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Retrieve orthogonal coordinate curves from interpolated surface

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Dimitrii Nikolaev
Dimitrii Nikolaev on 23 Jul 2021
Commented: Dimitrii Nikolaev on 11 Aug 2021
I'm trying to generate a pattern, which is well definable in cartesian coordinates on some curved survace, while longitudinal characteristics of the pattern would be preserved. (One could think for example on engraving a text on some surface)
EXAMPLE (straight line l=1 on the surface):
Assume a generic Spline-fitted surface
(for example, surface fit from spap2 documentation):
%% surface fit example spap2
x = -2:.2:2;
y = -1:.25:1;
[xx, yy] = ndgrid(x,y);
z = exp(-(xx.^2+yy.^2));
sp = spap2({augknt([-2:2],3),augknt([-2:2],3)},8,{x,y},z);
Result of the surface fit with spap2 allows us now to interpolate the cartesian z coordinate on the surface by calling fnval.
Line on this surface
Now let's define a straight line [0 0.5] -> [1 0.5] with length of 1 in cartesian xy coordinates, interpolate the z-coordinate and plot it in the same figure
%% plot line on the surface
hold on
lin_x = linspace(0,1,30);
lin_y = linspace(.5,.5,30);
lin_z = fnval(sp,[lin_x;lin_y]);
Both code section above results in following:
This line have a length of 1 only if z0->z1 is a straight line.
The possibility to define some line with a specific length in such plane woud be to use curvilinear coordinates where instead of straight orthogonal x and y vectors, (orthogonal) curves x(u) and y(v) are used, where u and v represent the distance along the x and y curve
how to use curvilinear coordinates properly under these circumstances?
What would be the best way to retrieve a set of orthogonal coordinate curves for describing coordinates in this plane in a curvilinear manner?
An approach I'm currently thinking about would be:
  1. Consider some origin point O on the plane
  2. Consider direction of the first coordinate curve x(u) in cartesian coordinates
  3. Sample a set of points in cartesian X0-plane from surface. fit a spline or use directly as linear interpolated curve
  4. Interpolate a point x(1) along this curve exactly 1 unit away from origin e.g. by using interparc
  5. Estimate the normal vector n to the surface on point x(1)
  6. compute orthogonal direction for y(v) as cross product from x(u) and n
  7. Sample a set of points in cartesian plane (where the curve O->y(v) is located) from surface. fit a spline or use directly as linear interpolated curve
  8. repeat Steps 4-7 for points between x(0) y(0) and x(1) y(1) for z-interpolation and plotting
However i'm not sure if I'm reinventing a bicycle rigth now and if there is some more elegant solution for 3D curvilinear coordinates in MATLAB.
Thanks for any Valuable input and considaretions in Advance!
Dimitrii Nikolaev
Dimitrii Nikolaev on 11 Aug 2021
@darova: well, for example I would like to describe two orthogonal lines with letgth of 1 unit along the surface and a circle, with opposite points on the surface exactly 1 unit far away from each other. Like that:
Please note: Of course all lengths slightly differ from 1, as I'm am unable to achieve my goal so far.
You can reproduce these plots using following code:
%% plot X-line on the surface
hold on
lin_x = linspace(0,1,30);
lin_y = linspace(0,0,30);
lin_z = fnval(sp,[lin_x;lin_y]);
%% plot Y-line on the surface
hold on
lin_x = linspace(0,0,30);
lin_y = linspace(0,1,30);
lin_z = fnval(sp,[lin_x;lin_y]);
%% plot circle of d = 1 with centre in .5 .5
r =.5 ;alpha = linspace(0,2*pi,30);
circ_x = r*cos(alpha)+.5;
circ_y = r*sin(alpha)+.5;
circ_z = fnval(sp,[circ_x;circ_y]);

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