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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.

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  • $\begingroup$ Note that despite what they teach you at school, in 2D space, there are only 4 types of straight lines. check this link. $\endgroup$
    – machine_1
    Apr 20 at 20:20
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    $\begingroup$ @machine_1 this concerns the approximation of arbitrary angled lines by a rectangular pixel grid, and has nothing to do with the topic at hand. $\endgroup$
    – paulina
    Apr 20 at 22:02
  • $\begingroup$ Relevant question: MSE Q1886890. $\endgroup$
    – Jam
    Apr 26 at 16:51

2 Answers 2

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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.

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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 $a$.

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    $\begingroup$ It is the definition given by Jean Dieudonné in Algèbre linéaire et géométrie élémentaire, Hermann, (3.3.1). You can even write it : $a+\mathbb R b$ $\endgroup$ Apr 20 at 11:20
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    $\begingroup$ yes, this is exactly the case from my answer when the christoffel symbols vanish, meaning a geodesic in flat space. $\endgroup$
    – paulina
    Apr 20 at 13:50

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