SESSION 3 OF COURSE 1 (Curvature)

Draw a line. It goes from one spot to another. Usually, when people refer to something as a "line", what they really mean is a straight line. Although truly straight lines are hard to draw in a sketch, these do exist in nature to a high degree of precision. With a compass, you can draw a line that is perfectly straight up to the precision of the thickness of that line. What may vary on a smaller scale is the thickness of the line due to your drawing utensil, or the smoothness of the straight edge that you used. When you use lines, you are involved with linear geometry. How do you determine if something is in a straight line? Take three of the points you are considering. Get out your straight edge. Try to have all three points even with the edge of the straight edge. If the points aren't all along the edge, then the points do not describe a line. Thus, the points are not linearly distributed.
A line describing a simple function does not change in curvature. Let's say that you had a horizontal x-axis and vertical y-axis. One line is perfectly vertical. This line has no real curvature. It could be said that its curvature is infinite, except that you can not divide by zero. Slope is the word used to indicate what a curvature is. Slope is equal to vertical change (rise) over horizontal change (run). If something rises with no run, then its slope is (something/0), which you can either look at as absolutely infinite or as a slope whose curvature is fictitious, since one can not divide by zero. A horizontal slope is zero, since (0/anything that is not zero or imaginary) is zero. A slanted line among these axes is real and non-zero, yet, if it remains constantly linear, then the curvature does not change. If a line is displaced, then it no longer is the same line. Such a change would have a change in curvature, yet this would be a different function, and what we discussed above was a constant function.
For instance, the identity function is a line where x=y all of the time, when you are talking about an x--y plane. If the line is constantly this function, then its curvature will not change. I will mention later of many functions where the curvature changes. You may even have a case where there are many axes for the dimensions of the particle, and there may be a straight line where each parameter for each axis is incremented to the same degree at each detectable level of measurement. This would be an even greater type of identity function.
A line may appear straight even when it isn't. It may be constantly in one plane, appearing to go in only one direction while it actually is jointal in many segments. This may be indicated by a change in the morphological appearance or texture that is shown by looking at the object. Sometimes, by distinguishing SINGULARITIES in segments of something's appearance, you may discuss how an object is not straight.