3D mesh generation: How to choose up-axis when extruding 2D shape along 3D curve?

Question

I have a 2D shape (a circle) that I want to extrude along a 3D curve to create a 3D tube mesh.

Currently the way I generate cross-sections along the curve (which form the basis of the resulting mesh) is to take every control point along the curve, create a 3D transform matrix for it, then multiply the 2D points of my circle by those curve-point matrices to determine their location in 3D space along the curve.

To create the matrix (from 3 vectors), I use the tangent on the curve as the up vector, world-up ([0,1,0]) as the forward vector, and the cross product of the up/forward vectors as the right vector. All three vectors are also orthogonalized during the process to create the final matrix.

The problem comes when my curve tangent is identical to the world-up axis. Ie, my tangent vector is [0,1,0] and the world-up is [0,1,0]....since the cross product of two parallel vectors is not explicit....the resulting extruded mesh has artifacts along those areas of the curve (pinching, twisting, etc).

I thought a potential solution would be to use the dot product of the curve tangent and the world-up as an interpolation value to shift my forward vector from world-up to world-right...in other words, as a curve tangent approaches [0,1,0], my forward vector approaches [1,0,0]...but that results in unwanted twisting along the final mesh as well.

How can I extrude my shape along a curve in a consistent manner that has no flipping/artifacts/twisting? I know it's possible since various off-the-shelf 3D applications can do it...I'm just not sure how.

Answer 1

One way I would approach this is to consider my tangent vector to the 3D curve as actually being a normal vector of the plane I am interested into.

Let's say, the tangent vector is

All you need now is two other vectors that are othoghonal to it, so let's.

Let's construct v like so:

(rotating the coordinates). Because v is the result of the cross product of u and something else, you know that v is orthogonal to u.

(This method will not work if u have equal x,y,z coordinates, in that case, construct the other vector by adding random numbers to at least two variables, rince&repeat).

Then you can simply construct w like before:

normalize and go.