Search code examples
matlabdifferential-equationsnumerical-integration

Finding a point on a surface given an arc length and direction/angle in Matlab

I need to find a point on a surface (given an angle relative to a starting point) where the arc length is a given value using Matlab.

Say I have a high order surface where z=f(x,y) which was been fitted from sampling points using Matlabs fit function. If I have a starting point, say a = (x_0, y_0, f(x_0,y_0)) and want to know the coordinate of a point along that surface at a user defined angle theta away in the xy plane so that the distance covered over the surface is a given value, e.g. 10mm.

I assume that what I need to do is solve this equation for the value of b, given we know a, s and function defining surface. But I'm unsure how to write this in Matlab. I'm assuming I need to use the solve function in Matlab.

Any help on how to write this in Matlab and in the most efficent form would be greatly appreciated!

Solution

  • here it is an example assuming dx=1,dy=1, incorporating arbitrary x and y step sizes should not be hard

    % //I am assuming here that you know how to get your Z
z=peaks(60); 
% //start point
spoint=[30,30];
% //user given angle
angle=pi/4;
% // distance you want
distance=10;
%// this is the furthes the poitn can be
endpoint=[spoint(1)+distance*cos(angle) spoint(2)+distance*sin(angle)]; 

%// we will need to discretize, so choose your "accuracy"
npoints=100;
%//compute the path integral over the line defined by startpoitn and endpoint
[cx,cy,cz]=improfile(z,[spoint(1) endpoint(1)],[spoint(2) endpoint(2)],npoints);

% // this computes distances between adjacent points and then computes the cumulative sum
dcx=diff(cx);
dcy=diff(cy);
dcz=diff(cz);

totaldist=cumsum(sqrt(dcx.^2+dcy.^2+dcz.^2));
%// here it is! the last index before it gets to the desired distance
ind=find(totaldist<distance,1,'last');

    enter image description here

    Visualization code

    imagesc(z);axis xy;colormap gray
hold on;
plot(spoint(1),spoint(2),'r.','markersize',10)
plot(endpoint(1),endpoint(2),'r*','markersize',5)
plot([spoint(1) endpoint(1)],[spoint(2) endpoint(2)],'b')
plot(cx(ind),cx(ind),'g*','markersize',10)