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Curve intersections

version (2.47 KB) by NS
Fast computation of intersections and self-intersections of curves using vectorization.


Updated 24 Sep 2010

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While a few other functions already exist in FEX that compute the intersection points of curves, this short piece of code was written with speed being the highest priority. No loops are used throughout, taking full advantage of MATLAB's vectorization capabilities

I welcome any comments, suggestions, bug reports etc.


INTERX Intersection of curves
P = INTERX(L1,L2) returns the intersection points of two curves L1
and L2. The curves L1,L2 can be either closed or open and are described
by two-row-matrices, where each row contains its x- and y- coordinates.
The intersection of groups of curves (e.g. contour lines, multiply
connected regions etc) can also be computed by separating them with a
column of NaNs as for example

L = [x11 x12 x13 ... NaN x21 x22 x23 ...;
y11 y12 y13 ... NaN y21 y22 y23 ...]

P has the same structure as L1 and L2, and its rows correspond to the
x- and y- coordinates of the intersection points of L1 and L2. If no
intersections are found, the returned P is empty.

P = INTERX(L1) returns the self-intersection points of L1. To keep
the code simple, the points at which the curve is tangent to itself are
not included. P = INTERX(L1,L1) returns all the points of the curve
together with any self-intersection points.

t = linspace(0,2*pi);
r1 = sin(4*t)+2; x1 = r1.*cos(t); y1 = r1.*sin(t);
r2 = sin(8*t)+2; x2 = r2.*cos(t); y2 = r2.*sin(t);
P = InterX([x1;y1],[x2;y2]);

Cite As

NS (2021). Curve intersections (, MATLAB Central File Exchange. Retrieved .

Comments and Ratings (128)

Maria Costa

Debarshee Ghosh

Works really well. Saved me a lot of time !



In addition to the comment by Patrik Forssén:

Change line 68 to:

if isempty(i),P = zeros(2,0);i=[];j=[];return; end;

so the modified function still works, even when the two curves are not intersecting


nicole fevola

nguyen nhan tinh

thanks NS.
Could you help me to understand the C1 and C2 in detail or the logical theory.

Ali Moshrefzadeh

Moayed Jumah

Why can't I use this function to find the intersection point of any two straight lines?

Patrik Forssén

Great submission! The one thing missing is the indices of the intersecting segments. However, this is easily fixed with some minor code modifications. Change the function header to,
[P, i, j] = InterXMod(L1, varargin)
and change row 76 to,
[P, ind] = unique([dx2(j).*S1(i) - dx1(i).*S2(j), ...
dy2(j).*S1(i) - dy1(i).*S2(j)]./[L L],'rows','stable');
P = P'; i = i(ind); j = j(ind);
Then you should have the intersecting segment indices for the first supplied polyline(s) in the output variable “i" and the ones for the second supplied polyline(s) in the output variable “j”.

Jesus Portilla

Marcus Buker

Adolfo Veliz

Valeri Disko

Siddhanth Shetty

Gayan De Silva

Works perfectly. Thank you very much.

Chuck Hayden

Guanting SU

Yang Lu

Hi NS,
I tried to translate it to fortran. Could you help me to understand the C1 and C2 in detail or the logical theory behind it?

Salman Khan

Great Function. I used it to automate reading of x-axis where my curves intersect a horizontal line. It worked flawlessly after following instructions for INTERX(L1,L2).


Highly resourceful function.


Thanks for this very useful function!
I need to determine only the intersection points that are above a set of decreasing line functions (i.e., with negative slope). How can I change this code to reach these points?

John Michael Hickey

Ned Qi

Joseph Lilek

Simon Chou

Mustapha El Hamdaoui

A very useful function. Thanks!

Tobias Kistler

Leopoldo Brasil

Maura E. Monville

I tried your code on finding if a square intersects a closed curve (polygonal). I get the following error:
Error using bsxfun
Non-singleton dimensions of the two input arrays must match each other.
Error in InterX (line 63)
C1 = feval(hF,D(bsxfun(@times,dx1,y2)-bsxfun(@times,dy1,x2),S1),0);
Error in Pixelize_Collimator_Aperture_v2 (line 107)
P = InterX([X;Y],[sqx;sqy]);

The polygonal is a closed curve made up of 42 points.
The square is a closed square curve made up of 5 points
Must the two curves have the same number of points? That is not specified in your description.
I can send my curves and squares if you like.
Thank you


InterX function returns empty list of poits, if self-intersection point is in input.
For example, InterX returns empty list, but self-intersection point is [1 1] in this code:
x=[0 1 2 1 1];
y=[1 1 1 2 0];
figure; hold('on');
for i=1:length(x)-1; plot(x([i i+1]),y([i i+1]),'LineWidth',i); end;

William Reed

no such command in R2019a

Jules Ray

It work but is not that fast as promised. After a test comparing against the "intersections" function give quite similar results:

InterX = Elapsed time is 10.502146 seconds.
intersections = Elapsed time is 10.567976 seconds.

both compared using the same two curves formed each by 1200 points.

Dominik Mattioli

saleem khan

Hi, I am new to Matlab, I am getting this error while trying to run this code, someone please help me.

parse error near line 176 of file C:\Users\Omar-PC\Desktop\Algo Problem\data Set\octave\code_samp

nested functions not implemented in this context

>>> function u = D(x,y)

any help will be Much appreciated, Thank you!

Kathryn Feller

Hi, i'm trying to calculate the intersection between a vertical line and a curve described by two vectors of data. This function seems to work, but its giving me two sets of coordinates, and there is only one possible intersection give the information I entered. Perhaps this is related to the fact that the curve is an array of data points and not a mathematical curve? Because the curve is from a list of entered datapoints, the x value I am trying to determine a y value for is between two points. Any help deciphering this output would be tremendous.

Amanda Bernardy

my matlab tells me "InterX" is an undefined function

Pim van Mil

Yang Lu

Hi NS,

The code works very good. I find the value of the intersections but now I want to find the indexes. When I use the find function, it does not find the right indexes of the intersections point but the empty value. Could you help me to find the way to find the right indexes of the intersections point?

Thanks in advance!




hello, I keep getting error as below.

Error using symengine (line 59)
Array sizes must match.

Error in sym/privBinaryOp (line 903)
Csym = mupadmex(op,args{1}.s, args{2}.s,

Error in .* (line 238)
X = privBinaryOp(A, B, 'symobj::zip', '_mult');

Error in InterX (line 60)
S1 = dx1.*y1(1:end-1) - dy1.*x1(1:end-1);

Error in solve_equation (line 28)
P = InterX(L1,L2) ;

Samuel Kelsey

David Stolnis

Does not work. Have 2 lines that indeed cross but I get no Xing info.... empty array! Looks like others had luck though.


Ben Mohankumar

Great script - works perfectly. I have a question - I have several intersections to compute and I was wondering how easy it would be to vectorise the calculation to improve code performance? Cheers, Ben

Michael Redling


Leopoldo Brasil

yenting chen

Syed Ammar Abbas

It is giving empty intersection point for these lines when they actually do interestect.
L1 = [1152.3 1069.6;
559.6 656.1]
L2 = [1570.3 1564.1;
884.1 937.2]

Ozan Celik

Hi, the code works very good. Thanks!

I have a question, is it possible to use this function to find the intersection of multiple curves in a vectorized fashion? Say we have a set of curves which are stored as the columns of matrix A and we have another set of curves which are stored as the columns of matrix B. Is it possible to find the intersection of A(:,i) with B(:,i) without using for loops?


Works perfect! Thanks so much for the code.

Mahta Mazloumi

It worked very well! thank you

Nedim Gökhan Aydin

Pretty fast, simple and useful.



curious what is 'D' in the given function? keeps giving error for the undefined variable D

Pushkar Mahajan

Great. Simple to use. Thank you very much!!!

Randika Vithanage

Cannot handle if the lines interests beyond the given points


Works great, would however be nice to know who the intersection is with when input contains multiple cureves

Ali Hassanzadeh

Ryan Meekins

This code works great, however, this code says that lines intersect even if they only overlap. For example:
xA = [0 0 1]; yA = [0 1 1]; LineA = [xA;yA];
xB = [0 0 .25 .25 0 0 .25 .25 .5 .5 1]; yB = [0 .5 .5 .75 .75 1 1 .75 .75 1 1]; LineB = [xB;yB];
plot(xA,yA, xB,yB);
P = InterX(LineA,LineB)

P =

0 0 0 0.2500 0.5000
0.5000 0.7500 1.0000 1.0000 1.0000

I'm looking for a code that would say the lines only intersect if they actually cross. Any ideas?

Michael Mann

Joshua Collins

Glenn Gomes

Alexander Herman

r p

David Onywoki

@JonathanCamilleri what was the problem. I am getting the same error

Cheng Zeng


Laura Berkowitz

Bing Sen Tay

Samuel Petitjean

Emrecan Yener

Awesome function, thanks!

Yuanjun Zhao

Great code, help me a lot! Thx!!!

Syed Hossain

Erik Manders

Great piece of Code! Thanks a lot!

Abrar Habib

Jonathan Camilleri

ignore my previous comment. realised what was wrong.

Jonathan Camilleri

hi, I am trying to use this function but i keep getting the following error:

Error in InterX (line 60)
S1 = dx1.*y1(1:end-1) - dy1.*x1(1:end-1);

Error in data_vis (line 66)
inter = InterX(f1, f2);

however, my curves, f1 and f2 are both the same size, 25525 x 1 double.#

any suggestions?





some problem in the Example:
t = 0:pi/200:2*pi;
r1 = 2; x1 = r1.*cos(t)+2; y1 = r1.*sin(1*t);
x2 = t; y2 = t-3;
P = InterX([x1;y1],[x2;y2]);
axis equal

Haidy Loh

Asif Arain

Is it possible to obtain intersection points along with a logical vector that indicates corresponding curves/lines involved in the intersection points? Later, I want to assign weights to the intersection points based on the lines/curves.
Any help is much appreciated. Thanks!

Far Bod


More detailed comment to come.

Tamás Balogh

Harish Babu Kankanala

Hojoon Choo

This is exactly what I was looking for. Thanks!

E. Cheynet


Brian Kim

by the way, i wonder whether this code is adequated at the non-linear.

farnaz hajipour

how i can download this file?


Comment on my comment (see below): When you are interested not only in the points but also in the indices then I now think, that it's the best not to use 'unique' or 'uniquetol' at all. (This is especially important when you are investigating self-intersection. Then one intersection point causes multiple entries in P at different indices in i and j.)
Otherwise you might loose the correct assignment from points to indices i and j.


Works well given that both of your plotted data series matrices are the same size. I think this is the problem the "it does not work" guy ran into. For my task at hand, my two data series are not the same size and so it gives a horzcat error. That would be a nice way to update the code if it could also work with two matrices/data series that aren't the same size. :)


Very good and very fast.
I think I found a minor bug which is relevant only if you are interested in the indices i and j, where the intersection is expected.
I suggest to replace line 76 and 77 by:
[P,ind]=uniquetol([dx2(j).*S1(i) - dx1(i).*S2(j),dy2(j).*S1(i) - dy1(i).*S2(j)]./[L L],eps,'ByRows',true);

In this way the sorting (coming from unique or better uniquetol) is assigned also to the indices. Additionally I increased the indeces by 1, but this is more or less a matter of taste.


Pouriya Zarbafian

Tillmann Stübler

Thank you for this fantastic piece of code.


Yongshou Liang

Yongshou Liang

Hi, NS. Thanks for this excellent code. It runs really fast. However, I also have the problem as Emil's. When the input points are vectors more than 20000 columns, it's out of memory. The version of my matlab is 2014a (64 bits on windows). Could you send me that modified code or give me some suggestion? My Email is: Thanks a lot.

Eric V

Thanks. That's great. Works fine.

Abel Brown


It does not work.

Chad Greene

This function has proven very helpful for me. Thank you for sharing, and thank you for writing such a nice, neat code.


THANK YOU SO MUCH. Great algorithm and works perfectly and super fast.



it would be helpful to show how to get the (all) values of "t", at which the intersections occur.

Namely, two values of the parameter "t" correspond to each self-intersection.


it would be helpful to show how to get the (all) values of "t", at which the intersections occur.

Matthew Arthington

I've searched for a fast intersections function and I can't find one faster than this.

I'm a bit surprised that repeated subfunction calls are as fast as they are for large curves. Vectorising some of the operations speeds up the function, but only for smaller curves.


To Emil: Thank you for your comment. Unfortunately there is not much that can be done about that, unless one uses for-loops, which are likely to delay the execution time considerably.

To make the execution time as fast as possible, I need to create N x M matrices, where N and M are the number of segments in each curve.

Perhaps you can try some other contribution on File Exchange that does not utilize vectorization, but it is likely to be very slow (there should be about 4e10 tests for intersections for vectors of that size!)

Alternatively, contact me to send you a modification to my code to test with your data. However, I cannot guarantee accuracy of the results, as I did not do extensive tests to it, but it seems to work.

Emil Olsen

I frequently use this function to analyze threshold crossings in locomotion data and I love it! However since using it in 2012b on a 64bit windows machine I keep running out of memory even if just testing 20000 column vectors. Any ideas for a fix for this?

Pie Mes

Really good, fast and versatile piece of code. BIG UPS.


Please ignore the above comment. I have found a way and using your code.

Best regards.



Thanks for the share.
In my case, i have two lines in the point data form. For example, I have 3 columns of the data Y,X1 and X2. first line L1 is obtained by plotting and connecting (X1,Y) and the second line L2 is obtained by plotting and connecting (X2,Y).
Can you please explain me how can i convert this point data to "two-row-matrices" as described here.



To Aviator: yes, it should handle these cases.


Does this code accommodate intersection points if a segment of the graph is vertical?


Thanks a lot for this function!!!Works correct where other intersection functions didn't!!

Jeroen van Nugteren


ayesha sohail

really very helpful..... many thanks

Zhang Yanxiang


Does exactly what it says. No problems.


useful, efficient, and fast

MATLAB Release Compatibility
Created with R2008a
Compatible with any release
Platform Compatibility
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