You said right, on the exploration path, we must be careful. In the plane, between two curve's distances to be vertical to two curve line segment length.
If a line (straight line either curve) to another line (straight line or curve) the distance maintains invariable, then these two lines (straight line or curve) are parallel.
Starts, the red straight line and the blue color straight line is parallel, afterward they became the spiral line, if their distance were invariable, therefore they were each other parallel. This is not the very simple truth?
Please note, my words were such say:Circumference is parallel to the center of the circle, so the circle became a circle, because the ellipse is not parallel, so the ellipse as the ellipse.
If a line (straight line either curve) to another line (straight line or curve) the distance maintains invariable, then these two lines (straight line or curve) are parallel.
If in the attached figure animation shows: If B plane and D plane parallel, the distance is E. A is a straight line, and A on B, therefore, A to the B distance is also E, when A becomes circumference C, C to the B distance is also E obviously, therefore, the distance is constant, therefore, C is...
If our original definition crow is the black, afterward we discovered that the crow also has the white, at this time, whether we should revise the original definition?
Newton's time's definition is different with Einstein's time's definition, after the theory of relativity appears, we should...
Circumference is parallel to the center of the circle, so the circle became a circle, because the ellipse is not parallel, so the ellipse as the ellipse.
Actually I have said many times, parallel is the constant distance. But some people just don't listen.
equivalent statements to the parallel postulate :
There exists a pair of straight lines that are at constant distance from each other.