Shortest range in between a factor and also a helix
I have a helix in parametric formulas that twists around the Z axis and also a factor precede. I intend to establish the fastest range in between this helix and also the factor, just how would certainly i deal with doing that?
I've attempted making use of Pythagorean theory to get the range and afterwards taking the by-product of the range function to locate the absolutely nos yet I can not appear to get a specific formula for T and also I'm stuck at that.
(I excuse the tags, not exactly sure just how to mark it and also I angle create new ones either)
Let the helix be offered by $(\cos t, \sin t, ht)$ (after scaling). If $P$ is your factor $(a,b,c)$, and also $Q = (\cos t, \sin t, ht)$ is the local factor on the helix, after that $PQ$ is vertical to the tangent at $Q$, which is simply $(-\sin t, \cos t, h)$ :
$-(\cos t - a)\sin t + (\sin t - b)\cos t + (ht - c)h = 0 $
This streamlines to $A \sin(t+B) + Ct + D = 0$ for some constants $A,B,C,D,$ as Moron claimed. Yet after that you need to address this numerically. There will certainly be greater than one remedy as a whole, yet (as Jonas Kibelbek mentioned in the remarks) you just require to examine the remedies with $z$ - coordinate in the interval $[c-\pi h, c+\pi h)$.