Look at this:

Region@ImplicitRegion[
  x^2 + y^2 <= (31981/10000 + z (9/10 + (-(9/250) + z/1000) z))^2 && 
   1/10000 <= z <= 181/20 && 
   Not[x^2 + y^2 <= (31/10 - (35 z)/976)^2 && 
     1/10000 <= z <= 122/25], {x, y, z}]

One of the edges is perfectly drawn on screen. Why not the other?

Also, Mathematica cannot discretize this very, very simple semialgebraic region, that arises from a cylindrical decomposition of our original object:

DiscretizeRegion[
 ImplicitRegion[(13689/1600 < x^2 + y^2 < 36616936723249/
     3810304000000 && 
    15128/175 - 976/35 Sqrt[x^2 + y^2] <= z < 181/20), {x, y, z}], 
 Method -> "Semialgebraic"]

You can read in step files, but you will need to make sure the extension is in lower case (i.e., "*.step"). I threw together a step file in SolidWorks and read in easily provided I changed the extension from STEP to step.

Needs["OpenCascadeLink`"]
Needs["NDSolve`FEM`"]
shape2 = OpenCascadeShapeImport["E:\\WolframCommunity\\lens.step"];
bmesh2 = OpenCascadeShapeSurfaceMeshToBoundaryMesh[shape2]
groups = bmesh2["BoundaryElementMarkerUnion"];
temp = Most[Range[0, 1, 1/(Length[groups])]];
colors = ColorData["BrightBands"][#] & /@ temp
bmesh2["Wireframe"["MeshElementStyle" -> FaceForm /@ colors]]

Hi Henrik,

As you may have guessed, I am new to Mathematica and I have been using my lockdown over Easter to dive in. Thank you for your help.You certainly have cleaned everything up.The code I posted had gone through so many edits and tweaks, as I tried more and more to get things to get it to work, that all variable names had become meaningless.

Antony

My guess here is that discretization (using DiscretizeRegion[]) is leading to the wrong track. Region is an abstract concept, and if you write (-> your notebook):

Clear["Global`*"];
z0 = 3.1981; z1 = 0.9; z2 = -0.036; z3 = 0.001;
lowz =  0.0001;
highz = 9.05;
highPinz = 4.88;
mainBody = ImplicitRegion[x^2 + y^2 <= (z0 + z*(z1 + z*(z2 + z3*z)))^2 && lowz <= z <= highz, {x, y, z}];
pin =  ImplicitRegion[x^2 + y^2 <= (3.1 - 0.175*z/highPinz)^2 && lowz <= z <= highPinz, {x, y, z}];
df = RegionDifference[mainBody, pin];

then the regions mainBody, pin and df have a clear and clean definition. In fact, we have

I do not know if it makes much sense to write/evaluate Region[df], because df is already a (kind of) region. For displaying regions there are functions like RegionPlot and RegionPlot3D, and here we see that the above issue simply arises due to unsufficient sampling of the regions when plotted:

Hi Yehuda,

The command

DiscretizeRegion[df, MaxCellMeasure -> 0.005]

works really well. Thank you, so much. I was really stuck.

My intention is to investigate the thermal performance of the lens with different LEDs. The prospect is that I will need to make the lens just once and once I have it, I can loop over many thermal simulations. The speed of execution for generating the lens is only expected to be a minor issue.

Again thank you.

Antony

@Gianluca - MaxCellMeasure -> 0.01 is not small enough. 0.005 provides much better results. I didn't encounter crashes for years using regions. What system do you use (I use Mac Pro and MacBook Pro by Apple).

@Anthony - you could think of several options to tackle this issue. First, the graphical representation on the screen has nothing to do with the real accuracy achieved by Region, since graphical representation is ALWAYS discretized. You could achieve a better behavior from the default Region by replacing all the approximate numbers with accurate ones, i.e., change all floating point numbers to rationals. This will improve the resolution at the wider part. Second, if the thickness is of no importance and speed is an issue, you could use (for both exact and approximate versions...)

BoundaryDiscretizeRegion[df, MaxCellMeasure -> 0.005]

which is a little bit more memory efficient than

DiscretizeRegion[df, MaxCellMeasure -> 0.005]

In your case, I would use discretization as late as possible in my computations to speed it up. HTH yehuda

Region computation is very unreliable in Mathematica, and it has been for several years. The kernel has just quit on me with DiscretizeRegion[mainBody]. I have found improvements only by setting a low MaxCellMeasure: DiscretizeRegion[df, MaxCellMeasure -> .01]