equal arc length and area
We want to determine the nonnegative differentiable real functions whose graph has the property that the arc length between any two points of it is the same as the area (http://planetmath.org/AreaOfPlaneRegion) by the curve, the -axis and the ordinate lines of those points.
The requirement leads to the equation
whence . In the case , the separation of variables yields
Consequently, the equation (2) has the general solution
and the singular solution
The functions defined by (3) and (4) are the only satisfying the given requirement. The graphs are a chain curve (which may be translated in the horizontal direction) and a line parallel to the -axis. Evidently, the line is the envelope of the integral curves given be the general solution.
|Title||equal arc length and area|
|Date of creation||2013-03-22 19:13:36|
|Last modified on||2013-03-22 19:13:36|
|Last modified by||pahio (2872)|
|Synonym||equal area and arc length|