Timeline for Metrics for lines in $\mathbb{R}^3$?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 24, 2013 at 11:51 | comment | added | Joseph O'Rourke | Very clever, Will! I see now that the discrepancy lies in my ambiguous second condition, "$d(L_1,L_2)$ increases with the degree of skewness between the lines." Your metric has the property that lines at, say, $\pi/2$ can be closer than lines at $\pi/4$, depending on how they sit w.r.t. the origin. But if you fix $L_1$ and rotate $L_2$, the distance is monotonic in $\theta$. So it satisfies the 2nd condition in one interpretation and fails in another. | |
| Jul 24, 2013 at 3:15 | history | edited | Will Sawin | CC BY-SA 3.0 |
added 41 characters in body
|
| Jul 24, 2013 at 3:10 | comment | added | Will Sawin | It's not the sum of the distances of the origin, sorry for the confusion. It's the distance between the two (closest points to the origin) of the two different lines. | |
| Jul 24, 2013 at 3:09 | comment | added | Vidit Nanda | $(1,1,0) + t(0,0,1)$ and $(1,-1,0) + t(0,0,1)$ will do: the distance between them is $2$ but the sum of the distances to the origin is $2\sqrt{2}$: being on the same plane doesn't help when the closest points are not collinear. | |
| Jul 24, 2013 at 3:07 | comment | added | Will Sawin | Can you show me a counterexample? It seems that the closest points on two parallel lines to the origin lie on the same plane, and so their distance is the same as the distance between the lines. | |
| Jul 24, 2013 at 2:54 | history | edited | Will Sawin | CC BY-SA 3.0 |
added 407 characters in body
|
| Jul 24, 2013 at 2:48 | history | answered | Will Sawin | CC BY-SA 3.0 |