction". That simply is not useful to apply globally on a sphere. It's very useful locally but globally it simply is not a "constant straight direction".

If you want to define a direction so that "straight lines do not change directions" you will need another definition. It can be done, but under the stronger definition, latititudes and rhumb lines do change directions and only great circles will not.

===== Huge unfinished essay and original answer below... I prefer the above as a final answer but the below may (or may not) have some useful expositions... but it's unfinished and uneditted.... ======

On a plane you have the following objects and concepts:

1) A line that is straight.

2) The concept of direction

3) A proposition that: A travelling point travels in a straight line if and only if its direction does not change.

You accept these without definition and/or proof and figure, if these apply to the plane they will apply to the surface of the sphere.

They do not.

First of all, we have to define each of these terms to the sphere.

I'm going to take of my mathematician "everything is abstract and a matter of how we define it" hat, and put my man on the street "things are what the are hat" to point out that the very first time a student comes across the idea of "lines in spherical geometry", their very first reaction SHOULD be "there are no lines in spherical geometry; nothing is straight so there are no lines-- if you draw a line between two points it would puncture the sphere and go through the interior; that is not on the sphere".

And that is the correct response. Any student who did not react that way is either an idiot or a sycophant.

But this is where I put my mathematician hat back on. A mathematician must ask "Well, than what is a line and what is straight?" (It's in my contract; question everything and define everything.) And there is no possible objective answer to that. After millenia of discourse it was determined that "line" and "straight" can only be defined axiomatically.

And there it is determined that whatever axioms we use do define a straight line in a plane can be used to axiomatically define a straight line on the sphere. Basically and loosely if a line is the shortest distance between to point, then applying geodesics the equivalent on a sphere are the great circles.

Now there are immediate difference that arise between the plane and the sphere. In the planes curves can be open and infinite in length or closed and finite and lines are open and infinite; on a spheres all curves including "lines" are closed and finite. On a plane lines and circles are two different things; on a sphere, straight lines are a subset of circles. Etc.

The two most important differences that will concern us are:

A) on a plane there are parallel lines and for a line and a point not on the line there is exactly one parallel line through the point. On a sphere there are no parallel great circles but there are parallel circles. (And for a circle and a point not on the circle there is exactly one parallel circle.)

B) Given a line on a plane, a point on the line and an angle measurement there is exactly one straight line with that angle to the original line. There will be an infinite number of circles with that angle though. On a sphere there will be an infinite number circles with that angle, but there will be exactly one great circle. (In this way the sphere and the plane are very much alike).

C) Given a line and two points on it, and two lines at those points and both lines have the same angle at those points, then in a plane those lines will be parallel. On a sphere those great circles will NOT be parallel. On a plane there will be an infinite number of circles through those points with the same angle but none of them will be concentric/parallel to each other. On a sphere there will be an infinite number of circles with the same angles and for every circle through one point there will be exactly one parallel cirlce with the same angle through the other point. Of these infinite pairs of parallel circles none will be both great circles. There will be exactly one great circle through the first point at the angle and a coresponding parrallel but not great circle through the other. (In this way, spheres and planes are very different. ANd this is TANTAMOUNT to your claim.)

2) The concept of direction

3) A proposition that: A travelling point travels in a straight line if and only if its direction does not change.

What is the definition of direction? Well there are two possible definitions:

a) Internal: (tangential) The angle difference between an object and its current immediate path.

b) External: In relation to an external coordinate system, the angular measurement of the immediate path in relation to the external coordinate system.

By definition a) a straight line on a plane always as direction $0$ and circle always has a constant non-zero direction.

By definition b) a straight line on a plane always has a constant direction.

Hence we can claim, with slight rewording, the proposition:

3) On a plane a straight line has $0$ internal direction and constant external direction.

Do those translate to a sphere?

There's something very strange about external definition of direction b) when applied to a sphere.

How can we apply an external coordinate system to a point? Well you must draw lines parallel to the x and y axis through the point and compare. Well, parallel on a plane has no problem, but on a sphere there are no parallel great circles so we can not do this. And there are infinite parallel small circles so that will not be unique.

The basic coordinate system on a sphere can be the two great circles, the equator and the prime meridian. And we can fix an angular direction based on that. Say, $27.5$ degrees so we have a great circle through the origin at $27.5$ degrees. Call that Circle M. That's fine. But how do we compare it to the possible direction to a different point. "March $27.5$ degrees" we say. How do we do that?

Do we take a small circle through our point that is parallel to the equator (or latitude line) and measure $27.5$ degrees through that and march ... in what. The great circle that is $27.5$ degrees at that point? Or the small circle that is parallel to circle M? Or what? Do we base this on the equator? Why not the prime meridian.

It's not clear.

So external definition b) for "direction" does not work without some fixing.

But we can fix it.

Definition of direction: A point has a circle of latitude parrallel to the unique equator and is on one of the infinite meridians (great circles perpedicular to the equator) An

• Comments are not for extended discussion; this conversation has been moved to chat. – Daniel Fischer Nov 27 '17 at 20:35
• @fleablood I answered your Rhumb line question in my answer. – enbin zheng Nov 29 '17 at 15:00

Rhumb line

The great circle is a straight line, and the direction is constant. The angle of the intersection of a great circle and a longitude line is obviously different. The angle between the longitude line and the Rhumb line is constant, so Rhumb line is not a straight line. The direction of Rhumb line is not constant. It also shows that in the sphere, a direction is determined from three points, and two points on the sphere can not determine the direction.

distance

The distance between the sphere and the plane is different. If we want to know the distance between two points on the sphere, we must know what direction the distance is, the distance from the direction of the great circle, or the distance from the direction of the small circle. In fact, there are infinitely many distances between two points on a sphere.

yellow line represents South to north

The direction in the sphere is determined by three points, not at two points. So the blue line represents the direction of the East and West, and the other lines do not represent the East - west direction. yellow line represents South to north, and the other lines are not South - North.

yellow line is South to north diagram

the parallel postulate

In a sphere, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.

So we have to be careful to avoid a simple understanding of the concepts on the sphere. Nor can we simply copy the concept on the plane.

• “The angle of the intersection of a great circle and a longitude line is obviously different.” – different from what? You make a good deal of use of the term “direction” without ever defining it. How do you picture 3 points defining a direction? Path length in the plane can be measured along a straight line or along a circular arc or along a zig-zag path. You are applying the same kind of argument for the sphere, but the distance should still be (defined as) the shortest path length in either geometry. Which means a great circle arc (geodesic) on the sphere. – MvG Nov 29 '17 at 21:08
• Your parallel postulate depends on your unconventional definition of “line”. It works if you say that only grid lines, i.e. meridian great circles and parallel small circles, are lines. If you allow arbitrary small circles as lines, then I fail to see where the claimed “at most one” should come from. For the conventional terminology, with only great circles as lines, there will be no parallel lines as all great circles must intersect. As I consider inventing your own definitions in conflict with established terminology and posting them as facts to be misleading, I'll downvote the post. – MvG Nov 29 '17 at 21:11
• Ah, found the explanation for how three points define a direction in your edited question. Which doesn't even read like a question any more, except for the first sentence. – MvG Nov 29 '17 at 21:23
• @MvG The definition of the direction is very simple. In the spherical coordinate system, if the latitude line represents the East - west direction, the longitude line represents the South - north direction. If a large circle is given, the angle of the intersection of the great circle and the longitude line is obviously different, unless the given great circle is the equatorial line. The three point is to decide one direction, that is to say the direction is determined by the circle. Because the three point determines a circle. – enbin zheng Nov 30 '17 at 5:11
• @MvG The distance on the sphere is special, that is, two points have different distances in different directions. The distance you say is only in the direction of the great circle. – enbin zheng Nov 30 '17 at 5:11