Heavily used in the world design of HyperRogue: This is a very nice presentation of the fact Shapes of the intermediate steps are inversions of ellipses, which are very similar to unions of two circles. We can also compose this with inversion, thus producing a transition from the Poincaré model to the inversion of Joukowsky model. (usually on a flat surface), while a "model" is a way to respresent the geometry mathematically.Īll the commonly used models of hyperbolic geometry work by representing the hyperbolic points as points in a subset of anĮuclidean plane, and thus they are projections (maybe except the hyperboloid model, which uses $\mathbb$ for intermediate values, A projection is a way to present the non-Euclidean surface Some authors use the word "model" instead of projection. Three-dimensional projections, and they all have their advantages and disadvantages. In the model might actually be very far away from each other. HyperRogue is just one of the possible projections, which does not tell the truth - objects very close The situation with the hyperbolic plane is similar. (e.g., Mercator, stereographic), all of which have their advantages and disadvantages. Maps can be areĬonformal (preserve angles) or equal-area (keep the area proportions), but not both. In such a way that all the distances are preserved. Maps aim to represent the surface of Earth on a flat piece of paper. Euclid's parallel postulate is replaced by another, equivalent to "Given a line $\ell$ and a point $p$ not on $\ell$, there exist two lines through $p$ that do not meet $\ell$." Pat Ryan's Euclidean and Non-Euclidean Geometry is an accessible treatment, including the hyperboloid model and the necessary geometry of Minkowski space, and requiring little more than one semester of linear algebra.Models and projections of hyperbolic geometry One method is to use Dini's surface, taking the edge of the disk to lie on the edge of the surface, and taking the center to lie "exponentially far away along the horn", so that the disk is tightly wrapped around the "axis".Ĭoda: Hyperbolic geometry also overlaps synthetic geometry. Qualitatively (thinking of crocheted models that start like a saddle and grow outward radially), a hyperbolic disk is "floppy", and the larger the radius, the more area must be accommodated in a Euclidean ball whose Euclidean radius grows like the hyperbolic radius of the disk "Euclidean space just can't keep up".Īs noted in the comments, however, a hyperbolic disk of arbitrarily large hyperbolic radius can be isometrically embedded in Euclidean three-space. The theorem of Hilbert mentioned asserts that the entire hyperbolic plane cannot be represented, even allowing self-intersection. Particularly, circumference grows exponentially with radius. The first three below naturally sit inside Minkowski three-space, the real Cartesian three-space equipped with the metric $dx^)$. It's possible that a hyperbolic model in the sense of the question is what I've called a hyperbolic plane above. In my experience, when people speak of the hyperbolic plane, they refer to the equivalence class of hyperbolic planes in the space of Riemannian manifolds up to isometry. It turns out that any two hyperbolic planes of curvature $-1$ are isometric as Riemannian manifolds. The rest of this answer assumes $K \equiv -1$. By scaling the metric, we may assume the curvature is $-1$ often when people say hyperbolic plane they're assuming $K \equiv -1$. In differential geometry, a hyperbolic plane is a complete, simply-connected Riemannian $2$-manifold equipped with a metric of constant negative (Gaussian) curvature. The difference between the model and the plane and their meaning really confused me. I don't know whether I understand it correctly. So what I think is that the hyperbolic model is like a framework where distance, geodesic, and something other like these can be easily defined and computed, and is easy to visualize, but hyperbolic plane is hard to visualize in our 3D Euclidean space. None of them "seem" to have negative curvature (they do not look like a saddle). Moreover, there exist several hyperbolic models such as poincaré disk, poincaré half-plane. Do the hyperbolic planes look just like those crochet work? However, I've also read that "Hilbert says that it is impossible to smoothly embed the hyperbolic plane in Euclidean three-space using the usual Euclidean geometry". And I've seen that hyperbolic planes can be visualized through crocheting. I've been taught that a hyperbolic plane is a plane with negative curvature, a visible example is a saddle. I'm new to geometry, and am confused by the difference between hyperbolic plane and hyperbolic model.
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