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.
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.
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.
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.
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.
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.
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.
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?
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.