MATLAB中3D轨迹曲线的切片

Question

I have numerical data that corresponds to a particle moving in three dimensions, sampled at n equally spaced times t = [1,2,3,4,5,...], resulting in three coordinate arrays, eg, for n=5:

x = [1,2,3,4,5]
y = [1,5,6,7,10]
z = [2,3,8,9,10]

I have been able to find a decent interpolation using cscvn, but I am now at a loss as to how I would find the x and y coordinates that correspond to a fixed value of z, ie, a constant z slice.

Answer 1

You can extract the individual cubics within the curve output by cscvn, then examine each one for if and where it crosses your chosen Z value within that piece's boundary.

x = randi(10,[1 5]);
y = randi(10,[1 5]);
z = randi(10,[1 5]);
zChosen = 5;

curve = cscvn([x;y;z]);

parametricCrossingPoints = struct('crossingPoints',[],'Xs',[],'Ys',[],'Zs',[]);
parametricCrossingPoints = repmat(parametricCrossingPoints,curve.pieces,1);
% for each peice of the parametric curve, find the points where it crosses
% your desired z
for i = 1:size(curve.coefs,1)/3
    % Find where parametric piece crosses zChosen
    pPoints = roots(curve.coefs(i*3,:)'-[0;0;0;zChosen]);
    % Filter out imaginary roots
    pPoints = real(pPoints(imag(pPoints) == 0));
    % Filter for points within the piece's boundary
    pPoints(pPoints < 0 | pPoints > curve.breaks(i+1)-curve.breaks(i)) = [];
    % Append points to list
    parametricCrossingPoints(i).crossingPoints = pPoints + curve.breaks(i);
end

% Get x, y, and z of each crossing point
for i = 1:numel(parametricCrossingPoints)
    xyzOfCrossingPoints = fnval(curve,parametricCrossingPoints(i).crossingPoints);
    parametricCrossingPoints(i).Xs = xyzOfCrossingPoints(1,:);
    parametricCrossingPoints(i).Ys = xyzOfCrossingPoints(2,:);
    parametricCrossingPoints(i).Zs = xyzOfCrossingPoints(3,:); % Should all equal zChosen
end

% Collate points
Xs = [parametricCrossingPoints.Xs];
Ys = [parametricCrossingPoints.Ys];
Zs = [parametricCrossingPoints.Zs];

% Sometimes breakpoints aren't chosen, add them manually if necessary
xsToAdd = x(z == zChosen);
ysToAdd = y(z == zChosen);
for i = 1:numel(xsToAdd)
    if ~any(Ys(Xs == xsToAdd(i)) == ysToAdd(i))
        Xs(end+1) = xsToAdd(i);
        Ys(end+1) = ysToAdd(i);
        Zs(end+1) = zChosen;
    end
end

% Plot results
fnplt(curve);
hold on
plot3(x,y,z,'o')
patch([min(x)-1,max(x)+1,max(x)+1,min(x)-1],...
    [min(y)-1,min(y)-1,max(y)+1,max(y)+1],...
    [zChosen,zChosen,zChosen,zChosen], ...
    'green',...
    'FaceAlpha',0.5)
plot3(Xs,Ys,Zs,'*')
hold off

There's some funky calculation to do around the parametric independent parameter (which doesn't match your time = t) for each piece of the curve and the specified breakpoints, but other than that it's pretty straightforward.