This is the partial result I have.
Let $m = N-1$, the number of non-degenerate tetrahedron $\mathcal{N}_N$ is given by:
$$\begin{align}
\mathcal{N}_N &= 3\binom{m}{2}^2 + 48\binom{m}{2}m^2 + 15m^4 - 3\mathcal{O}_N\\
&= \frac34 m^2 (53m^2 - 34 m + 1) -3\mathcal{O}_N
\end{align}$$
where $\mathcal{O}_N$ is something which I didn't have a close form expression.
$\mathcal{O}_N$ is the number of degenerate tetrahedra where the four vertices are taken from two pairs of opposite edges of original tetrahedron. The factor $3$ before $\mathcal{O}_N$ is there because there are 3 possible choices for the two pairs.
It can be shown that $\mathcal{O}_N$ is the number of solutions for the following equation:
$$\frac{u}{N-u}\frac{z}{N-z} = \frac{v}{N-v}\frac{w}{N-w}$$
where $u, v, w, z \in \{1,\ldots,m\}$. $\mathcal{O}_N$ is something of order $O(3N^2)$
and can be computed with following algorithm in $O(N^2 \log N)$ steps.
histogram = {};
for( u = 1; u < N; u++){
U = u/(N-u);
for( z = 1; z < N; z++){
Z = z/(N-z);
histogram{U*Z}++;
}
}
number_degen = 0;
foreach fraction in histogram {
number_degen += histogram{fraction}^2;
}
return number_degen;
For example, when $N = 37$, $\mathcal{O}_N = 4836$ and $\mathcal{N}_N = 65575980 - 3*4836 = 65561472$.
EDIT
Let me explain the logic behind the algorithm and why its complexity is $O(N^2 \log N)$.
Consider a simpler example where $f$ is a function which sends
$1, 2, 3, 4, 5$ to $0, 0, 1, 1, 1$ respectively.
To count the number of solutions for $f(x) = f(y)$,
you don't need to loop over $x$ and $y$ twice. Instead, you loop over $x$ once to find
$f(x) = 0$ twice and $f(x) = 1$ three times.
For $f(x) = f(y) = 0$, $(x, y)$ can be any one of the $2^2$ combinations:
$$(1,1), (1,2), (2,1), (2,2)$$
For $f(x) = f(y) = 1$, $(x, y)$ can be any one of the $3^2$ combinations:
$$(3,3), (3,4), (3,5), (4,3), (4,4), (4,5), (5,3), (5,4), (5,5)$$
So there are $13 = 2^2 + 3^2$ solution for this simpler problem. The key is once
the histogram of the range of $f$ is known. One can compute the total number of
solutions by summing the square of the counts.
If one apply this to count the solutions for
$$\frac{u}{N-u}\frac{z}{N-z} = \frac{v}{N-v}\frac{w}{N-w}$$
you can construct the histogram in $O(N^2)$ steps.
On the surface, the complexity of the algorithm is $O(N^2)$ instead of $O(N^2\log N)$.
But the truth is the possible values of fractions are not that uniform and not that direct to represent in computer. One need to use some higher-level data structure to hold the histogram for fast access and manipulation.
I use an associated hash in Perl to do the dirty work. In other language, you may need to implement the data structure yourself. In the worst case, you can always implement the histogram using a binary tree. During the lifetime of the loop, the histogram will be holding the counts of $O(N^2)$ fraction. So each access/modification of the binary tree itself contribute a $O(\log(N^2))$ factor. The final complexity is $\sum_{O(N^2)} O(\log N) \sim O( N^2 \log N)$.
EDIT2
For people requesting actual code, a Perl implementation is given below.
use Math::BigRat;
my $N = 37;
my %histogram = ();
for( my $u = 1; $u < $N; $u++ ){
my $fU = Math::BigRat->new(sprintf('%d/%d',$u,$N-$u));
for( my $z = 1; $z < $N; $z++ ){
my $fZ = Math::BigRat->new(sprintf('%d/%d',$z,$N-$z));
$histogram{ ($fU*$fZ)->bstr() }++;
}
}
my $number_degen = 0;
foreach my $count (values %histogram){
$number_degen += $count*$count;
}
printf "number_degen = %d\n", $number_degen;
An $\epsilon-\delta$ proof would be difficult since you don't have any definition of distance from infinity. So limits would be your best bet.
For a sequence $\mathbf x_n$ on the sphere converging to the north pole, show that $\liminf_{n \to \infty}|\pi(\mathbf x_n)| = \infty$.
Going the other way, if you have a sequence $\mathbf y_n \in \mathbb C\cup \infty$ so that $\liminf_{n \to \infty} |\mathbf y_n| = \infty$, then show that $\psi(\mathbf y_n)$ converges to the north pole.
For a sequence $\mathbf z_n \in \mathbb C$, saying that $\liminf_{n \to \infty} |z_n| = \infty$ means that for any positive real number, only finitely many of the $\mathbf z_n$ are within that distance from the origin. It is, in this setting, the most sensible way to define "converging to infinity".
Best Answer
“General position” means “no special position”. This is the opposite of the specification that the cube edges are parallel to the coordinate axes. In this second task, you should assume that your tetrahedron is inscribed in the sphere, but may be rotated arbitrarily. If you want to do this in terms of coordinates, you should introduce parameters describing that rotation. But a more geometric approach might work as well, and give a description of what those projections will look like no matter the rotation.