# Geometry  ### Key Ideas * Geometry fundamentally deals with the **closeness** of things. * If we can find/invent a notion of **distance**, we have a geometry. * Geometry is about **relationships**, particularly how close two things are to eachother. * **Geometry** *requires* a **[Metric](Metric.md)**. * Geometry requires **points** and **lines** (see Jkun pg 139). ### Key Quotes To assign a patch of land, or a set of people, or a set of horses a **geometry** is, at bottom, to assign a number to any two points, which we interpret as the distance between them. And a fundamental insight of modern geometry is that there are many different ways to do this, and a different choice means a different geometry. Geometry is the *language* of shape. ### Metric A **metric** is the assignment of a distance to each pair of points. ### Geometries * Note that a saddle, a type of hyperbolic geometry, is *not* of constant curvature. * In positively curved space, say the surface of a sphere, straight lines are called *geodesics* ### Biggest Ideas in the Universe * One way to quantitatively characterize the curvature of a surface is to determine the circumference of a circle: * No curvature: $c = 2 \pi r$ * Positive curvature: $c < 2 \pi r$ * Negative curvature: $c > 2 \pi r$ * We can take this idea and extend it as follows: * at every point in 2d space draw a little circle (in 3d space draw a little sphere) and compare how the area of the sphere or the circumference of the circle depends on the radius * compare how this to what would happen in flat space * This yields a number at every point which tells us how much we are deviating from flat euclidean geometry * **Extrinsic vs Intrinsic** * extrinsic - looking from the outside * intrinsic - looking from the inside * we really want to be able to: * characterize the geometry intrinsically * characterize all different ways the space you are in is curved * [Riemannian Geometry](Riemannian%20Geometry.md) # Space vs Geometry In ML we often think about a [Space](Space.md) as a set of points with coordinate structure (e.g. $\mathbb{R}^D$). Geometry is then the rules for measuring distance, angles and volume in this is space — defined via a [Metric](Metric.md). Say I have a space, $S_1$. I want to provide it with a metric. Let's look at three in particular: the euclidean metric, the [Mahalanobis Distance](Mahalanobis%20Distance.md) metric, and the [Radial Basis Function](Radial%20Basis%20Function.md). Regardless of the metric I chose, the space $S_1$ does not change. It's *geometry* changes, but the space (the points in the set tha make up the space) do not transform, they do not move. Frequently, researchers may informally say that a space has become "nonlinear" after they have endowed the space with a nonlinear metric, providing it with a nonlinear geometry. However, the space itself has not changed—the only thing that has changed is it's geometry. Say we have a space $S_1$ with a metric $M_a$, where $M_a$ is [Mahalanobis Distance](Mahalanobis%20Distance.md). One common thing to do is take our space $S_1$, transform in a specific way via a function $f$ into $S_2$ and then measure distances in $S_2$ via euclidean distance. Done correctly, for any two points $x_1, x_2 \in S_1$, their [Mahalanobis Distance](Mahalanobis%20Distance.md) in $S_1$ equals their euclidean distance in $S_2$. --- Date: 20211006 Links to: [Mathematics MOC](Mathematics%20MOC.md) [Shape](Shape.md) [Right Representation](Right%20Representation.md) Tags: #review References: * Shape, Jordan Ellberg * Against the gods, Risk * See summary in Eulers Gem * [The Biggest Ideas in the Universe | 13. Geometry and Topology - YouTube](https://www.youtube.com/watch?v=kp1k90zNVLc&list=PLrxfgDEc2NxZJcWcrxH3jyjUUrJlnoyzX&index=26&t=2717s)