A Proposal of a Genuine Definition of a Straight Line

2010 © Cameron Rebigsol

    

Fundamentally, in Euclidean geometry the definition of a straight line is given by “the shortest distance between two points in space.” However, a distance itself is a segment of straight line. So this definition for a straight line literally just says that a straight line is “the shortest straight line between two points in space”.  How much genuineness can we learn from this definition about a straight line?  

 

We need a more solid definition to fill this fissure found in the grand auditorium of Euclidean geometry.  Let's try the following:

Any given curve of infinite length in space is a straight line if it satisfies the following conditions:

     1. Find a segment from this curve between any two randomly chosen points A and B. On this curve, no matter how B may infinitely travel away from A, B can never have a chance to meet A.

     2.  Reproduce an exact replica of AB. Name this replica A’B’.

     3.  Flip and attach this replica to the original segment so that A’ falls on B, and B’ falls on A.

     4.  Between segment AB and the flipped replica B’A’, if we can find a space in which a point can be found belonging to neither AB nor B’A’, the curve from which AB is taken is not a straight line. If we cannot find such a space between AB and B’A’, this curve is a straight line.

     The condition of “any two points” in 1 above is to guarantee that the geometric property shown in any segment is exactly shared by any other segment of this curve, no more and no less. The no meeting condition is to prevent two situations: 1. allowing A and B to be the same; 2 allowing the curve to be a closed loop. The “any two points” condition can also help to exclude any curve similar to a sine wave from being considered as a straight line. In case AB happens to be a segment of a sine curve of exact one period between two nodes of zero amplitude, AB and its replica B’A’ can seamlessly coincide with each other, leaving between them no space.   Condition 4 alone without the restriction of “any two point” would allow it to be called a straight line. However, the “any two points” condition allows us to re-examine the sine curve by taking a segment of a ¾ period. This ¾ period segment and its replica must create some space between them and therefore fails the one period segment to be regarded as a straight line.

     In a two-dimensional plane, in case AB is an arc from a circle, condition 4 must reject this arc being from a straight line regardless of the length of its radius. The arc and its flipped replica B’A’ must embrace some space between them. In a three dimension space, even though the end points of the same arc of AB and B’A’ anchored each other, the rest of each of them can move away from the plane they have been in. Such movement allows B’A’ to have one chance to coincide seamlessly with AB when only both of them meet in the same plane and on the same side of the center of the circle. When this meeting happens, they would embrace no space between them. However, when they are not in the same plane, space must appear between them, inevitably failing the arc to stand as a straight line according to condition 4.

     Suppose we can find a space between AB and B’A’. In this space we must be able to find a point belonging to neither AB nor B’A’. Let’s call it C.  The existence of C must allow at least one curve to be constructed between A and C as well as one curve between B and C.  Obviously, curve ACB cannot be part of AB, neither is it part of B’A’, except sharing end points with AB and B'A'. No one can rule out that curve ACB may have a chance to be a straight line. That ACB may be a straight line must reject that AB can have any chance to be a straight line. On the other hand, in an effort locating C, we may come to one unique case. In this case, we can find no point independent of AB and B’A’, but every point so located must be commonly shared by AB and B’A’. In other words, AB allows its flipped replica no deviation from it. This uniqueness of AB defines AB from a straight line.   This quality of no deviation determines that between any two points in space there can exist only one straight line. Any curve not having such uniqueness must fail to stand as a straight line.

     Now, we can come across a far more concise definition of straight line:

     First, in plane geometry:

     A curve in a plane is a straight line if the reversely flipped replica of any arbitrarily chosen segment from this curve can seamlessly coincide with the curve at wherever it falls on.

    The drawback of this definition is that a plane in space still needs two straight lines crosing each other to define. Therefore, we need a definition that can be even more primitive than the above one and be applicable in space that contains any plane.  Such more primitive definition can be given as following:

    A curve in space is a straight line if any arbitrarily chosen segment from this curve can seamlessly coincide with the curve at wherever it falls on.  Such coincidence is said seamless only if the movement of this chosen segemnt in any manner can never produce any space between it and the original curve while the two end points of this segment stay on the original curve all the time, wherever these two points happen to fall on. 

     After all these descriptions, let’s examine the shortfall of some contemporary definitions about straight line.

    General verbal description from dictionary: A straight line is a line traced by a point traveling in a constant direction; a line of zero curvature; "the shortest distance between two points is a straight line."

     In all these definitions, “direction” can be defined only after a straight line is found. A “constant direction” is even more so. “Curvature” is something that must rely on the validity of direction, which by itself awaits the definition of a straight line. “Distance” cannot be determined unless a straight line has been defined. All these can only tell us that a straight line must rely its definition on another straight line, but this definition waiting on another straight line can be endless.  

     The failure of proposing a sound definition about a straight line is a misfortune to Euclidean geometry. Its fifth postulate therefore allows a weakness for various non-Euclidean geometry theories to come in to decline its completeness in validity. They even further declare that Euclidean geometry loses its universal validity in the universe. Of course, in doing so, these non-Euclidean geometries bring to themselves even bigger misfortune, because they must find themselves impossible to establish credit without Euclidean geometry at the first place. No theory can be valid if it finds itself necessarily relying on something invalid.

     Besides the verbal description, there are also many definitions that are presented in mathematical form, such as y=ax+c in analytic geometry, r=OA+ƳAB in vector field, geodesics…  It can be said that none of these definitions would stand firm without Euclidean geometry. All these definitions must rely on the recording of spatial coordinates, while such recording can be done only if a coordinate system composed of straight line(s) has been ascertained. Simply, for example, one of the non-Euclidean geometry is developed based on the property of hyperbola. Without a spatial backdrop that is described by Euclidean geometry, based on what is a hyperbolic curve or surface constructed? Can any segment of a hyperbolic curve and its flipped replica seamlessly coincide with each other?

     Nevertheless, let alone the discussion on validity; are all the “advanced” definitions from the non-Euclidean geometry necessary and practical to a person who just begins the study of geometry? But, then, after further study or many more years, people can only tell him that what he learns is not universally valid in a world of more advance study? Isn’t it obvious that he is barred outside of an ivory tower until he accepts the emperor’s new robe?  

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