Background: I've started reading Miles' "Undergraduate Algebraic Geometry" (Link) recently though struggling a lot.
I'm stuck at processing the following paragraph...
Sec. 1.7 Parametrisation of a conic
Corollary 1.6
Here is my question list;
- How can we picture the new coordinates $(X+Z, Y, Z-X)$? I tried to visualise it on some 3D visualiser, eg., GeoGebra, but it gives me error as it's not a function...
- What are the variables $U, V$? Can I think of it as some form of parameteric functions controlled by a variable like $t$?
- To understand the bit
$C$ is projectively equivalent to the curve $(XZ = Y^2)$, I graphed C and the curve here but I can't seem to find what the equivalence means....
Sorry for listing questions but can somebody help me?
This is not easy to visualize; this is a coordinate transformation $\mathbb{R}^3 \to \mathbb{R}^3$. In fact, this is a linear transformation. Can you find the matrix representing it? Examples of these kinds of transformations include dilations and rotations
$U$ and $V$ represent real numbers so that $(U, V)$ is a point of $\mathbb{P}^1$. (That it, $U$ and $V$ are not both zero.) The parametrization is telling you, for example, that $\Phi((1,2)) = (1, 2, 4)$.
There is a mistake in your visualization. In particular, $eq2$ should be $x^2 + y^2 = z^2$ instead of $x^2 + y^2 = 1$. Once you make this change, it'll look quite possible to move one cone to the other using rotations and dilations.