I am trying to model the join of the leading edge of this 'wing'.
The tubes are inflated, each section is conical in shape. I know the radius of the tubes at each joint. Given a vector in 3D space of each joint, the radius of the tube, the direction of the tube (another vector) how can I model the joint?
I am coding this in Python.
My thoughts are:
- If these were tubes, a cross section of the join would be an ellipse
- I don't know how to angle the ellipse (see the picture) if the tubes are NOT on the same plane
- I don't know how to approach the joint if the tubes taper
- tubes will share ellipse focus points
The intersection between two cones is, in general, very complicated. But the intersection between a cone and a plane is an ellipse (if the inclination of the plane is small, as in the case at hand) and it is not difficult to construct two cones having a given elliptical section in common (see the details here, for instance).
If, in particular, the cones have the same aperture the construction is very easy (see figure below). Suppose the first cone has vertex V and is cut by a plane $\alpha$, the intersection being an ellipse of major axis $AB$. Consider the line perpendicular to the plane and passing through the center $E$ of the ellipse and let $V'$ be the reflection of $V$ about this line: a second cone, with the same aperture as the first one, vertex $V'$ and axis the bisector of $AV'B$, will also intersect plane $\alpha$ in the same ellipse. The angle between the axes is twice the inclination of plane $\alpha$ with respect to the base of the first cone.