N-Dimensional Spheres
27.10.23
This week I thought about calculating the limit of functions which go
from
to
because it is very interesting to see how they behave for some
when
goes to zero.
In
there is that interesting trick where you use the trigonometic functions
und then you only have one variable which you then can let move to zero.
So that you only have to deal with one variable in you limit
expression.
Definition
If you always want to have only one parameter in your limit
expression you always want to approximate your limit with a sphere,
where the center of it is your limit and the radius is the delta.
Spheres have a pretty simple property:
Where
is the radius and
is the number of dimensions.
Trigonometric Representation
We now want to prove that the following set defines some arbitrary
sphere
.
To do so we define
recursivly:
Squaring Properties
So right now we only need to prove that the definition of our
above holds the definition of a sphere. But first we will prove that
where
to be more readable. We will prove now this hypothesis with
induction.
Base Case
Our Base case is
so it follows that
by the definition of
Induction Step
Prove of Trigonometric
Representation
Given some
we need to show that
which can be also written as
.
Use Case
Right now the only use-case I can find for this formula, is to find
the limit of a N-Dimensional function. But the problem with this
representation is that it gets quiet long because the number of
which are multiplied to
grow proportional to
in
.
Maybe there is some way to structure the expression to make it more
useful and powerful for real use-cases.
Further Readings
If you want to read or view more about spheres and trigonometry:
- Trigonometric
Functions, Wikipedia
- Sphere,
Wikipedia
- n-sphere,
Wikipedia
- Strange
Spheres in Higher Dimensions, Numberphile - Youtube