On the plane, 2 points are needed to define the straight lines, on the sphere, do we need 3 points to define straight lines?
closed as unclear what you're asking by David K, Aqua, Arnaud D., 5xum, Stefan4024 Nov 20 '17 at 11:47
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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.
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.
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.
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.
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: $(a_1,a_2,...a_n)$.
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.
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.
In any euclidean space, a straigth line is defined by two points. Or your question refer to a no euclidean space, like spheric geometry?