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On the plane, 2 points are needed to define the straight lines, on the sphere, do we need 3 points to define straight lines?

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closed as unclear what you're asking by David K, Aqua, Arnaud D., 5xum, Stefan4024 Nov 20 '17 at 11:47

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    $\begingroup$ No. Certainly not, as you may have 3 points in triangular alignment ,say for instance, floating inside the sphere. And that does not make them collinear. $\endgroup$ – Shatabdi Sinha Nov 16 '17 at 12:24
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    $\begingroup$ @ShatabdiSinha I don't understand what you mean. To me the sphere in 3D space is a (2D) surface, what do you mean by a "point floating inside the sphere"? $\endgroup$ – N.Bach Nov 16 '17 at 12:47
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    $\begingroup$ Regarding your question, do you mean "straight lines on the sphere", where you look at the sphere as a 2-dimensional manifold? Or "straight lines" in the regular 3D space? If it is the former case, I'm not qualified enough to answer, but intuitively the answer would be no. In the latter case, see the many other answers/comments. $\endgroup$ – N.Bach Nov 16 '17 at 12:49
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    $\begingroup$ What N.Bach said. Are you asking about Euclidean lines, or geodesics on the sphere, i.e., great circles? If the latter, then in general 2 points suffice, unless those points are an antipodean pair, in which case you do need a 3rd point to define a unique line. $\endgroup$ – PM 2Ring Nov 16 '17 at 12:51
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    $\begingroup$ @N.Bach I asked about the sphere.My problem is the problem of spherical geometry. $\endgroup$ – enbin zheng Nov 16 '17 at 12:52
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In spherical geometry, in general two points are sufficient to define a straight line (a great circle), unless those two points are an antipodean pair, (that is, the points are exactly opposite each other, like the North and South poles of the Earth), in which case you do need a 3rd point to define a unique line.

This is a little bit annoying. :) One way of dealing with that is to define a "point" in spherical geometry as an antipodean pair of Euclidean points, and in that way every pair of non-identical points defines a unique line, and also, every pair of non-identical lines intersect at exactly one point.

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  • $\begingroup$ Always need three points, the sphere is the big circle, also need three points. You say the big circle, this is a hidden point. $\endgroup$ – enbin zheng Nov 17 '17 at 12:54
  • $\begingroup$ @enbinzheng When doing spherical geometry & trigonometry we don't normally worry about the center of the sphere. We are only concerned with the points on the surface of the sphere. If you like, you can think of that sphere in terms of 3D Euclidean space, in which case it's common to use a sphere of unit radius centered at the origin. You might use a different radius if (for example) you're using a sphere to model the surface of the Earth, and you want to convert between (latitude, longitude) coordinates and (x, y, z) coordinates. But then you aren't really doing spherical geometry anymore. $\endgroup$ – PM 2Ring Nov 17 '17 at 13:20
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    $\begingroup$ @enbinzheng Your last two comments are completely wrong, and you have been told this numerous times already. I am beginning to wonder if you really mean to learn anything or if you come here just to say things that will annoy people so you can get a reaction. $\endgroup$ – David K Nov 18 '17 at 1:33
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    $\begingroup$ The last comment is true, but that is not what people mean when they say "determine a great circle." If you choose three points at random, the chances are there will be no great circle through all three. If you use mathematical language in a non-standard way you will generate misunderstandings. Is that what you want? $\endgroup$ – David K Nov 19 '17 at 16:22
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    $\begingroup$ No, no, a thousand times no, not every circle on the sphere is a straight line even in a geometry where the great circles are straight lines. Either you are a troll or you refuse to learn anything. I'm done here. $\endgroup$ – David K Nov 20 '17 at 7:28
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In a curved space like a sphere, a "straight line" is called a geodesic. Most of the time, two points on a sphere are enough to define a geodesic in the sense that only one geodesic passes through both points. If the two points are diametrically opposed, like the north and south pole, then there are infinitely geodesics passing through both points, so picking a third point identifies a unique geodesic passing through all three. However, if you randomly choose three points on the sphere, they are probably not colinear, so no geodesic passes through all three.

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  • $\begingroup$ I mean, small circles are straight lines, too $\endgroup$ – enbin zheng Nov 16 '17 at 14:15
  • $\begingroup$ How do you define straight line on a circle? $\endgroup$ – Alex S Nov 16 '17 at 14:18
  • $\begingroup$ I mean on a sphere. $\endgroup$ – Alex S Nov 16 '17 at 14:25
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    $\begingroup$ @enbinzheng No, small circles are not straight lines: they're circles. The only straight lines in spherical geometry are the great circles. $\endgroup$ – PM 2Ring Nov 16 '17 at 14:44
  • $\begingroup$ @ Alex S On the sphere, the line in which the direction does not change is the straight line. On a sphere, this type of line is a circle. $\endgroup$ – enbin zheng Nov 17 '17 at 13:32
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On the sphere, a line is a segment of a great circle. A great circle is the intersection of a plane containing the center of the sphere and the sphere. Thus, any two non-antipodal points on the sphere and the center of the sphere determine a plane, which in turn determines a great circle.

Therefore, on the sphere, any two non-antipodal points determine a straight line.


Clarification

Although the center of the sphere can be used to construct the great circle containing two non-antipodal points, the great circle exists whether or not we use the center to construct it. Starting at any point, we can generate a great circle by starting out in any direction and heading straight forward, not veering left or right; that is, a great circle is a curve on the sphere whose curvature is perpendicular to the surface. It is simply a property of a great circle of a sphere that it lies in a plane which contains the center of that sphere. We used this property to construct a great circle passing through two non-antipodal points on the surface of the sphere.

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    $\begingroup$ @enbinzheng What do you mean "1 point center"? Spherical geometry only deals with the points on the surface of the sphere, any interior points of the ball are irrelevant. $\endgroup$ – PM 2Ring Nov 16 '17 at 14:48
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    $\begingroup$ @enbinzheng: The center can be used to construct a great circle, but that great circle exists whether or not we use the center to construct it. We can also construct a great circle by drawing a curve on the sphere whose curvature is perpendicular to the surface; this is analogous to driving a car straight forward on the surface of the Earth, neither turning right or left. It is simply a property of a great circle that it lies in a plane which contains the center of the sphere. Knowing this property, we can show that two points on the sphere determine a straight line on the sphere. $\endgroup$ – robjohn Nov 16 '17 at 18:51
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    $\begingroup$ @enbinzheng Great circles are geodesics, so they have parallel transport. Whether or not the direction changes depends on how you want to define direction. Eg, if you use compass headings to define direction then the direction changes constantly as you traverse any great circle that's not a meridian or the equator. If you traverse a small circle of latitude the compass heading remains constant (exactly east or west), but a latitude circle (apart from the equator) is certainly not a geodesic. $\endgroup$ – PM 2Ring Nov 19 '17 at 16:19
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    $\begingroup$ @enbinzheng: As I mention in my answer: the center can be used to construct a great circle, but the center does not need to be used to construct a great circle. It is simply a property of a great circle that it contains the center of the sphere, and that property can be used to simplify the construction. Your question says: "on the sphere," so a sphere is given, which means we only need to specify two non-antipodal points to construct a unique great circle containing them. The center can be used in one construction. There are other constructions not using the center. $\endgroup$ – robjohn Nov 19 '17 at 18:32
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    $\begingroup$ @GerryMyerson: as I said, it means "a curve on the sphere whose curvature is perpendicular to the surface". To a person on the surface, it curves neither right nor left because the curvature is into or out of the surface. $\endgroup$ – robjohn Nov 20 '17 at 3:47
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In 3D, or in general Euclidean space, only 2 points suffice to form a line. If you have 3 points in 3D or in other dimension, it is not necessary for them to be collinear. The only difference is in the dimension and the representation of the corresponding point. Say in n-dimension you use a n-tuple: $(a_1,a_2,...a_n)$.

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A line cannot pass through three different points on sphere.Imagine one sphere and two points on it.If you draw a line from that two points,all other points except that two points on that line either lie inside or outside the sphere.

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  • $\begingroup$ I asked about the sphere.My problem is the problem of spherical geometry. $\endgroup$ – enbin zheng Nov 16 '17 at 12:56
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Pick the equator and North Pole on globe. They are not on one line, so picking any two non-opposite points on equator and a North Pole doesn't define any line on the sphere.

So the answer is no.

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  • $\begingroup$ Technically, 3 points on the sphere is enough to define a circle that lies on said sphere. So visually it can be confused with a straight line on the sphere. Problem is, that this seemingly "straight line on the sphere" is not (in general) a geodesics on the sphere. It only is "straight" if it is a great circle. $\endgroup$ – N.Bach Nov 16 '17 at 13:05
  • $\begingroup$ The 1 point center, 2 points on the surface of the ball, 3 points too. $\endgroup$ – enbin zheng Nov 16 '17 at 14:13
  • $\begingroup$ @N.Bach What are straight lines? On the plane, the straight lines are the trajectories of the fixed points where the direction does not change. The same is true on the sphere, so any circles on the sphere are straight lines. Great circles are straight lines, and small ones are straight lines.Only by overthrowing my reasoning can you explain that small circles are not straight lines. $\endgroup$ – enbin zheng Nov 19 '17 at 16:27
  • $\begingroup$ @enbinzheng In that case, you should start by properly defining what a direction on the sphere is. Often, straight line segments are defined as the shortest path on the surface between two points (aka geodesics). "Small circles" are not geodesics. If we stick to your definition of "no turn", I'll refer you to this extract from vsauce: youtube.com/v/Xc4xYacTu-E?start=1050&end=1210 Note that "latitude lines" are small circles. $\endgroup$ – N.Bach Nov 19 '17 at 16:41
  • $\begingroup$ @enbinzheng Afaik, one reason why geodesics are used instead of the "no turn" definition, is because precisely defining what a direction is, is fairly annoying. EDIT: just noticed my youtube video link is kinda broken. The full video can be found at youtu.be/Xc4xYacTu-E?t=17m30s and this link should start at 17m30s. The end of the relevant extract is around 20m26s. $\endgroup$ – N.Bach Nov 19 '17 at 16:43
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In any euclidean space, a straigth line is defined by two points. Or your question refer to a no euclidean space, like spheric geometry?

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    $\begingroup$ This seems to be better suited to a comment on the question that as an answer. $\endgroup$ – robjohn Nov 16 '17 at 13:43
  • $\begingroup$ spheric geometry $\endgroup$ – enbin zheng Nov 16 '17 at 13:56

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