There are many contexts where defining either 1/0 = -1/0 = ∞ or 1/0 = +∞ and –1/0 = –∞ is better than the alternatives, especially when working in an approximate number system like floating point.
In geometric modeling kinds of applications, I would say that these definitions are typically desirable, with 1/0 = undefined only better in unusual cases. As a simple example, it is typically much more useful for the “tangent” of a right angle to be defined as ∞ than left undefined.
But anyhow, there are no “facts” involved here. Only different choices of mathematical models, which can be more or less convenient depending on context / application.
In geometric modeling kinds of applications, I would say that these definitions are typically desirable, with 1/0 = undefined only better in unusual cases. As a simple example, it is typically much more useful for the “tangent” of a right angle to be defined as ∞ than left undefined.
But anyhow, there are no “facts” involved here. Only different choices of mathematical models, which can be more or less convenient depending on context / application.