f_catenary()
 
Calculate and return catenary related values. The aim is to provide general
functionality around which one might build hanging chains or whatever.
 
The general herein approach works by halves with the lowest (or highest) point
not offset from the x axis, but rather sitting on it with the curve scaled by
the typical 'a' constant to otherwise end on the second passed coordinate.
Full curves to be constructed symmetrically with abs(x) or asymmetrically, by
multiple pieces.
 
Where using abs(x) for x, it's best to define the x1,y1 - x2,y2 coordinates
where x1 is zero and all coordinates are already in the +x,+y coordinate
plane. While the code internally normalizes to a working +x,+y space and
de-normalizes results, it cannot maintain continuity across zero with respect
to results.
 
To use, first solve for 'a' with mode 0. After which use that 'a' constant
when calling the f_catenary() with any functional mode > 0.
 
Reference: https://en.wikipedia.org/wiki/Catenary
 
Note. The function's internal solver will numerically fail to find the
necessary set up constant 'a', if the ratio of the width to the length the
rectangle formed by (x1,y1), (x2,y2) is too small. As a rule, keep it larger
than 1/1000.
 
Inputs:
 
1)  x1 - The first, lowest x value at the start of the curve.
2)  y1 - The y value at x1.
3)  x2 - The second, highest x value at the end of the curve.
4)  y2 - The y value at x1.
 
5)  The mode of operation.
        0 - Calculate and return the 'a' constant for the passed bounding
            range. Expensive to calculate. It should be run once prior using
            the f_catenary() function generally. Inputs (6 and 7) are ignored.
        1 - Calculate and return the x value for the given [0..1] 't' value
            passed as input (7). The calculation is likely too slow for other
            than parse time use.
        2 - Calculate and return the y value for the given [0..1] 't' value
            passed as input (7). The calculation is likely too slow for other
            than parse time use.
        3 - Vertical y value at x(7) between x1 and x2.
        4 - The first derivative, y', at x(7).
        5 - The second derivative, y'', at x(7).
        6 - The path / arc length at x(7) from x1. If x==x2, this returns the
            total path length.
        7 - Return the 't' value at x(7). (arc len / total path len)
        8 - The radius of curvature at x(7).
        9 - Slope at x(7).
       10 - The angle of the tangent at x(7) relative to the x axis. This can
            be negative if the y1,y2 values decrease over the range. Return
            value in radians.
 
6)   a - Constant pre-calculated once by this function. Ignored, if calculating
         the value itself. This is herein effectively a scaling factor the
         fundamental catenary curve.
 
7)   x or t - If the input is x, this is a value between >=x1 and <=x2 and
              inputs are clamped to this range. If the input is a 't' value
              representing the arc length along the total path length, this is
              a value in the [0..1] range.
 
 
// SPDX-License-Identifier: CC-BY-SA-4.0