This might be an easy one. Motivated by this answer. I would like *rational* pointts on the unit sphere of bounded height:
$$ \{ (x,y,z)\in \mathbb{Q}^3: x^2 + y^2 + z^2 = 1 \}$$
This seems straightforward since I could take any three integers $(a,b,c) \in \mathbb{Z}^3$ and re-scale to put them in the unit sphere. However,
$$ \frac{(a,b,c)}{\sqrt{a^2 + b^2 + c^2}} \notin \mathbb{Q}^3 $$
Instead if I have a rational point on the sphere I could clear denominators and get an integer solution to:
$$ a^2 + b^2 + c^2 = N^2 $$
just the same way rational points on the circle are related to Pythagorean triples.

The standard solution is to start with the point $(0,0,1)\in \mathbb{Q}^3$ since: $$ 0^2 + 0^2 + 1^2 = 1^2 $$ and to introduce a line of rational slope cutting through the sphere. And get more rational points. : $$ L = t\,(0,0,1) + (1-t) (a,b,c)$$

I forget the exact formula. And how do we control the height this way?

In modern language this could be in terms of hyperplane divisors, but the question I am asking here is quite ancient and basic.