I've asked this to many mathematicians but I don't get a conclusive answer.
Regarding origins (centers): - I understand that the origin in spherical geometry is the equidistant to all the points on the plane, but how was the center (origin) of the hyperbolic geometry decided?
Regarding straight lines: - In euclidian space lines are not defined regarding the origin, why is it so important in spherical and geometrical spaces? Why the straight lines of this other geometries need to be in planes that go through the origin.
I hope I've been clear enough ^_^
Thank you!
There is a presupposition to your question which is not really true: Geometries do not necessarily even have origins.
For example, if you go back to the original text on geometry, Euclid's "Elements", you will see that there is no origin at all. There are points, lines and planes, there are line segments and angles, and many other aspects of geometry as we know it, but there's no origin. None at all.
Perhaps the first appearance of an origin in geometry was with Descartes' introduction of coordinate geometry, in which numbers are used to model geometry. When one chooses coordinates, one must choose an $x$-axis and a $y$-axis, and as you may have learned in pre-calculus or in elementary physics, and the idea is to choose those axes, and the origin where they cross, using whatever physical considerations make for the simplest possible formulas to model your problem.
Eventually (quickly?) Cartesian coordinate geometry became the most important model of geometry. It has an origin by means of its very construction, i.e. the unique point all of whose coordinates are zero. Nonetheless, from the point of view of Euclidean geometry, there is nothing special about the origin.
Regarding your first question: you are presupposing that hyperbolic geometry has an origin. But, it does not. Various models of hyperbolic geometry use Cartesian coordinate geometry to aid in their construction, and those Cartesian coordinate geometries have their origin as explained earlier (I'm thinking here about the Poincare disc model, the Poincare upper half plane model, the hyperboloid model; and others). But in none of those models is there anything special about the origin of the Cartesian coordinates with respect to hyperbolic geometry itself.
Regarding your second question: I'm not sure why you say straight lines must go through the origin; they do not have to. Perhaps you are confusing geometry with vector spaces. In the theory of vector spaces, there is a vector addition operation, that operation has an identity element called the zero vector. The zero vector does indeed play an important role, somewhat like an origin, in the theory of vector spaces. For example, a linear subspace of a vector space must pass through the origin, which is perhaps what you might have been remembering when you wrote that lines must pass through the origin.