@Kitonum 

Math put that way in only 2 lines is really nice.
S<=0 seems to be a quite broad definition for solids with a closed surface S=0 as a boundary. 

However, extending your method of defining an intersection with inequalities and max to polyhedrons does not work straight away. I tried to plot a cube and an infinite slice with planes without immediate success. At least with one plane it can be used to chop of the head of an egg.

z^2/9 + (x^2 + y^2)/(4*(1 - z/5)) - 1;
plots:-implicitplot3d(max(%, z - 2), x = -3 .. 3, y = -3 .. 3, z = -3 .. 4, style = surface, color = red, grid = [100, 100, 100]);

Thanks again.

@acer 
Thanks for the clarification

@acer 

Vertical adjustments become tricky. 

If I understand correctly, the question about linking such a user defined symbol to a name should be asked separately.
I am never quite sure when to branch off.

Thank you.

@acer 
HTML is not exactly straightforward either. Is it possible to nudge the Y up a bit?

Can this typesetting output now be liked to a name or user defined operator?

@mmcdara 

Maybe with plottools,extrude its easier to plot a solid.

@mmcdara 

, as you said, is as difficult

restart;
with(plots);
ce := (p, q, r, a, b) -> ((x - p)^2 + (y - q)^2 + (z - r)^2 + a^2 - b^2)^2 - 4*a^2*((x - p)^2 + (y - q)^2);
T1 := ce(1, 1, 1, 2, 1);
T2 := ce(1, 6, 1, 2, 1);
implicitplot3d(min(T1, T2), x = -4 .. 4, y = -2 .. 9.5, z = -1 .. 1, style = surface, color = "Red", grid = [100, 100, 100], scaling = constrained, axes = normal, orientation = [15, 80]);

@Dkunb
There might be some real valued solutions the plot command misses.

Plotting your assumptions reveals that they are "over constrained" (i.e. too many).

assumptions := 1/5 < x, x < 1, 0 < beta, beta < 1, 1/2 < x/(beta*x^2 + 1);
plots:-inequal({assumptions}, x = 0 .. 1, beta = -0.1 .. 1, axes = frame);

I have tried to work with refined assumptions

assumptions_a := 1/2 < x, x < 1, 0 < beta, beta < 1; 
assumptions_b := x < 1, 0 < beta, beta < (2*x - 1)/x^2;

without success.

Maybe someone knows a magic trick to show that roots number 3 an 4 are imaginary

[{x = (-sqrt(3)*5^(2/3)*(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625)^(3/4)*sqrt(beta) + sqrt(3)*5^(5/6)*sqrt(beta)*sqrt((-5*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + (34*beta - 100)*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + (-5*beta^2 + 130*beta - 125)*5^(2/3))*sqrt(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625) - 450*(sqrt(beta) + (4*beta^(3/2))/25)*sqrt(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)*5^(2/3)*sqrt(3)) + 45*beta*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/6)*(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625)^(1/4))/(150*beta*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/6)*(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625)^(1/4))}, {x = (-sqrt(3)*5^(2/3)*(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625)^(3/4)*sqrt(beta) - sqrt(3)*5^(5/6)*sqrt(beta)*sqrt((-5*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + (34*beta - 100)*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + (-5*beta^2 + 130*beta - 125)*5^(2/3))*sqrt(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625) - 450*(sqrt(beta) + (4*beta^(3/2))/25)*sqrt(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)*5^(2/3)*sqrt(3)) + 45*beta*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/6)*(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625)^(1/4))/(150*beta*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/6)*(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625)^(1/4))}]

  Even then only the first root fits to your assumptions (see the plot above).
 

More observations that were repeatable at least 3 times:

• A successfully loaded classic worksheet (*.mws) saved under to the *.mw format could always been opened by double-click with 2023.
• When running Maple 2021 (or other instances of Maple) in parallel while the computer was on battery, the classic worksheet could be opened by double-click. Without other instances running in parallel Maple 2023 froze at the fist double-click.
• Back in office running the computer plugged in did not show a dependency on other instances of Maple running in parallel. This time Maple 2023 froze on the second double-click after a first successful attempt to open by double-click (Maple was closed before the second double-click).  The second double-click to freeze Maple 2023 was a new behavior.
• After logging out and in, no freezes occurred. At this point, I saw myself in my mind being taken away in a straitjacket, screaming, but it really happened. I decided to discontinue my investigations and opened other applications. For some reason I double clicked one last time and Maple 2023 froze again. Finally, it turned out that Maple froze when a pdf reader (Foxit) was running in parallel.   
• I decided a reboot and (unplanned) Windows was updated. Since then, I can’t force a freeze anymore.

If this case were repeatable, I would ask differently now: Why does Maple 2023 freeze under certain states of the computer system when opening this particular classic worksheet

With the update of Windows on my computer, this case is now a cold case

@mmcdara 

I get squares on my computer with 2023. Can you send a screen shot how it should look like?

Update: It looks like this after opening

@all: Is that reproducible on other computers or is it only my local installation?

@Nicole Sharp 
I do not think it’s a bug its more an enhancement you are looking for which I would also welcome for text passages. If you want to do math with it, you could try in the meatime the following:

Download userdefined_symbols.mw

And yes, fun is important in life! 🙂

@mmcdara 
Kitonum is correct. I used jargon from CAD where intersection is the operation to drill holes into a body, for example.
Thank you for providing the intersection line that I needed as well. Both answers fit nicely together.
@Kitonum 

To express my reaction in HTML: &#x1f632
Can you give me hint what max does with implict geometries?

I am missing something important here.
min, by the way, leaves out the inner surfaces which is nice as well.

Nice, but not reproducible for me (with modified color option)

a := 5;
b := 1;
h := 7;
alpha := Pi/2 + 1/tan(3/2);
plot3d([(a + b*cos(u))*cos(v) - b*cos(alpha)*sin(v)*sin(v), a + b*cos(u)*sin(v) + b*cos(alpha)*sin(v)*cos(v), b*sin(alpha)*sin(u) + h*v], u = -2*Pi .. 2*Pi, v = -2*Pi .. 2*Pi, numpoints = 2000, style = surface, color = "DarkRed");

Could you share the plot command you used?

As vv and sand15 indicated in an earlier post it would be nice to get a little more background.

@Nicole Sharp 

I agree, renaming (and hiding labels) would clean up documents without the need of unnecessary temporary variables.

What might interest you: Hiding is hiden but possible.

@Thomas Richard 

It's a specific file. I will send it.

The freeze looks like that (with a Windows hourglas not visible on the screen shot):

The taks has to be killed.

Thank you

@Thomas Richard 

One reason it is less popular may be the description. I am one of those users for whom "evaluate in an algebraic number (or function) field" does not indicate that this command is useful for simplification tasks of expressions.
I will experiment with it.