Hyperbolic plane9/28/2023 The metric tells you the length of a “line element”, essentially the length of a tiny, infinitesimal line segment. The metric in euclidian space is which tells us, for a small displacement dx in x, dy in y, dz in z, the total distance traveled squared ( ) is – which is essentially just the 3d pythagorean theorem. A metric essentially gives you a formula for lengths in a space. So how is Minkowski space different from regular Euclidean space? Distance in Minkowski space is measured differently than Euclidean space- it has a different Metric, which is an idea from differential geometry. Minkowski hyperboloid used in HyperRogue, with the Poincare disc projection shown below Yes, we are representing hyperbolic space by a not-hyperbolic 2D hyperboloid inside of a not-hyperbolic 3D space. This is because the point does not exist in regular euclidian 3 dimensional space, it exists in Minkowski space, which isn’t euclidian but isn’t perfectly hyperbolic either. But it is even less clear at first how the hyperboloid model works- the surface seems to have positive curvature everywhere, not negative, but this isn’t true. If we say x is the vertical axis (you’ll see why later), the point lies on the hyperboloid if. The Minkowski hyperboloid is used in representations of spacetime, and is used internally in HyperRogue (and probably Hyperbolica also). The Poincare disc view is just a projection of the hyperboloid onto the plane (analogous to stereographic projection of a sphere onto the plane). It turns out that hyperbolic space in math is represented by the one surface of a hyperboloid of 2 sheets, called the Minkowski hyperboloid. But points aren’t represented on the 2d disc. Most of the visualisations of hyperbolic spaces we’ve seen so far display the whole space compressed into a disc- called a Poincare disc. Hyperbolic paraboloid, and Hyperboloid of one sheet Minkowski space, metric, and paraboloid So are we forced to represent hyperbolic space by a horrible crumpled up mess? Yes and no. Secondly, as we go farther from the origin in the z axis, the hyperboloid becomes tangent to a cone, which has zero gaussian curvature- so it doesn’t have constant negative curvature. Firstly, it is topologically congruent to a cylinder, not a plane. Perhaps hyperbolic space is better represented by a hyperboloid of one sheet? Clearly not. Even though the hyperbolic paraboloid has negative curvature everywhere, the magnitude of curvature decreases as the distance from the origin increases, so it isn’t perfectly hyperbolic. The first thing that came to my head as a model of hyperbolic space was a hyperbolic paraboloid- the classic saddle shaped Pringles chip surface. a way of giving a coordinate system and parametrisation to points on the space, I ran into trouble. When I was first trying to find an effective model of hyperbolic space, i.e. Maybe I know all the math and definitions, but how am I supposed to put numbers on things? How am I supposed to represent coordinates and to transformations?Ģd Hyperbolic plane embedded in 3d euclidian space- every point is a saddle point There are tons of formulas and relations in differential geometry that gives you ways of measuring curvature and etc, but at this point I was stumped. The constant negative sectional curvature is the tricky part, and the only factor that makes hyperbolic space different from euclidian and spherical space. What on earth does that mean? Well, maximally symmetric means that all points and all orientations are essentially the same, and simply connected means that there are no holes or defects in the space, and N-dimensional Riemannian Manifold says that there is some way of measuring vector lengths, and zooming in yields an approximately flat space. What is hyperbolic space? According to wikipedia, hyperbolic space in N dimensions is a maximally symmetric, simply connected, N-dimensional Riemannian Manifold with constant negative sectional curvature. "Hyperbolic Paraboloid."įrom MathWorld-A Wolfram Web Resource.Trailer for HyperRogue- I highly recommend giving it a try (it’s free on his website) Penguin Dictionary of Curious and Interesting Geometry. "Spezielle algebraische Flächen." Encylopädie der Math. Surfaces from Differential Geometry: Hyperbolic Paraboloid.". "The Hyperbolic Paraboloid."ĭifferential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Modelle aus den Sammlungen von Universitäten und Museen, Kommentarband.īraunschweig, Germany: Vieweg, pp. 3-4, 1986. CRC Standard Mathematical Tables, 28th ed.
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