Definitions straight line

Question

In wikipedia, the notion of straight line is described as a basic notion, a primitive that is not defined. I wonder if there're any formal definition for a straight line in any specificular context so far, for instance in $\mathbb R^n$, in differential geometry .. ? Thanks.

Answer 1

In geometry, one usually starts with a set of axioms on which appropriate definitions for the objects at hand can be based. This also means that "the" definition for a straight line does not exist. For example, in differential geometry on curved manifolds, a straight line is known as a geodesic, and is a line which describes the shortest possible path between two points. Let $s\mapsto\gamma(s)$ be a curve in $\Bbb R^n$ equipped with a metric. Then $\gamma(s)$ is a geodesic if its components $\gamma^i$, $i \in \{1,\ldots,n\}$ solve the differential equation (also known as the geodesic equation) $$ \frac{d^2\gamma^i}{ds^2} + \sum_{j,k}^n\Gamma^i_{jk}\frac{d\gamma^j}{ds}\frac{d\gamma^k}{ds} = 0 $$ where the $\Gamma^i_{jk}$ are known as the Christoffel symbols, and describe the curvature of the space in terms of partial derivatives of the metric. If your space is flat, meaning all the $\Gamma^i_{jk}$ vanish, the geodesic equation reduces to $\frac{d^2\gamma^i}{ds^2} = 0$, a familiar property of linear functions, whose graphs are straight lines in flat space. On a sphere, a geodesic is a great circle, meaning a circle which has the same radius as the sphere itself, so that any "straight line" indeed closes in on itself.

Answer 2

I'm not sure if this counts as a "definition" but I usually use

$$\{a + bt : t \in \mathbb{R}\}$$

where $a, b \in \mathbb{R}^n$. This is equivalent to @Jan E.'s answer (and, I assume, is a special case of @paulina's answer) but I find it easier to thinking about.

Edit: @Federico Poloni is right. We need $b \neq 0$ for the above to make sense. Otherwise, the above is just the point $a$.