Formula for a parametric helical curve centered on the z axis that passes through 2 given points.

Question

I need to make a parametric helical curve on the perimeter of a cylinder centered on the z axis, defined by two points on the perimeter of the cylinder.

Answer 1

Suppose your points are $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$. You know that $x_1^2+y_1^2=x_2^2+y_2^2=R^2$. You can force your helix to go through $P$ by doing $$ x(\theta)=R\cos(\theta+\theta_0);\qquad y(\theta)=R\sin(\theta+\theta_0) $$ where $\theta_0$ is such that $x_1=R\cos\theta_0$ and $y_1=R\sin\theta_0$. Next, you need to choose the "step" of the helix. Suppose $x_2=R\cos\theta_1$ and $y_2=R\sin\theta_1$. We can assume WLOG that $0\le \theta_0< \theta_1\le 2\pi$. Then we need the step to be such that, over an angle $\theta_1-\theta_0$, the curve advances $z_2-z_1$. The step is thus $b=(z_2-z_1)/(\theta_1-\theta_0)$ and the last parametric equation should be $$ z=z_1+b\theta. $$ Then you get the point $P$ when $\theta=0$ and the point $Q$ when $\theta=\theta_1-\theta_0$.