Helix along vector in 3D space

Question

Let's say I have a random vector, for example <1, 3, 5>.

What would the function be for a helix that spirals around/along this vector with a given radius?

Answer 1

Let $\mathbf{W}$ be a unit vector in the direction of the given vector. So, in your case $\mathbf{W} = (1,3,5)/\sqrt{35}$. Let $\mathbf{U}$ and $\mathbf{V}$ be two other unit vectors perpendicular to $\mathbf{W}$ and to each other, so that $\mathbf{U}$, $\mathbf{V}$, $\mathbf{W}$ form a mutually orthogonal set. In less fancy language, $\mathbf{U}$, $\mathbf{V}$, $\mathbf{W}$ form a coordinate frame. Constructing $\mathbf{U}$ and $\mathbf{V}$ is not very difficult. One approach is to let $\mathbf{U}$ be the unit vector in the direction of $\mathbf{W} \times (1,0,0)$, and then let $\mathbf{V} = \mathbf{W} \times \mathbf{U}$.

Suppose we want a helix with radius $a$ and pitch $b/a$. We can get this by rotating the standard equation to the $UVW$ coordinate system. The standard equation is: $$ P(t) = (a\cos t, a \sin t , bt) = (a\cos t)\mathbf{I} + (a \sin t)\mathbf{J} + (bt)\mathbf{K} $$ The rotated equation is $$ P(t) = (a\cos t)\mathbf{U} + (a \sin t)\mathbf{V} + (bt)\mathbf{W} $$