Calculus objects via scaling cross sections

[professorryan]

The previous forum post (Calculus Cross Section) was useful in seeing a couple of nice techniques, however, I am trying to do the following.  Given a base region and a particular type of cross section (i.e. square, semicircle, equilateral triangle, isosceles triangle, etc)perpendicular to either the x or y axis, create the solid (for 3D printing).

Specifically, I was attempting to make one of these with the base being the region enclosed by a parabola, a vertical line, and the x-axis. I've attached how I managed to (mostly) create these but doing non-square, non-equilateral cross sections required me to approximate using loft and a lot of individually created sketches. For squares, I intersected perpendicular extrudes and for the equilateral triangle I intersected two lofts so those were fairly easy.

I feel like I should be able to use loft or sweep to do this but I haven't been able to get it to work properly.  The problem is that Loft seems to linearly scale the z-height of the cross section to a point (whereas it should be parabolic in it's z-y plane projection).  I also tried using Sweep but because of the region, I couldn't get it to create a solid in many cases (because the three curves were considered one rail I think?).

Ideally, I would be able to draw a single cross section and move that cross section along a perpendicular path being scaled in all dimensions based on fitting between the sketch curves.

Any advice/solution would be exceptionally welcome!

[TrippyLighting]

First, let me say that you'd better had added this to the existing thread.

Can you describe what the end goal is ?

The two tools of choice in this case, loft and / or sweep create NURBS surfaces. Those are not able to create mathematically accurate representations of the geometry you are trying to create. They can approximate but the enclosed volume, for example, will not be as precise as calculated by analytical methods e.g. using mathematical formulae.

Then, I note that  none of your sketches are properly dimensioned or constrained.

The error message is correct. None of your rails touch any of the profiles (or vice versa) that is easy to observe.

[TrippyLighting]

Attached is the model with the profiles touching the rails. I used the sketch project intersection tool to project the pierce point of the parabolic spline and the line into each profile sketch and then added coincidence constraints between the half circles and also the circle center and the connecting line.

That creates a loft into a singularity and the curvature map shows that the surface is wavy to a degree. That, for example would not be the case with a proper analytical surface.

[TheCADWhisperer]

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@professorryan wrote:

...cross section along a perpendicular path 
Any advice/solution would be exceptionally welcome!

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I suspect that one of the Project tools or maybe Split might get you where you need to go.

But I don't really follow your Design Intent from this description.

[professorryan]

Sorry for being unclear, I am trying to create a model to be 3D printed to help my students visualize the following type of problem:

Given the region enclosed by the curves y=x^2, x=3, and the x-axis find the volume of the solid whose cross sections perpendicular to the y-axis are semicircles. 

Other problems would have different functions describing the base region or would have cross sections being squares or equilateral triangles. I’m just trying to understand the general workflow and also to avoid having to create several cross section slices and connecting them (as I rigged together in my posted file). Without the intermediate cross sections, loft from the base semicircle to the intersection point (3,9) - scaled to (30,90) - using the parabolic and linear rail does not preserve the cross section being a semicircle. 

[professorryan]

I see, sorry for the rough nature of my design. I’ve only used CAD software a few times before to make very very simple things. I was using a different software before but thought Fusion 360 might be more appropriate for this type of thing. I couldn’t determine how to force the semicircles to be precisely on the other sketch lines. Thank you for explaining that to me!

The good news (for me) is that in small scale printing, the curvature errors are small enough not to be visually obvious.

I was hoping not to have to draw the other intermediate profiles. Is there a way to have the original semicircle cross section follow the straight line and scale down so that the diameter is the distance between the two curves on the initial sketch?

[davebYYPCU · Accepted solution]

Finishing to the point fails, (sketches modified for coincidence as mentioned.

Peter would know why the solid Sweep fails but surface is not so bad.

Out of time, for now - needs a top centre line rail for a Loft to work as expected.....

Won't help but sorta works.....

[professorryan]

You're right on on the top centerline rail but that's hard to generate.  It's an easy parametric equation but I can't figure out how to sketch from equations directly.

[professorryan]

After the suggestions and messing around a bit more, doing some combination gives the best outcome. All of you were right that the non-coincidence of the points was critical to why I was getting the sweep errors!

I was playing around and came to the same conclusion as davebYYPCU that the main problem with sweep was when trying to reach a point.  Combining the sweep with a final loft gave a reasonable result.  I do wonder why Fusion 360 can loft to a point but not sweep?

Thank you to all for the suggestions and help!

[davebYYPCU]

Not hard to get top rail, use helper bodies and not Intersection Curves

I have not used conic curves, so no idea - used ellipse as demo.

Might help

[TrippyLighting]

If you attach the model I’d love to see how you created a top rail. In my understanding, that top rail would need to precisely touch the apex of each profile in order  to create an even halfway accurate body.

it will also possibly no do so much in improving the quality of the surface. 

[davebYYPCU]

Sorry I can't drive a conic curve. (Required)

The pic denotes an extrude bottom centre line with a vertical cutting ellipse, for the helper edge.

Sweep with scale option, does not need intermediate profiles, as the initiating  profile is a semi circle, so therefore is any other position of the object. Sweep to almost 100% path, has been adopted, so far.

[professorryan]

In addition to failing at the singularity, when using a parabola, the sweep fails at the start of the sweep (perhaps because it is tangent to the guide rail?). I fixed this by drawing a circle just away from the singularity point and using it as the profile for the sweep. Here's the file without the starting and ending lofts.

[TrippyLighting]

With these three inputs:

1. Half circle as profile

2. Straight surface edge as rail

3. Curved surface edge as guide rail

I can create a surface sweep 99.9% toward the end point. That last ~0.1mm is below what most 3D printers can print.

Then I create patches for the tiny front and the end and stitch them together (with 0.01mm stitch tolerance.

Unfortunately, when creating the bottom patch I am confronted with this error message, but I suspect If I do the sweep to 99% that will succeed.

Edit: Nope, not even at 90%. That means this pointy end isn't the problem, but the other "pointy" end is!

Edit 2: OK. A little bit more fiddling around and I got it to patch, stitch and now we have a solid body that is more representative of the shape than the lofted version.