Intersection of two 3D parametric curves

Quaoar

Hi, I have two parametric curves defined in three dimensions, which are functions of a variable t, like so:

x1 = f1(t)
y1 = f2(t)
z1 = f3(t)

x2 = f4(t)
y2 = f5(t)
z2 = f6(t)

I am trying to find the intersection of these two curves, but I am having some difficulty with the mathematics. In two dimensions, I simply solve for t as a function of x, and then plug that value of t into my y function to obtain y as a function of x. With three equations, I cannot do this.

Any idea of how I should approach this problem? Thanks!

mathman

In general, 3 dimensional curves won't intersect. You need more information.

Quaoar

Well, I do know that they have a single intersection, no more and no less. I'll flesh out the actual equations here:

x1 = v_e * cos(theta_e) * cos(phi_e) * t
y1 = v_e * cos(theta_e) * sin(phi_e) * t
z1 = v_e * sin(theta_e) - g/2 * t^2

x2 = x_m + v_m * cos(theta_m) * cos(phi_m) * t
y2 = y_m + v_m * cos(theta_m) * sin(phi_m) * t
z2 = v_m * sin(theta_m) * t

v_e, theta_e, phi_e, x_m, y_m, v_m, theta_m, and phi_m are all constants. The first equation is a parabola, the second equation is a line.

HallsofIvy

At the point of intersection, the x, y, z values for each set of parametric equations has to be equal but the values of the parameters do not. I recommend you use t for one, s for the other.

You have
x1= v_e * cos(theta_e) * cos(phi_e) * t= x_m + v_m * cos(theta_m) * cos(phi_m) * s= x2,

y1 = v_e * cos(theta_e) * sin(phi_e) * t= y_m + v_m * cos(theta_m) * sin(phi_m) * s= y2

z1= v_e * sin(theta_e) - g/2 * t^2= v_m * sin(theta_m) *s

You have three linear equations to solve for the two unknown parameters. You should be able to do that using only two of the equations. Then try putting those parameters into the third equation to see if they are the same. If so that is the point of intersection. If not, then they do not intersect.

mathman, you don't need "more information". Either they intersect or they don't!

Quaoar

Well, actually, I know that s = t at the point of intersection.

What I'm trying to do is find an analytical solution for t so that I can solve for theta_m and phi_m. In the 2D case, we just have to solve for theta_m, and I found that the solution was transcendental, requiring the solution for theta_m to be computed numerically.

So in essence, we have three unknowns: t, theta_m, phi_m. In the 2D case, I was able to find an analytical solution for t for both curves, which I then set equal to each other, and then used a numerical computation to determine the value of theta_m. It appears that in the 3D case I cannot do this?