# Intersections of hyperboloids and affine maps

Consider the cartesian product $$\mathbb{H}_{n}$$ of $$n$$ $$m-1$$-dimensional forward hyperbolae in $$\mathbb{R}^{mn}$$ as given by the parametrization:

$$\mathbb{H}_i: \ \ x_i=\sqrt{(\vec{x}_{i+1}^2+1)}, \hspace{1cm} i=1,3,...,2n-1, \ (x_i,\vec{x}_{i+1})\in\mathbb{R}^{m}$$

and consider now the intersection of $$\mathbb{H}_{n}$$ with a backward hyperboloid (rather, when thought in $$\mathbb{R}^{mn}$$, an hyperbolic cylinder), given by the equation

$$\mathbb{H}': \ \ c+\sum_{j \ \text{odd}} A_j x_j=-\sqrt{\Big(\sum_{j \ \text{even}} A_j \vec{x}_j \Big)^2 +1 }$$

where $$A_j=A_{j+1}$$ and $$A_j\in\{-1,0,1\}$$ and $$c\in\mathbb{R}$$.

The first question is: what is the nature of this intersection? From what I can understand, when restricted to the subspace $$(x_i,\vec{x}_{i+1})\in\mathbb{R}^m$$, this intersection is $$\mathbb{H}_i$$ if $$A_i=0$$, a lower dimensional hyperboloid if $$A_i=-1$$ and an ellipse if $$A_i=1$$, provided that $$c<-1$$. In case $$A_i=\pm 1$$ the position of the foci of the ellipses and hyperboloids might depend on the value of the other $$(x_j,\vec{x}_{j+1}) \ j\ne i$$. Is there a name for this kind of manifolds?

The second question is: what can one say of the image of this intersection by an affine map (whose linear part is an isomorphism)? I would espect that the general structure in preserved, since an ellipse is mapped in an ellipse by an affine map and the same holds for an hyperboloid.

• How is defined a "forward hyperbola" ? – Jean Marie Dec 27 '18 at 15:20
• Given the implicit equation, e.g in two dimensions, $x^2-y^2=1$, one can solve for x, obtaining two sheets, i.e. $x=\pm\sqrt{1+y^2}$. Then $x=\sqrt{1+y^2}$ is the forward hyperbola and $x=-\sqrt{1+y^2}$ the backward hyperbola. – Tanatofobico Dec 27 '18 at 15:23
• OK, the branch of hyperbola situated in the first quadrant. – Jean Marie Dec 27 '18 at 15:27