> For the complete documentation index, see [llms.txt](https://enerzion.gitbook.io/enerzion/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://enerzion.gitbook.io/enerzion/introducing-enerzion/gradient-of-line.md). # Gradient of Line **In general, why is it assumed there is only one line passing through two points with a given slope, rather than an infinite amount of overlapping lines?** A line in space is obtained by placing a ruler in space at some fixed point with a fixed direction, and drawing the line it gives. That is, every point of a line can be described in the form `x(t)=x0+tv𝑥(𝑡)=𝑥0+𝑡𝑣` where x0𝑥0 is the starting point in space, and t𝑡 is the "time it takes to get to x𝑥 in the direction of the velocity vector v𝑣." If a line given parametrically by `x(t)=x0+tv𝑥(𝑡)=𝑥0+𝑡𝑣` passes through P𝑃 and Q,𝑄, we may without loss of generality assume it passes through P𝑃 at time 0 and Q𝑄 at time 1--we may scale and/or shift t as necessary to accomplish this. So we have `x(t)=x0+tv𝑥(𝑡)=𝑥0+𝑡𝑣` `x(0)=P𝑥(0)=𝑃` `x(1)=Q𝑥(1)=𝑄` from which we conclude `x0=P𝑥0=𝑃` `v=Q−P.𝑣=𝑄−𝑃.` `That is, x(t)𝑥(𝑡) must given by` `x(t)=P+t(Q−P),𝑥(𝑡)=𝑃+𝑡(𝑄−𝑃),` and there is a unique line between P𝑃 and Q.\
Based on the well-established mathematical principle, we propose the following understanding: > **A straight line defined by two specific points can include infinitely many other points. This is because only one straight line can pass through any two fixed points.** For instance, consider a straight line named Line X that passes through two given points, A and B. If you choose any other two points on this line, say C and D, and remove the original points A, B, and Line X, you can still redraw the same straight line through points A and B using the new points C and D. By repeating this process, it becomes evident that a straight line contains infinitely many points. This concept is supported by mathematical proofs available on platforms like Math Stack Exchange.