Solving a 'skew' quadrilateral for vertex position.

[athuss]

I've been scratching my head over this one. I'm trying to find a system of equations to solve for points R and S. The known parameters are: Point Q, tangent vector t and the axis vector a.

The following vectors are perpendicular to each other:
a,d
a,b
b,c
c,t
d,t

The other known parameter is that vector c has a magnitude of 0.0625. This creates a quadrilateral that is skew, with vector c and a being at an angle to each other and pulling the sides out of plane.

I'm trying to find the mathematical way of solving this across a number of points. SolidWorks can find one valid answer for the sample (and fully defines the topology) but I cannot get a system of equations to solve to match in Mathcad.

Any method would be appreciated? I'm trying to get it so that I don't have to rely on the SolidWorks sketch engine to reduce computation times.

 

[chiro]

Hey athuss and welcome to the forums.

So looking at your diagram, here is what we know:

Q + c = R
R + b = S
T + a = S
Q + d = T

t^unit x c^unit = d^unit

The approach I am going to take is to use the tangent vector to get the plane that contains the points Q, R, and T as well as the vectors c and d.

We know the plane equation since we have a point on the plane Q and our normal vector t. If we use n to be the normalized vector for t we get our plane equation to be n . (r - Q) = 0.

Now from this point we need to establish how the direction of a affects the direction of either c or d directly.

Now we know that a and c are parallel since a,b orthogonal and b,c orthogonal which means a^hat = c^hat. But because we know this, it means that we can calculate d^hat by calculating d^hat = t^hat x c^hat where the hat vectors are normalized.

Now we have the vector c since we have its magnitude so we get c = c^hat*||c||.

From this we have R by using R = Q + c. So that's R down.

Please tell if I've screwed up anywhere with regards to assumptions.

 

[athuss]

Hello chiro, thanks for the help.

I worked through your reply and I think that there is one assumption that isn't true. That is:

t^unit x c^unit = d^unit

t^unit and c^unit are perpendicular, as are t^unit and d^unit, but c^unit and d^unit are not.

This means that a and c are not parallel and the vector c is still unknown.

I've attached another view to show the skew of the quadrilateral. I think there is enough information to solve this but I haven't yet found it.

 

[chiro]

Thanks for that athuss! I'll take a look later though it's getting a little late here.

 

[blue_raver22]

I understand your frustration and desire to find a mathematical solution to this problem. Unfortunately, the complexity of the problem may make it difficult to find a single, straightforward solution. However, there are a few potential approaches you could consider.

One option is to use the known parameters to set up a system of equations and then use numerical methods to solve for the unknown points R and S. This could involve using techniques like Newton's method or gradient descent, which are commonly used in optimization problems. However, this approach may still require some trial and error to find a solution that satisfies all of the given constraints.

Another option is to use geometric constructions and principles to find a solution. For example, you could use the given tangent and axis vectors to construct a circle, and then use the known magnitude of vector c to determine the radius of that circle. From there, you may be able to use relationships between the sides and angles of the quadrilateral to solve for the positions of points R and S.

In either case, it may be helpful to break the problem down into smaller, more manageable pieces. This could involve solving for one point at a time, or finding a solution for a simplified version of the problem before tackling the full complexity.

Overall, solving this type of problem may require a combination of mathematical techniques and creative problem-solving. I wish you luck in finding a solution that meets your needs.