A midpoint-radius representation is widely used in interval computations, so it is desirable to be able to transform each interval from inf-sup form to midpoint-radius form. In line with the general spirit of interval computations, we need the new midpoint-radius interval to enclose the original one, and to make the excess width as small as possible.
Thus, once a midpoint m is fixed, it is reasonable to define the radius r as the smallest value for which [m - r, m + r] encloses the original interval.
For midpoint, in mathematics, the ideal value is m = (a + b)/ 2. Since this number is not always computer representable, we would like to select a computer representable number which is the closest to it. For finite a and b, it does not matter that much which of the two nearest computer representable values we select. However, when e.g., a is finite and b is infinite, the situation is different. Theoretically, we could select +oo or we could select MaxReal. The problem with +oo is that for this choice of a midpoint, we will never get the desired enclosure no matter what radius r we select:
* for finite r, we will not get an enclosure since the resulting interval will be [+oo, +oo], and
* for an infinite r = +oo, the value m - r is not defined
Thus, for an interval in which the left endpoint is finite and the right endpoint is infinite, we should select m = MaxReal
Similarly, for an interval in which the left endpoint is infinite and the right endpoint is finite, we should select m = -Maxrreal
For an interval (-oo, +oo), natural symmetry ideas (symmetry w.r.t. x --> -x) necessitate the selection of a midpoint which is invariant w.r.t. this symmetry, i.e., m = 0.