How can I generate and plot a helix that passes through a given point?

Question

I need to create and graph an equation of a spiral helix that will pass through the point {54, 0, 12.5), but I am not sure how to attempt this. Any help is appreciated.

Answer 1

Manipulate[
 Column[
   {TextGrid[{{"x", "y", "z"}, 
   v = {Cos[tt], Sin[tt], tt/10}}, 
   Frame -> All], 
   Show[
       ParametricPlot3D[{Cos[t], Sin[t], t/10}, {t, 0, 30}], 
       Graphics3D[{Red, PointSize[0.05], Point[v]}]]},
   Alignment->Center], 
   {tt, 0, 30}]

Answer 2

Consider that

54 {Cos[t], Sin[t], t/10 + 12.5/54} /. t -> 0

gives

{54, 0, 12.5}.

Since

54 {Cos[t], Sin[t], t/10 + 12.5/54}

is the parametric expression of a helix with radius 54, it represents a helix that solves your problem. This is visualized by

Show[
 ParametricPlot3D[54 {Cos[t], Sin[t], t/10 + 12.5/54}, {t, -10, 10}],
 Graphics3D[{Red, Sphere[{54, 0, 12.5}, Scaled[.015]]}]]