# What is Straight line: Definition and 241 Discussions

In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth. Lines are an idealization of such objects, which are often described in terms of two points (e.g.,

A
B

{\displaystyle {\overleftrightarrow {AB}}}
) or referred to using a single letter (e.g.,

{\displaystyle \ell }
).Until the 17th century, lines were defined as the "[...] first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which [...] will leave from its imaginary moving some vestige in length, exempt of any width. [...] The straight line is that which is equally extended between its points."Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry).
In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it.
When a geometry is described by a set of axioms, the notion of a line is usually left undefined (a so-called primitive object). The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus in differential geometry, a line may be interpreted as a geodesic (shortest path between points), while in some projective geometries, a line is a 2-dimensional vector space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line.

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1. ### I Is My Calculus of Variations Approach Correct?

I would like to use the Calculus of Variations to show the minimum path connecting two points is a straight line, but I wish to do it from scratch without using the pre-packaged general result, because I'm having some trouble following it. Points are ##(x_1,y_1),(x_2,y_2)##. And we are to...
2. ### B Basic Straight Line Permanent Magnet Accelerator

I stumbled upon this video on YouTube: Here is a screenshot with some colored lines added of the part that generated a few questions in my head that I hope some of you smart folk can answer for me. In the video, the spherical magnet (at the end near the two big magnets) appears to increase...

46. ### B What is the definition of a straight line in mathematics?

Really basic question. I was talking with my friend and we started to get onto discussing lines when I said in three dimensions a straight line can be curved. He thinks of a straight line as a line with no curve, whilst I see it as the shortest possible distance from A to B, which in 3...
47. ### I Prove that only one straight line passes through two point

I was just thinking of basic definitions of geometry and i came to this question, so how could i prove that only one straight line passes through two distinct points.
48. ### Distance between point coordinates in a straight line

Homework Statement Let A = (1,2,5) and B = (0,1,0). Determine a point P of the line AB such that ||\vec{PB}|| = 3||\vec{PA}||. Homework EquationsThe Attempt at a Solution Initially, writing the line in parametric form\vec{AB} = B - A = (0-1,1-2,0-5) = (-1,-1,-5)\\ \\ \Rightarrow \vec{v} =...
49. ### Integral of a area under a straight line as summation

Homework Statement Homework Equations Summs: $$1+2+3+...+n=\frac{n(n+1)}{2}$$ $$1^2+2^2+3^2+...+n^2=\frac{n(n+1)(2n+1)}{6}$$ The Attempt at a Solution $$\Delta x=\frac{b}{n}$$ S_n=f\left( \frac{\Delta x}{2} \right)\Delta x+f\left( \Delta x+\frac{\Delta x}{2} \right)\Delta x+...+f\left(...
50. ### N-body simulation - straight line orbits

Homework Statement The problem is your typical N-body simulation, implemented using Python and Numpy. The implementation specifically calls for using the Euler-Cromer method. For this particular case I used the Sun and the first 4 planets of the solar system. Essentially the problem is I'm...