Calculate area between irregular unknown curve and straight line

Hi,

I have a series of data that are trajectories of cursor movements. Below are some of the trajectories that I have.

I need to find the area between each trajectory and its ideal straight line trajectory (a straight line from the first point to the last point). Both the mouse trajectories and straight line have been normalized to have 101 sets of coordinates each. The trajectories always start from (0, 0) and go to the top left or right of the screen. I have attached the coordinates of the 2 trajectories and lines in an Excel file.

My question is: How do I find the area between the curves and the straight lines? I do not have the functions of the curves, so I can't use any of the integral functions. I have tried to use trapz with the y-coordinate as the independent variable, i.e.,

trapz(traject1_y, traject1_x) - trapz(str_line1_y, str_line1_x)

While it returns me a value, I'm not really sure how that value is derived; the Matlab documentation on trapz does not provide a lot of details. Moreover, my understanding is that the functions for integration require the independent variable to increase monotonically, and that isn't the case for me, as you can see from the plots that I have attached. Is there another way to find the area?

Thanks in advance!

[KSSV]

YOu may also check with polyarea.

[elin]

Thanks for the quick response. I have another question: does polyarea work on intersecting polygons?

I tried using polyarea to calculate the area enclosed by the trajectory and the straight line in the top image attached. This is my code:

traject1_x = -traject1_x   %flipping the figure along the y-axis to have positive values for x
str_line1_x = -str_line1_x;
flipped_y = fliplr(str_line1_y);       %flipping the elements of the straight line so the last coordinate of the trajectory joins the first coordinate of the straight line
flipped_x = fliplr(str_line1_x);
x_area = [traject1_x flipped_x];
y_area = [traject1_y flipped_y];
A = polyarea(x_area, y_area)

The area returned was 0.3013. Just to make sure, is the area that was calculated the one in the image attached, i.e., the sum of the positive area and negative area?

Thanks again!

[Rik]

The area returned by polyarea is the area enclosed by you curve, as you can read in the documentation. I can't open your image (yet?), but polyarea with not recognize an intersecting line to mark an area as negative area.

[elin]

Sorry I didn't realise the image was not uploaded; I had apparently exceeded my daily limit of 10 uploads even though I have only uploaded 3 items to the thread so far. I'll have to wait for 24 hrs before I can upload the image.

I managed to open the Matlab function itself for polyarea and it says "The polygon edges must not intersect. If they do, POLYAREA returns the absolute value of the difference between the clockwise encircled areas and the counterclockwise encircled areas." So looking at the first image that I uploaded in the original post that is flipped along the y-axis, does this mean that the value I got from polyarea was the area of the bigger part minus the area of the smaller part?

[David J. Mack]

Hey Elin, are you stuck with the area approach? Otherwise you could simply compare the path lengths of the two curves to get a measure of trajectory curvature... Greetings, David

[elin]

Hey David, I think I figured it out now. Thanks!

[Stijn]

Hi Elin,

I'm coping with the same problem. How did you solve the intersection and the small area being negative -issue?

[elin]

Hi Stijn,

Sorry for the late reply. For me that was not a problem, because I wanted the small area to be calculated as negative. So I just took the net area for that figure. If you want to have the small area calculated as positive, then you can try polyarea. If I'm not wrong, polyarea takes the absolute value for the figure, so you always get a positive value for the figure. I'm just not sure if it counts the negative area as negative or positive in its calculation. Hope this helps!

[Stijn]

Hi Elin,

Thanks for your response. I indeed found out that Polyarea retracted the negative values, so it didn't suit my needs. However, I found and modified a solution proposed by Ahmed Owaid. It's sort of a 'hack', but it works :)

[Abdallah Abu Aisha]

Hi Elin and Stijn, did you, by any chance, find a more "practical" way to find the sum of all areas between the two paths ( regardless positive or negative)? Since I am working on something very similar but for huge number of trips, so the solution proposed by Owaid doesn't work for my case. I have been struggling finding a function/code to solve this issue but all of them return only the "net" area. Best Regards, Abdallah

[elin]

Hi Abdallah,

Unfortunately I did not manage to find such a method. But perhaps someone else reading this thread has done so?

Good luck!

Answer 1

[KSSV]

trapz(x,y) should work...

[elin]

Yup I realized that trapz and polyarea basically implements the same equation, although the way I was doing it like in the original thread would give a negative value. I now used trapz with x_area and y_area as the input arguments and it gave me the same answer as polyarea. Thanks!