How can I graph stream lines in 3d?

[Fox5]

Homework Statement

Given a set of points in 3d space (x,y,z) each having a velocity vector (u,v,w), create a general algorithm for tracing out a stream line.
The points are locked to a grid, so data points are evenly along an axis. (ie, increment along x only and it's always the same change in x to the next point, and the same holds for y and z) The points are monotonically increasing, so steady-state has already been achieved.
You are allowed to pick your starting point.

Homework Equations

From wikipedia:
en.wikipedia.org/wiki/Streamlines%2C_streaklines%2C_and_pathlines
Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. This means that if a point is picked then at that point the flow moves in a certain direction. Moving a small distance along this direction and then finding out where the flow now points would draw out a streamline.

Also, wikipedia states dx/u = dy/v = dz/w.

The Attempt at a Solution

I have no idea. I would imagine that given point(x1, y1, z1), I could search through a list of other points (maybe only adjacent points?), and then see if (x2-x1)/u2 = (y2-y1)/v2 = (z2-z1)/w where (x2,y2,z2,u2,v2,w2) are the position and velocity components of the 2nd point. If the check passes, then connect the two points and continue the process for the next point (ignoring the current one) and continue until no points satisfy the algorithm.
I don't know if that works though, and it seems like it would generate non-unique solutions anyway. After all, couldn't more than one adjacent point satisfy the condition?

 

[mighty2000]

Your proposed solution is a good starting point, but it may not generate unique solutions as you mentioned. Here is a more detailed algorithm that takes into account the monotonicity of the points and the grid structure:

1. Choose a starting point (x1, y1, z1) and its corresponding velocity vector (u1, v1, w1). This can be any point on the grid, but it may be helpful to choose a point with a high velocity to start with.

2. Calculate the step size for each coordinate based on the velocity components: dx = u1*dt, dy = v1*dt, dz = w1*dt. The value of dt can be chosen based on the desired accuracy and the magnitude of the velocity vectors.

3. Move to the next point by adding the step size to the current point: x2 = x1 + dx, y2 = y1 + dy, z2 = z1 + dz.

4. Check if the new point (x2, y2, z2) is within the grid boundaries. If it is outside, stop the algorithm.

5. Check if the new point is adjacent to the previous point (x1, y1, z1). This means that only one coordinate should have changed by the step size, and the other two coordinates should be the same. For example, if x1 = 1, y1 = 2, z1 = 3 and the step size is dx = 0.5, then the adjacent points are (x2, y2, z2) = (1.5, 2, 3), (0.5, 2, 3), (1, 2.5, 3), (1, 1.5, 3), (1, 2, 3.5), (1, 2, 2.5).

6. If the new point is not adjacent to the previous point, stop the algorithm.

7. Check if the velocity vector at the new point (u2, v2, w2) satisfies the condition dx/u2 = dy/v2 = dz/w2. If it does, then this point is part of the streamline.

8. If the condition is not satisfied, stop the algorithm.

9. Repeat steps 2-8 for the new point (x2, y2, z2) until the algorithm stops.

10. Connect all the