what's the equation of helix surface?

Solution 1:

We can use a local orthonormal basis of a parametrized curve to get a surface of the desired type.

A helix running around the $x$-axis has a parametrization like $$ \vec{r}(t)=(ht,R\cos t, R\sin t). $$ Its tangent vector can be gotten by differentiating $$ \vec{t}=\frac{d\vec{r}(t)}{dt}=(h,-R\sin t,R\cos t). $$ We note that this has constant length $\sqrt{h^2+R^2}$. With a more general curve this is not necessarily the case, and we would normalize this to unit length, and switch to using the natural parameter $s=$ the arc length. This time $ds/dt=\sqrt{h^2+R^2}$, and we can keep using $t$ as long as we remember to normalize.

We get a (local) normal $\vec{n}(t)$ vector by differentiating the (normalized) tangent $$ \vec{n}(t)= \frac{\frac{d\vec{t}}{dt}}{\left\Vert\frac{d\vec{t}}{dt}\right\Vert}=(0,-\cos t,-\sin t). $$ As the name suggests this is orthogonal to the tangent vector (in the direction of change of the tangent). The third basis vector is the binormal $$ \vec{b}(t)=\frac1{\Vert\vec{t}\Vert}\vec{t}\times\vec{n}=\frac{1}{\sqrt{R^2+h^2}}(R,h\sin t,-h\cos t). $$ This is, of course, orthogonal to both $\vec{t}$ and $\vec{n}$.

The key is that we get the desired surface by drawing (3D-)circles with axis direction determined by the direction of the curve, i.e. the tangent. Equivalently, we draw a circle of radius $a$ in the plane spanned by $\vec{n}$ and $\vec{b}$. Hence we get the entire surface $S$ parametrized as $$ S(t,u)=\vec{r}(t)+a\vec{n}(t)\cos u+ a\vec{b}(t)\sin u $$ with $t$ ranging over as many loops as you wish, and $u$ ranging over the interval $[0,2\pi]$. In terms of individual coordinates this reads (barring mistakes due to lack of coffee): $$ \begin{aligned} x(t,u)&=ht+\frac{Ra\sin u}{\sqrt{R^2+h^2}},\\ y(t,u)&=R\cos t-a\cos t\cos u+\frac{ha\sin t\sin u}{\sqrt{R^2+h^2}},\\ z(t,u)&=R\sin t-a\sin t\cos u-\frac{ha\cos t\sin u}{\sqrt{R^2+h^2}}. \end{aligned} $$

Here's what Mathematica-output looks like with parameters $h=1$, $R=3$, $a=0.4$:

Here's the effect of the change to $a=1.0$. The lines on the surface correspond to constant values of $t$ and $u$. These are now more clearly defined.