Angle of intersection of two curved lines.

[DaveW]

I'm not a mathematician by any stretch. I've done a little simple research, and I've seen formulae and calculators for determining the angle of intersection of two curved lines, but these work when one is starting with the formulae for those curves; my question concerns a practical issue where I don't know the formulae for those curves. I need to be able to determine the angle of intersection of two curved lines. These lines already exist. They are called Wallner lines and fracture undulations, and appear on the surface of fractured metal. The Wallner lines intersect the fracture undulations at an angle related to the speed of the fracture. I wish to determine the speed of the fracture, so I need to measure the angle of intersection.

Is there an app or program that would allow me to draw two intersecting curved lines (say, as an overlay over an image of these Wallner lines and fracture undulations) and tell me the angle of intersection of those lines?

 

[RalphD]

I don't think such an app exists, but there's an easier way. First of all, if you want to solve this problem programmatically and you're familiar with and have access to MATLAB, then read up on https://nl.mathworks.com/help/image...ection.html?requestedDomain=www.mathworks.com. That really solves exactly the problem you describe.

However, if you aren't required to find the angles of intersection through a program, you could simply do it by hand. Before we get into this, you must realize that you can find the angle of intersection of two straight lines. That means that to find the intersection angle of any two curves, you simply have to draw the tangent lines of those curves at the point of intersection.

The formula that gives you the angle of intersection when the two straight lines are known can be found through a Google search. I've re-written the expression described at https://www.easycalculation.com/analytical/learn-angle-between-two-curves.php to arrive at the following formula:

\(\displaystyle \displaystyle \theta = \tan^{-1}\left(\frac{\left|m_2 - m_1\right|}{1 + m_1 \cdot m_2}\right)\)

Where \(\displaystyle \theta\) is the angle of intersection and \(\displaystyle m_1, m_2\) are the slopes of the tangent lines of the curves at the point of intersection.

 

[Otis]

read up on https://nl.mathworks.com/help/image...ection.html?requestedDomain=www.mathworks.com. That really solves exactly the problem you describe.

The steps at that URL describe a sequence of commands for extracting information from an image file (i.e., "using image acquisition and image processing techniques" and pixel locations). It doesn't seem like that's "exactly" what the OP describes.

I didn't study your work at the other URL, but that approach seems more helpful.

 

[RalphD]

The steps at that URL describe a sequence of commands for extracting information from an image file (i.e., "using image acquisition and image processing techniques" and pixel locations). It doesn't seem like that's "exactly" what the OP describes.

I didn't study your work at the other URL, but that approach seems more helpful.

The steps include choosing two line segments from the image data and then finding the angle of intersection and its coordinates. Probably redundant, since the OP wants to solve the problem manually, but it does in fact solve the problem described if I'm not terribly mistaken. The OP could stop at "Step 7: Find the Angle of Intersection" since the remaining two steps consist of finding the coordinates and plotting the results, which may not be required.