# Definitions straight line

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.

• Note that despite what they teach you at school, in 2D space, there are only 4 types of straight lines. check this link. Apr 20 at 20:20
• @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. Apr 20 at 22:02
• Relevant question: MSE Q1886890.
– Jam
Apr 26 at 16:51

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.
$$\{a + bt : t \in \mathbb{R}\}$$
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 $$a$$.
• It is the definition given by Jean Dieudonné in Algèbre linéaire et géométrie élémentaire, Hermann, (3.3.1). You can even write it : $a+\mathbb R b$ Apr 20 at 11:20