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# polyxpoly

Intersection points for lines or polygon edges

## Syntax

[xi,yi] = polyxpoly(x1,y1,x2,y2)
[xi,yi,ii] = polyxpoly(...)
[xi,yi] = polyxpoly(...,'unique')

## Description

[xi,yi] = polyxpoly(x1,y1,x2,y2) returns the intersection points of two polylines in a planar, Cartesian system. x1 and y1 are vectors containing the x- and y-coordinates of the vertices in the first polyline, and x2 and y2 contain the vertices in the second polyline. The output variables, xi and yi, are column vectors containing the x- and y-coordinates of each point at which a segment of the first polyline intersects a segment of the second. In the case of overlapping, collinear segments, the intersection is actually a line segment rather than a point, and both endpoints are included in xi, yi.

[xi,yi,ii] = polyxpoly(...) returns a two-column array of line segment indices corresponding to the intersection points. The k-th row of ii indicates which polyline segments give rise to the intersection point xi(k), yi(k). To remember how these indices work, just think of segments and vertices as fence sections and posts. The i-th fence section connects the i-th post to the (i+1)-th post. In general, letting i and j denote the scalar values comprised by the k-th row of ii, the intersection indicated by that row occurs where the i-th segment of the first polyline intersects the j-th segment of the second polyline. But when an intersection falls precisely on a vertex of the first polyline, then i is the index of that vertex. Likewise with the second polyline and the index j. In the case of an intersection at the i-th vertex of the first line, for example, xi(k) equals x1(i) and yi(k) equals y1(i). In the case of intersections between vertices, i and j can be interpreted as follows: the segment connecting x1(i), y1(i) to x1(i+1), y1(i+1) intersects the segment connecting x2(j), y2(j) to x2(j+1), y2(j+1) at the point xi(k), yi(k).

[xi,yi] = polyxpoly(...,'unique') filters out duplicate intersections, which may result if the input polylines are self-intersecting.

## Examples

Use the polyxpoly function to find the intersection points between a rectangle and a two-part polyline.

% Define and fill a rectangular area in the plane
xlimit = [3 13];
ylimit = [2  8];
xbox = xlimit([1 1 2 2 1]);
ybox = ylimit([1 2 2 1 1]);
mapshow(xbox,ybox,'DisplayType','polygon','LineStyle','none')

% Define and display a two-part polyline
x = [0 6  4  8 8 10 14 10 14 NaN 4 4 6 9 15];
y = [4 6 10 11 7  6 10 10  6 NaN 0 3 4 3  6];
mapshow(x,y,'Marker','+')

% Intersect the polyline with the rectangle
[xi, yi] = polyxpoly(x, y, xbox, ybox);
mapshow(xi,yi,'DisplayType','point','Marker','o')

% Display the intersection points; note that the point (12, 8)
% appears twice because of a self-intersection near the end of
% the first part of the polyline.
[xi yi]
ans =

3     5
5     8
8     8
12     8
12     8
13     7
4     2
13     5
% You could suppress this duplicate point by using the 'unique'
% option.
[xi, yi] = polyxpoly(x, y, xbox, ybox, 'unique');
[xi yi]
ans =

3     5
5     8
8     8
12     8
13     7
4     2
13     5

Use the polyxpoly function to find the intersection points between the state of California and a small circle.

'UseGeoCoords', true,...
'Selector',{@(name) strcmpi(name,'California'), 'Name'});
usamap('california')
geoshow(california, 'FaceColor', 'none')

lat0 = 37; lon0 = -122; rad = 500;
[latc, lonc] = scircle1(lat0, lon0, km2deg(rad));
plotm(lat0, lon0, 'r*')
plotm(latc, lonc, 'r')

[loni, lati] = polyxpoly(lonc, latc, ...
california.Lon', california.Lat');
plotm(lati, loni, 'bo')