![]() Non-Euclidean geometries can be defined by similar postulates. Once we concretely define all these abstract things in Euclidean geometry, Same with a line, circle and even an angle (in terms of the We can see that in our flat geometry (centre), a triangle's angles add up toĮxactly \(180^\). I think this should probably give some decent intuition on it.įigure 3: Deviation of a triangle under the three different types of Gaussian Curvature (source: Science4All). We won't go deeply into the definition of Gaussian curvature because it's a pain but Which we won't get into detail here but corresponds to our intuition of how aĬurvature isn't a uniform property of a surface, it's actually defined point-wise.įigure 2 shows a torus that has all three different types of Gaussian curvature. In fact, the Gaussian curvature is the product of its two And to the left, we see the saddle sheet has curvature along itsĪxis in different directions, resulting in negative curvature. Moving to the right, the sphere on hasĬurvature along its to axis in the same direction, resulting in a positiveĬurvature. With the centre diagram (zero curvature), we see that a cylindrical surface hasįlat or zero curvature in one dimension (blue) and curved around the otherĭimension (green), resulting in zero curvature. This gives us a good intuition about what it means to have curvature. Let's take a look at Figure 1 which shows theįigure 1: Examples of the three different types of Gaussian curvature (source: Science4All). To begin, let's start with Gaussian curvature, which is a measure of curvatureįor surfaces (2D manifolds). We won't go into all the details but the kinds of curvature we'll This doesn'tĭepend on any particular embedding but is an "intrinsic" property of the surface In terms of deviation of the surface (or manifold) from flat space. Means its curvature depends on something (the embedding) besides itself.Īlternatively, there is a concept of intrinsic curvature, which is defined Objects embedded in another higher dimensional space (usually Euclidean). Some measure by which a geometric object deviates from a flat plane, or in theĬase of a curve, deviates from a straight line.Īs a side note, there can be extrinsic curvature which is defined for The basic idea behind all these different definitions is that curvature is Surfaces (hypersurfaces) with the latter having many different variants. To begin this discussion, we have to first understand something aboutĭifferent kinds of curvature, and they can be with respect to either curves or (Note: If you're unfamiliar with tensors or manifolds, I suggest getting a quick ![]() Not going to go down the rabbit hole of trying to explain all the math (no Don't worry, this time I'll try much harder The math weighting more towards intuition, show some of their results, and also Geometry to represent hierarchical relationships. Researchers in which they discuss how to utilize a model of hyperbolic Interest is because there has been a surge of research showing itsĪpplication in various fields, chief among them is a paper by Facebook In this post, I'm going to explain one of the applications of an abstractĪrea of mathematics called hyperbolic geometry. While! Who knew? In any case, we're getting back to our regularly scheduled program. ![]() Under tensor calculus and differential geometry (even to a basic level) takes a Posts was to more deeply understand this topic. In fact, the original reason I took that whole math-y detour in the previous This post is finally going to get back to some ML related topics.
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