If in the text to write "#" in front of the approximate solution x0] 1], x0 [2], and remove the "#" from any version of Optimization package then no one variant will not work.
OPT_DIF.mw 

@Kitonum  Any quadrilateral and so on is divided into triangles

@vv  No words

@Kitonum  Thanks, but, yes, unfortunately, this only applies to the simplest cases.

     Can anybody explain to a "local specialist', that he had to teach math to understand that in the denominator of the parameter should not be? This smooth curves in 3d on the all set of definitions.

    This parametric curve for second-order surface and for plane in the common form equations.
   

         PLAN_CURVE_3d_COMMON.mw.

@tomleslie  I remain in the dark about whether I was able to answer your question: "Purpose ???"

For those who have not used the program text. This parametric equations of the curves in the second and third Figs respectively.

 x1^2+0.1*x2^2+x3^2-9=0;
 x1+3*x3+1=0;
x1(s)= -(135121896351/50000000000)cos(2s)+(8407313781/10000000000)sin(2s)-4999999999/50000000000;
x2(s)= -(45040632117/5000000000)sin(2s)-(2802437927/1000000000)cos(2s);
x3(s)= (45040632117/50000000000)cos(2s)-(2802437927/10000000000)sin(2s)-14999999997/50000000000;
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x1^2-0.1*x2^2+x3^2-9=0;
x1+3*x3+1=0;
x1(s)= -(390000000009/200000000000)exp(2s)-(205384615389/200000000000)exp(-2s)-10000000001/100000000000;
x2(s)= -(130000000003/20000000000)exp(2s)+(68461538463/20000000000)exp(-2s);
x3(s)= (130000000003/200000000000)exp(2s)+(68461538463/200000000000)exp(-2s)-30000000003/100000000000;

@tomleslie   Here is the purpose: parametric equation of second-order curve in 3d.
For example, the first curve equations:
x1(s)= -135000000013/5000000000+(166055512773/10000000000)exp(s)+(101773194523/10000000000)exp(-s);
x2(s)= (7828707271/10000000000)exp(-s)-10000000001/10000000000;
x3(s)=-(166055512773/10000000000)exp(s)-(54800950897/5000000000)exp(-s)+140000000013/5000000000;
And where are your equations? Read carefully, please...

@Kitonum  The programs should work for any three points, and in the texts there is a check.
Creation of procedures in Maple yet not mastered by me.

Another way of constructing a circle by three points

CIRCLE_3_POINTS_geom3d_0.mw

@vv 
“I thought that you want to know how your surfaces really look (globally)”
You really thought so?

@vv 

Then what is the point in Par3d, if the question was associated with this procedure? And what about the plane is not a joke.
(graph of  z-1/2 * exp (sin (x + 5/2 * y + z)) = 0.; continuous)

Merry Christmas!

@vv 

 Please show graphs of these functions:
(x^2+y^2-0.4)^2+(z+sin(x*y+z))^4-0.1=0.; (or can simply (x^2+y^2-2.)^2+z^2-1.=0.;)
x+y+z(+const)=0.;
 If possible, and this function too:
  z-0.5 * exp (sin (x + 2.5 * y + z)) = 0.;

@vv 
 Is this the rotated ellipsoid?
x1^2+4.*x2^2+x3^2+x1*x2+x1*x3+0.1*x2*x3-2.=0.;