Intersection of Discretized Curves

Hi,

I have a scatter plot of two curves.

The curves intersect each other and ideally they should have one or more intersection points.

However, as I have discrete points and two curves do not intersect at any of those points, I can not get the point of intersection.

I'm attaching a figure for more clarity.

I believe I will have a set of points here instead a single point.

How do I get those set of points from the scatter plot data or the data points I have now?

Thanks..

[Tony]

The pair of points on the two curves with the smallest Euclidean distance between them will be the intersection point. If you are looking for multiple, then you can put a threshold; e.g. find the pairs of points whose distance is less than 1e-6

[Prasanna Routray]

Hi.. thanks for the reply. How do I do that?

Answer 1

[Star Strider]

It would help to have the data.

TThe approach to these problems is straightforward. In this instance, the independent variable values are different, so the second curve needs to to be interpoalted to the values of the first independent variable (or the reverse of this, depending on what you want to do). After that, it is relatively straightforward to interpolate to find the intersection. (If there are more than one intersection, it will first be necessary to find all the approximate indices of the intersections, and then loop to find the exact values of each intersection.)

Try this —

x1 = linspace(0, 0.06);
y1 = -x1.^2;

x2 = linspace(0, 0.058);
y2 = 1E-3 - x2.^2 * 1.5;

figure
plot(x1, y1, '.')
hold on
plot(x2, y2, '.')
hold off

y2i = interp1(x2, y2, x1, 'pchip', 'extrap');                                      % Values Of 'y2' Interpolated To 'x1' 

intxx = interp1(y1-y2i, x1, 0)                                                      % Y-Coordinate Of Intersection

intxx = 0.0447

intxy = interp1(x1, y1, intxx)                                                      % X-Coordinate Of Intersection

intxy = -0.0020

figure
plot(x1, y1, '.-')
hold on
plot(x1, y2i, '.-')
plot(intxx, intxy, 'cs')
hold off
text(intxx, intxy, sprintf('$Intersection\\ at\\ (%.4f, %.4f) \\rightarrow$',intxx,intxy), 'Horiz','right', 'Interpreter','LaTeX')

This should work with your data, as written. I will of course help with any problems if necessary in adapting this to your data.

.

[Prasanna Routray]

Hi..

Thanks for your answer.

[Prasanna Routray]

I was also wondering about another way this could be tackled.

How about finding a set of points instead of a single exact point.

I'm not very sure about the method but what if the points represent a 2D normal distribution kind of representation? Just curious about that.

[Star Strider]

As always, my pleasure!

I am not certain what you intend by ‘set of points’. Since my code detects and plots the exact intersection, one option for ‘set of points’ could be to consider all the points within a specific Euclidian distance of the intersection, perhaps with the knnsearch function.

Consider something like this (untested):

edist = 1E-3; % Distance Threshold (Limit)

[C1_idx,D1] = knnsearch((x1-intxx), (y1-intxy));

[C2_idx,D2] = knnsearch((x2-intxx), (y2-intxy));

Then, compare ‘D1’ and ‘D2’ separately with ‘edist’ to return a logical vector that can then be used with the index vectors to return the points referenced to them.

.

[Prasanna Routray]

So, I was casually trying to get intersection of curves and tried intersecting two lines usng same script.

But, it gives error about the unique point thing. How do we sort this?

x1 = -100:1:100;

y1 = 5*ones(length(x1),1);

y2 = -100:1:100;

x2 = 5*ones(length(y2),1);

y2i = interp1(x2', y2', x1', 'pchip', 'extrap'); % Values Of 'y2' Interpolated To 'x1'

intxx = interp1(y1-y2i, x1,0) % Y-Coordinate Of Intersection

intxy = interp1(x1, y1, intxx)

the error is expected,

Error using matlab.internal.math.interp1

Sample points must be unique.

How to solve this so that we can use it for all types of curves?

Best,

[Star Strider]

@Prasanna Routray —

It would be helpful to have a plot call or something similar, so I understand the result you want. (I added onee.)

Also, the code is incomplete. For example, the vectors ‘x1Coord’ and ‘y2Coord’ seem to be missing. Also, how would they be plotteed (since I assume they should be)?

Please provide the missing information, and explain what you want to do.

EDIT — (5 Oct 2024 at 16:16)

For two orthogonal lines, one with a zero slope and another with an infinite slope, the eassiest way would bee to use the ismember (or ismembertol) —

x1 = -100:1:100;
y1 = 5*ones(length(x1),1);

y2 = -100:1:100;
x2 = 5*ones(length(y2),1);

Lv = ismembertol(x1, y1, 1E-10);
Nv = find(Lv)

Nv = 106

intxx = x1(Lv)

intxx = 5

intxy = y1(Lv)

intxy = 5

% y2i = interp1(x2', y2', x1', 'pchip', 'extrap'); % Values Of 'y2' Interpolated To 'x1' 
% intxx = interp1(y1-y2i, x1,0)                    % Y-Coordinate Of Intersection
% intxy = interp1(x1, y1, intxx)


figure
plot(x1, y1)
hold on
plot (x2, y2)
plot(intxx, intxy, 'cs',  'MarkerSize',10)
hold off
grid

The typical intersection approaches, such as in my original answer, will not work in this instance. They will only work with finite slopes.

.

[Prasanna Routray]

Hi...

The idea was to plot the two lines, get the point of intersection and plot the intersection point.

yes, the code had some typo. I edited the above one and posted the new code below.

x1 = -100:1:100;

y1 = 5*ones(length(x1),1);

y2 = -100:1:100;

x2 = 5*ones(length(y2),1);

y2i = interp1(x2', y2', x1', 'pchip', 'extrap'); % Values Of 'y2' Interpolated To 'x1'

intxx = interp1(y1-y2i, x1, 0) % Y-Coordinate Of Intersection

intxy = interp1(x1, y1, intxx)

% the following should plot everything if all goes well

figure

plot(x1, y1)

hold on

plot (x2, y2)

plot(intxx, intxy,'or')

hold off

grid

Answer 2

[Bruno Luong]

Use this file exchange https://www.mathworks.com/matlabcentral/fileexchange/22441-curve-intersections

This code is robust, it doesn't break for repeating x or y coordinates.

The curves are assumed to me line segments connectings data points. Intersection coordinates of two line segments that meet are returned, and all of them.

x1 = -100:1:100;
y1 = 5*ones(size(x1));
y2 = -100:1:100;
x2 = 5*ones(size(y2));
X = InterX([x1; y1],[x2; y2]);
plot(x1,y1,'-',x2,y2,'-.',X(1,:),X(2,:),'or')

[Prasanna Routray]

Great. It works. Thanks Bruno..