An example of solving a system of nonlinear equations, when all (real) solutions are found for one initial approximation. To make it easier to understand the program text, a description of the solution method is provided separately in PDF format.

Draghilev’s_method_pdf.pdf

3_EQUATIONS.mw

Using the curve, the projection of which from the space R^4 to R^3 we see in the figure, we find good approximations to the solution. For each solution, the program prints out the norm of the discrepancy of the entire system of equations and shows a simple graph when the additional variable intersects R^3 . At this moment of intersection, the solution of our system is fixed.
System of equations and initial approximation.
 

f1 := x1^2+x2^2-x3^2; 
f2 := (x1-1)^4+x2^4+(x3-2)^4-81; 
f3 := x3-3*tan(.4*x2); 
x01, x02, x03 := 1, 1, 1

Old link to a similar example, where the dimension of the system of equations is 2, and we can see the auxiliary curve completely in R^3.

This is a task from one forum:  “Let's mark an arbitrary point on the circle. Let's draw a segment from this point, perpendicular to the diameter, and draw a circle, the center of which is at this point, and the radius is equal to this segment. Let's mark the intersection point of the segment connecting the intersection points of the circles with the perpendicular segment. Prove that the locus of all such points is an ellipse.”
I wanted to get a picture of a numerically animated "proof" using Maple tools.
МАTH_HЕLP_PLANET.mw
 And in fact, it turned out that AB=2AC, or AC=BC.

The flag of Germany on the strip of the German mathematician August Ferdinand Möbius. Basically, it's just one way to represent flags of a certain type. It seemed that the flag looked good on the Mobius strip.
FLAG.mw

This is another attempt to tell about one way to solve the problem of inverse kinematics of a manipulator.  
We have a flat three-link manipulator. Its movement is determined by changing three angles - these are three control parameters. 1. the first link rotates around the black fixed point, 2. the second link rotates around the extreme movable point of the first link, 3. the third link − around the last point of the second link. These movable points are red. (The order of the links is from thick to thin.) The working point is green. For example, we need it to move along a circle. But the manipulator has one extra mobility (degree of freedom), that is, the problem has an infinite number of solutions. We have the ability to remove this extra degree of freedom mathematically. And this can also be done in an infinite number of ways.
Let us give two examples where the same manipulator performs the same movement of the working point in different ways. In one case the last red point moves in a straight line, and in the other case it moves in an ellipse. The result is the same. In the corresponding program texts, the manipulator model is described by a system of nonlinear equations f1, f2, f3, f4, f5 relative to the coordinates of the ends of the links (very easy to understand). The specific additional connection that takes away one degree of freedom is highlighted in blue. Equation of a circle in red color.
1.mw

2.mw


And as an elective. The same circle was obtained using a spatial 3-link manipulator with 5 degrees of freedom. In the last text, blue and red colors perform the same functions as in the previous texts.
3.mw

 

Circles inscribed between curves can be specified by a system of equations relative to the coordinates of the center of the circle and the coordinates of the tangent points. Such a system can have 5 or 6 equations and 6 variables, which are mentioned above.
In the case of 5 equations, we can immediately obtain an infinite set of solutions by selecting the ones we need from it. 
(See the attached text for more details.)
The 1st equation is responsible for the belonging of the point of tangency to one of the curves.
The 2nd equation is responsible for the belonging of the point of tangency to another curve.
In the 3rd equation, the points of tangency on the curves belong to the inscribed circle.
In the 4th and 5th equations, the condition is satisfied that the tangents to the curves are perpendicular to the radii of the circle at the points of contact.
The 6th equation serves either to find a specific inscribed circle or to find an infinite set of solutions. It is selected based on the type of curves and their mutual arrangement.

In this example, we search for a subset of the solution set using the Draghilev method by solving the first five equations of the system: we inscribe circles in two "angles" formed by the intersection of the exponent and the ellipse.
The text of this example, its solution in the form of a picture,"big" option and pictures of similar examples.
INSCRIBED_CIRCLES.mw


 

Addition 09/01/24, 
One curve for the first two equations in coordinates x1,x2 and x3,x4
f1:= x1^2 - 2.5*x1*x2 + 3*x2^2 - 1;
f2:= x3^2 - 2.5*x3*x4 + 3*x4^2 - 1;