How to Determine When a Moving Point Intersects a Moving Edge?

[FranBoltzmann]

Im new to the forum, so I didnt know where to post this.

(x is cross-product, . is dot-product and * is multiplication)

Consider point P with linear velocity Pv.

Consider points A and B that define the edge AB of a square with C as center.

Consider that C has linear velocity Cv and angular velocity Cw.

Av = Cv + Cw x (A-C)
Bv = Cv + Cw x (B-C)

(Considering this is 2D, you can see Cw as (0,0,angle) and all linear velocities as (Vx,Vy,0) )

Now, let's add time as a variable.

P(t) = P(0) + Pv*t
A(t) = A(0) + Av*t
B(t) = B(0) + Bv*t

(consider velocities to be constant)

Now the fun begins:

A(t) + k*[ B(t) - A(t) ] is a point along the edge AB where 0 ≤ k ≤ 1

Which means:

A(0) + Av*t + k*[ B(0) + Bv*t - A(0) - Av*t ]

So far so good, but what happens if we want to know when P(t) intersects AB(t)?

P(0) + Pv*t = A(0) + Av*t + k*[ B(0) + Bv*t - A(0) - Av*t ]

or

A(0) + k*B(0) - k*A(0) - P(0) = t
(Pv - Av - k*Bv + k*Av)

or

P(0) + Pv*t - A(0) - Av*t = k
B(0) + Bv*t - A(0) - Av*t

Finally, I should state that all variables are known except k and t.

The objective is to know t and k that satisfy the equation BUT then discard any pair (t,k) where k doesn't satisfy 0 ≤ k ≤ 1.

Can anyone help me find a solution or better way to solve this problem? Or is this impossible?

Thanks in advance!

 

[mighty2000]

Dear newcomer to the forum,

Welcome! It's great to see a fellow scientist interested in solving complex problems. From what I understand, you are trying to find the time and position at which a point P will intersect with the edge AB of a square, taking into account the linear and angular velocities of points A, B, and C. This is definitely a challenging problem, but it is not impossible to solve.

One approach you can take is to use the concept of parametric equations. In this case, you can treat the position of P and the edge AB as two separate parametric equations, each with its own set of variables. Then, you can equate the two equations and solve for the variables t and k, as you have already done. The only additional step would be to discard any solutions that do not fall within the range of 0 ≤ k ≤ 1.

Another approach would be to use numerical methods, such as the Newton-Raphson method, to approximate the values of t and k that satisfy the equation. This method involves repeatedly guessing values for t and k and then refining them until you reach a satisfactory solution.

I hope these suggestions help you in finding a solution to your problem. Good luck! And don't hesitate to reach out to the forum for further assistance or discussion.