Re: Line of sight

Francesco wrote:

how can I calculate if a point on a 2D function (e.g. x^2) is visible from an arbitrary point set somewhere in my axes? The mapping toolbox contains the
los2 function, which requires latitude and longitude as input parameters and
is therefore useless for me since I am not anlysing terrains but simple functions.

Let f(x) be 1 if x is an "odd perfect number" (a number whose factors, including
one but excluding itself, add up to the number itself); and let f(x) be 0 otherwise,
but define f(infinity) to be 1.

Question: is f(infinity) visible from any arbitrary point (0,y) 0 < y < 1 ?

If you had an algorithm that could answer the question definitely either way
(definite yes or definite no), then that algorithm would have to find an
existence proof or existence disproof for "odd perfect numbers", which is
something that there is no known proof or disproof of.

Similarly, the algorithm would have to be able to find existence proofs
or disproofs for any existence test that was representable by mapping onto
the real number line.

You could run multiple related existence tests simultaneously by mapping
to different heights. For example, you could map the first root of an expression
to 1, the second root to 2, the third root to 4, and so on, and define f(infinity)
to be 1/2: then you could test how many roots the expression had by finding
the smallest height on the y axis from which f(infinity) was visible.
You can see from this that the algorithm would have to be capable of finding
all of the roots of an arbitrary univariate expression -- which, of course,
it is known there is no analytical solution to even just for polynomials of
degree five or higher, let alone more complex operations.


So.... maybe I just haven't had enough coffee yet this morning and so am missing
a wonderful algorithm that is too large to write in the margin of this posting,
but I believe that what you are asking for is literally mathematically impossible.

What you are asking for -might- be possible if you were to severely restrict
the class of functions to be manipulated... e.g., to polynomials of degree 4 or less.
(Possibly it could be extended to multiples and ratios of polynomials each of
degree 4 or less, when the degree-4 form was available for simple inspection
rather than having to be factored out of a larger polynomial.)
.

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