Intersection of ray and not quite helicoid?

Question

I have a shape that I'm not sure what to call. Basically, given two line segments $p$ which lives on 3d points $(a,b)$, and $q$ which lives on 3d points $(c,d)$, the shape is defined by the surface created when creating line segments from every point starting from $a$->$c$ all the way to $b$->$c$, for every point between $a$ and $b$ on the line segment $p$.

What I need is a way to take a ray defined by $origin + t * direction$ or $o + t*d$ and find the intersection if it exists on the surface.

I'm not sure what this shape is called, it is sort of like a helicoid, but it has a different equation, it isn't defined by lateral lines with respect to $p$ like a helicoid is, and twisting the line segments completely around does not produce a 3d shape like a helicoid would, it produces a plane. I can't look up what the intersection would be because of this.

here is the type of shape and intersection I'm talking about:

(there would be no holes in the surface, the lines have spaces to illustrate where each line exists).

I know the set up for the equation must look something like:

$$o + t*d = EquationForSurface(p,q,(o + t*d))$$

but I'm not entirely sure what the other equation is. I know its an equation that spits out a different equation of a line segment depending on the location, but I don't know how to parameterize it.

I'm thinking I can rotate the point to make $p$ a line oriented on the origin laying flat, and considering that if you move a point along $p$ half way, the resulting line segment will end up half way on $q$, I can use that to take the x position of the transformed point to directly correspond to a point on $p$, and knowing the percentage along $p$, I'll also know the point on $q$, and then it is a simple test for point line-segment intersection.

so the equation for the surface should be something like (excluding bounds checking and edge cases):

def EquationForSurface(p:linesegment, q:linesegment, point:point3d):
    #calculate the transformation matrix to orient p at 0,0 on middle to x,y plane. 
    transpx = calcorientxmat(p)
    px = transform(p, transpx)
    qx = transform(q, transpx)
    pointx = transpx * point
    px_percent = pointx.x / p.b.x + 0.5 #oriented at zero, so either side has side length
    px_start = find_point_on_segment(px, px_percent)
    qx_end = find_point_on_segment(qx, px_percent)
    intersecting_line_segment = LineSegment(px_start, qx_end)

    #orient line segment back to the origional space
    reoriented_line_segment = transform(intersecting_line_segment, inv(transpx))
    return equation_for_linesegment(reoriented_line_segment, point)

but having a hard time trying to bring this back into a parametric form I can solve for so I can figure out how to find the intersection with a ray.

Answer 1

The problem is that you over-complicate the problem. Let's start with your definition of the "surface". If you connect every point $\vec p$ on $\vec{ab}$ with every point $\vec q$ on $\vec{cd}$ you will not get a surface, but a volume (you have a tetrahedron). To see that, just connect $\vec a$ to $\vec c$ and $\vec b$ to $\vec d$.

Since you are talking about a surface, my guess that what you really want is to move a fraction $f$ of the way between $\vec a$ and $\vec b$ and connect with the point at the exact same fraction between $\vec c$ and $\vec d$. So that means $$\vec p=\vec a+(\vec b-\vec a)f\\\vec q=\vec c+(\vec d-\vec c)f$$Now write the equation connecting $\vec p$ and $\vec q$ as $$\vec i=\vec p+(\vec q-\vec p)g$$where $g$ is a parameter. Now all you need to do is to say that $\vec i$ is on the ray $\vec o+\vec v t$. I've used $\vec v$ here as not to confuse with the position vector of the $d$ point. Since you have three components, you have three equations. The unknowns are $f$, $g$, and $t$.