Geometry, and as we will see in subsequent chapters, physics, currently utilizes a point-based descriptive system. All of the descriptions that we use and even the calculations that we make are extremely dependent on non-dimensional points and the specific locations that they describe. This works great for building things that are basically rectilinear objects like houses, roads and geodesic domes. One can just connect the dots of the outlines of the structure with straight lines and everything comes out fine. Roads that seem like gentle curves can be easily made from a series of straight lines, because the sectional changes are so small in relation to the overall roadway, that even if there is no true curve, the overall effect is indistinguishable to the motorist driving down the road.
The problem comes in when you start trying to describe and create true curves. Curves require rotation to be properly outlined, but as put forward in the last chapter, rotation is not one of the officially included dimensions in standard geometric constructions, and therefore, it is rarely used in the description of a physical object.
As an example, there has been a particular problem for the machining of metal parts, and it has to do both with the equations and the tools that are utilized to do the job. Milling machines that cut metal, are mostly built on the Cartesian axis model. This is great for making accurate, straight line cuts, but not so good for curves. Curves require complex mathematical equations. For example, the equation for a circle in Cartesian terms is:
X2 + Y2 = some constant, which is the radius, assuming that the center of the circle is at 0.
Of course, a circle is a very simple curve. Imagine the calculations necessary for a hypersonic inlet or the curves of a beautiful automobile. If one were trying to program them into a Computer Numerical Control (CNC) for a milling machine, no matter how hard you tried, the best construction that you could get without the use of a rotating holder is a series of small, straight lines with the length proportional to the resolution of the machine in terms of measurement control; not a smooth, curved surface. Of course, this problem has been largely solved by more modern modeling and machining techniques, but this is the reason that in the not too distant past, all of the parts for shapes like this were made by hand, whereby the machinist or model maker him or herself could approximate these smooth curves by hand. In fact, this practice is still in use in the automotive industry where hand made models are used to create molds that are in turn used for stampings and castings that can produce complex shapes with modern production techniques.
The point is this: X2 + Y2 = C is the equation for a polygon, not a circle.
This is because of the very definition of points and lines that was discussed way back in the chapter about Euclidean Geometry. Since points have no dimension of their own, and take up no space, and all lines are straight, a circle created from this equation will always fundamentally be a collection of straight line segments. By the definition of the geometric system in use, one has limited the objects that you can accurately describe to those that can be created to those that are rectilinear in nature, that is; polygons.
If, on the other hand, we decide to include our good friend rotation into the formulation, the equation for the circle (or any curve, really) becomes simple, and more accurate. In rotational terms the equation for a circle is:
RΘ where Θ = 3600 (or 2Π in radians)
This is an exact equation for a circle. This is also the way one uses a compass to draw one. This equation is shorthand for:
Select a center
Select a radius
Attach one end of the radius to the center and rotate the radius completely around.
No points of the circle are missed. Every portion of the construct is a true curve and a continuous part of the true circle. And, it is much more elegant. There is no need to calculate second order equations to obtain the outline of the shape (more on this in the next chapter). Calculating the length of any one curve section is easy, just use the above equation and substitute the desired angular measurement of the sweep in radians for Θ and you have your desired measurement. Try that with the Cartesian equation. Good luck with that.
The real point of this chapter is this: the choice to use a Cartesian coordinate system for the case of the circle means that you can never accurately, exactly describe the desired physical shape. The language, the form, of the mathematics makes it impossible to describe the most prevalent shape in the entire macro and microscopic universe as we know it. Yet, somehow Renee, we still love ya!