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Refined the algorithm and added examples

Hi. This solver is great, it's very stable and works in cases where I can't figure how it do things right. In the attached, I just clean the code a little, but the application it's entirely the same.

DistMod 2.sm (47 KiB) downloaded 68 time(s).
DistMod 2.pdf (167 KiB) downloaded 67 time(s).

Best regards.
Alvaro.

Thank you very much, Alvaro, for modifying the program. In your version I reduced the number
of arguments in the Dist and al_nleqsolve functions. Your program began to work much faster.

DistRazonar.sm (55 KiB) downloaded 79 time(s).

Hi Ber. Added Lagrange multipliers for refine the last nearest point in the curve. this enables to decrease the number of steps, up to a factor of ten.

I guess that the al_nleqsolve calls could be implemented as Draghilev method too.

DistBer.sm (53 KiB) downloaded 76 time(s).
DistBer.pdf (181 KiB) downloaded 68 time(s).

Best regards.
Alvaro.

Wrote

I guess that the al_nleqsolve calls could be implemented as Draghilev method too.

Is that what you mean ?

Contour_10 Draghilev [Electron].sm (25 KiB) downloaded 77 time(s).

Wrote

Wrote

I guess that the al_nleqsolve calls could be implemented as Draghilev method too.

Is that what you mean ?

Contour_10 Draghilev [Electron].sm (25 KiB) downloaded 77 time(s).

Hi Jean. Yes, that, and the use of Dragilev (without h, I forgot that) method for solve nonlinear equations, as uni teach sometime ago.

Best regards.
Alvaro.

Wrote

Hi Jean. Yes, that, and the use of Dragilev (without h, I forgot that) method for solve nonlinear equations, as uni teach sometime ago.

From recollection, original Uni [Viacheslav]
further exploited by Ber7 [smart cat !]
Buenos dias, Amigo.

Finding Distance of a Point from Surface
The difference from the program distance to the curve in only one expression:
in the top line instead of i: = (1 ... 2)) we have i: = (1 ... 3).
A is a given point, a is its projection to the surface, Аa (k) is the length of the segment Аa (k)
to the surface, the h-step of decreasing the distance, a (k) is the intermediate coordinate of
the point touch.The red line is the normal, the green is the trajectory of the movement of the
point a (k) from the initial the position of a (1) to the final a.I borrowed the trajectory
demonstration from Razonar,to whom I express my gratitude.

Dist3d.sm (63 KiB) downloaded 80 time(s).

You may have interest in that stuff:

[url=https://www.geometrictools.com/Documentation/DocUpdateHistory.pdf]

Many thanks for the link.I will study this material.
https://www.geometrictools.com/Documentation/Documentation.html

Wrote

Finding Distance of a Point from Surface
The difference from the program distance to the curve in only one expression:
in the top line instead of i: = (1 ... 2)) we have i: = (1 ... 3).

Hi. Attached version have some reformulations, but is essentially the same, just added Lagrange multipliers for refine last point and join 3D and 2D in only one procedure. Maybe don't works in all cases, I guess that because use less points, but is more pedagogical from my point of view.

Dist.sm (172 KiB) downloaded 68 time(s).
Dist.pdf (500 KiB) downloaded 72 time(s).

Best regards.
Alvaro.

Thank you, Razonar, for improving the program. I would like to
have a criterion to verify the accuracy of our calculations. For example,
you can use the vector product of unit vectors dist and normals,
whose modulus with perfect accuracy will be equal to to zero.
he magnitude of the deviation from zero characterizes the accuracy.
Agree Are you with this criterion?

Cross.sm (65 KiB) downloaded 74 time(s).

Wrote

Thank you, Razonar, for improving the program. I would like to
have a criterion to verify the accuracy of our calculations. For example,
you can use the vector product of unit vectors dist and normals,
whose modulus with perfect accuracy will be equal to to zero.
he magnitude of the deviation from zero characterizes the accuracy.
Agree Are you with this criterion?

Cross.sm (65 KiB) downloaded 74 time(s).

Hi Ber. Thanks for say "our" calculations, but actually are yours. I just make some few modifications. Some general notes: It's a numerical procedure, so, guess values are part of the solution. Because of that, and the fact that you take a random external point I guess that my major modification can't get the global solution, and sometimes get only local ones. Each curve or surface need an specific general expression for the guess value in such case.

This is a small modification of your implementation, with some notes:
Cross.sm (73 KiB) downloaded 72 time(s).

This is the modified implementation, and shows that the introduction of the Lagrangian multipliers for refine the last point get a very small cross third component value in all the cases where can find a solution.
Dist.sm (190 KiB) downloaded 70 time(s).

This shows the importance of the initial guess value for seek the global solution
Dist_InitialGuess.sm (71 KiB) downloaded 69 time(s).

So, answering your question, my conclusion is: Yes, you can use cross to show that, in 2D case the circle is locally tangent to the curve and, for the 3D case, a sphere is. But guess that for construction of your algorithm can't be used as a stop criteria because in your setup you need to move the point along the curve for seek the better approach.

But what I see is that I can find only local solutions, but not the global one. For that, given a random external point A to a curve or a surface you must to get the initial guess xo as a function of the given function and the point A, that's specific for each function.

Dragilev method can help here for finding the global distance because with that you know that you have a curve parametrization, so guess that only need to know in which direction you need to move, and estimate for how many 'time' you need to do that. Next, you use your algorithm and refine the last point with the Laplace multiplier to ensure that the circle centered in A is tangent to the curve.

But for the 3D case I don't see any help here, because with two parameters you lost the sort property of reals: plane points can't be sorted.

Hope that this notes helps.

Best regards.
Alvaro.

“Finding Distance of a Point from curve or surface” - not far the best application of the Draghilev method.
An example of the projection of points from one surface to another using the optimization methods in Maple.

[albumimg]1589[/albumimg]

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“Finding Distance of a Point from curve or surface”

If Draghilev method wouldn't have been discovered, it could be solved otherwise,
by more conventional method ... like in the attached
1. Distance between user random triplets pairs.
2. Specified "user random" from one pair triplet [surface/solid]
3. Direction cosines plane [normal to solid] as one pair triplet
the 2nd triplet pair from solid.
Easy but not included: distance between helix/random-Brownian.
I'm lost in that thread, further latest SS don't read previous stable SS.
If Mathcad 11 and earlier can do w/o Draghilev, Smath earlier as well.

Cheers ... Jean

Vector Maths Direction Cosine [Dist-in-Space] Copy.sm (82 KiB) downloaded 80 time(s).

It seems to me that the Draghilev method is intended for a “large” number of variables with a “small” number of equations.
For example, as in the case of solving the inverse problem of kinematics for a three-link manipulator with a variable length of the working link. This manipulator has 6 degrees of freedom.
[albumimg]1590[/albumimg]

Hi. The distance for 2D parametrizing plane curves by Dragilev method for seek a good guess for using with Laplace multipliers method for minimizing the distance.

Dist2Dragilev.sm (71 KiB) downloaded 73 time(s).
Dist2Dragilev.pdf (374 KiB) downloaded 72 time(s).

Best regards.
Alvaro.

Point Orthogonal Projection onto a flat or spatial Curve

Calculation procedure
1. We will use Draghilev's method
a) to parameterizethe curves and find a good initial guess.
We will use the point a0, found by the minimum distance
from the curve to a given point P.
b) to calculate the unit vector of the tangent T (x) to a
point on the curve.
2 To the system of equations we add an equation that
expresses the conditions of perpendicularityof the tangent
at the point x of the curve and the segment connecting the
points P and x. This means that the inner product of the unit
tangent vector to the curve f (x) and the unit vector Px is
equal to zero. As a result, we obtain the system of equations
3. We solve the system with the initial assumption a0

distantion.sm (124 KiB) downloaded 65 time(s).
Trascidental curves.pdf (394 KiB) downloaded 75 time(s).

Draghilev's method and an approach to modeling platforms (parallel manipulators) with the number of degrees of freedom from 1 to 6.
Forum MaplePrimes 1 and 6

Distance and projection of a point onto a parametric surface

ParSur.sm (74 KiB) downloaded 73 time(s).
ParSur.pdf (619 KiB) downloaded 69 time(s).

Hi. Thanks for the post. Here, my attempt for reproduce your gamma "point of view". I guess that it could be more interesting to change it following the matlab function view(), but I'm not sure if the attached reproduce the matlab's view behavior

https://la.mathworks.com/help/matlab/ref/view.html

https://la.mathworks.com/help/matlab/creating_plots/setting-the-viewpoint-with-azimuth-and-elevation.html?lang=en

Gamma matrix for rotations.sm (74 KiB) downloaded 70 time(s).
Gamma matrix for rotations.pdf (284 KiB) downloaded 70 time(s).

Best regards.
Alvaro.