In the Betrami-Klein model of hyperbolic geometry, geodesics are represented as straight lines. Hence the exponential map of a tangent vector $\mathbf{v}$ at a point $\mathbf{p}$ is $\mathbf{p} + \lambda \mathbf{v}$, where $\lambda$ is a scalar that depends on $\mathbf{p}$ and $\mathbf{v}$. For example, suppose $\mathbf{p} = 0$. Then the exponential map is $$ \exp_\mathbf{p}(\mathbf{v}) […]

I was given the following question regarding hyperbolic geometry: In the hyperbolic geometry in the upper half plane, construct two lines through the point $(3,1)$ that are parallel to the line $x=7$. How do I go about doing this? I am very new to non-Euclidean geometry. Thank you.

Preamble Tarski’s axioms formalize Euclidean geometry in a first-order theory where the variables range over the points of the space and the primitive notions are betweenness $Bxyz$ (meaning $y$ is on the line segment between $x$ and $z$, inclusive) and congruence $xy\equiv zw$ (meaning the line segment between $x$ and $y$ is the same length […]

I learned that in two-dimensional geometry, there are Euclidean geometry, hyperbolic geometry and spherical geometry. These geometries are homogeneous and isotropic. Is there another kind of two-dimensional geometry that is homogeneous and isotropic?

This question was triggered by a series of others and reading some references: Keshav Srinivasan & Euclid Eclid’s Elements As quoted from the last reference: GEOMETRICAL ALGEBRA. We have already seen [ .. ] how the Pythagoreans and later Greek mathematicians exhibited different kinds of numbers as forming different geometrical figures. [ skip text ] […]

In the first introductory chapter of his book Gravitation and cosmology: principles and applications of the general theory of relativity Steven Weinberg discusses the origin of non-euclidean geometries and the “inner properties” of surfaces. He mentions that distances between all pairs of 4 points on a flat surface satisfy a particular relation: $$\begin{align} 0 &= […]

There are many things which are called “elliptic” in various branches of mathematics: Elliptic curves Elliptic functions Elliptic geometry Elliptic hyperboloid Elliptic integral Elliptic modulus Elliptic paraboloid Elliptic partial differential equation I take it that most of them relate to ellipses in one way or another, but the relation is often unclear to me. Does […]

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