Calculate curve to make two curved panel fit together

[franksbedford]

Version: 3.29.4.0 (3.29.4)
Internet: QCAD.org
Build Date: Feb 7 2024
Revision: 36a6423
Qt Version: 5.14.2
Architecture: x86_64
Compiler: Clang 10.0.0

Hey all,

I'm designing a pizza oven mold. The mold's dome is covered by eight identical curved pieces that need to meet to from a smooth, straight seam. Those pieces follow a Qcad arc (two point and height). I'm struggling to come up with a shape for the cover piece where the edges meet without overlapping or gaps.

Here is what the dome structure looks like. It's a hexagon.

IMG_2394.jpeg (827.62 KiB) Viewed 997 times

IMG_2325.jpeg (867.01 KiB) Viewed 997 times

IMG_2343.jpeg (1.05 MiB) Viewed 997 times

My approach has been to measure the radius of curve at multiple points, draw a hexagon with that radius to get the width of the width of the panel at those points. It's close, but not quite right, and I have to manually cut/adjust to make the 8 dome pieces fit. I want the CNC cut panels to line up in a straight line.

I've attached a DXF with the calculations and the shape of the dome cover piece.

I hope I made this clear. Any suggestions?

Thanks!

[CVH]

FB,
This is called unfolding or unrolling, developing and all is based on geometric projections.
There are some ventures to do this in QCAD.
Already a most basic one is a standard example of unfolding a box, see script Libray Items CubeCuttingOut.js

A more elaborate trial was posted by tukuyomi.
The story starts about here: https://www.qcad.org/rsforum/viewtopic. ... 741#p22875

He included several basic methods an some functions that can be molded to any case.
The math regarding that is horrible ... 2023 passed and I still haven't found the time to look into this deeper.

Breaking up the problem.
The sides follow a circular path with R=11.85768452
Every curved side is thus a part of a cylinder.
And that is cut by yet another cylinder of the same size.
The axis are 135° apart or both cylinders are joined to from a 45° bend.
Or the cylinder is cut by a plane at 45°. ERRATUM : 'plane at 22.5°'

And now it is more easy ... Your are looking for a part of a cosine curve.
The length of a full period is the circumference of the pipe which is pi*D.
The amplitude is D/2=R. ERRATUM : Amp = tan(22.5°)*D/2
In an expression: y = Amp*cos(x/R) where x = 0 ... pi*D

Put this is a spreadsheet and calculate for example 49 points (48 intervals)
Or more if you want to be more precise than 12 points for about 1/4 turn.
Mold the X and the Y value to strings and concatenate that separated by your QCAD application based list separator.
Copy the list of textual coordinate pairs.
Now start the spline from fitpoints tool (SL), opt for not closed and past the coordinates list to the Command Line.
Terminate the spline tool.

Now it is the matter of matching and trimming that shape in regards with your cylinder axis = arc center, bottom and top and mirroring.
One could setup the spreadsheet to limit coordinates within the range what is required.
One can swap x and y to have it vertical.

This is for the inner surface
To be exact:
For the outer surface you need to use R + material thickness and the sides must beveled at varying angles.

Regards,
CVH

[Husky]

I want the CNC cut panels to line up in a straight line.

Means what exactly? Metal? Wood? Styrofoam? Stripes welded / clued together for each of the 8 sites? What is the thickness of the material? Final shape? A real curve or a semi "curve" built out of straight segments?

[CVH]

FB,
Here are the Fit-Points coordinates of the curve projected to vertical in Dot-Comma style:

Code: Select all

4.9116135,0.0
4.89292332777584,1.03477818224748
4.83699505459943,2.06955636449497
4.74425432840004,3.10433454674245
4.61540696210245,4.13911272898994
4.45143356196433,5.17389091123742
4.2535820645706,6.20866909348491
4.02335823928241,7.24344727573239
3.76251422842315,8.27822545797988
3.47303521241739,9.31300364022736
3.15712430136914,10.3477818224748
2.81718576806369,11.3825600047223
2.45580675,12.4173381869698
2.07573755971215,13.4521163692173
1.67987075323029,14.4868945514648
1.27121911598265,15.5216727337123
0.852892733679293,16.5564509159598
0.428075322681924,17.5912290982072
0.0,18.6260072804547
-0.428075322681925,19.6607854627022
-0.852892733679294,20.6955636449497
-1.27121911598265,21.7303418271972
-1.67987075323029,22.7651200094447
-2.07573755971215,23.7998981916922
-2.45580675,24.8346763739396
-2.81718576806369,25.8694545561871
-3.15712430136914,26.9042327384346
-3.47303521241739,27.9390109206821
-3.76251422842316,28.9737891029296
-4.02335823928241,30.0085672851771
-4.2535820645706,31.0433454674245
-4.45143356196433,32.078123649672
-4.61540696210245,33.1129018319195
-4.74425432840004,34.147680014167
-4.83699505459943,35.1824581964145
-4.89292332777584,36.217236378662
-4.9116135,37.2520145609095
-4.89292332777584,38.2867927431569
-4.83699505459943,39.3215709254044
-4.74425432840004,40.3563491076519
-4.61540696210245,41.3911272898994
-4.45143356196433,42.4259054721469
-4.2535820645706,43.4606836543944
-4.02335823928241,44.4954618366419
-3.76251422842315,45.5302400188893
-3.47303521241739,46.5650182011368
-3.15712430136914,47.5997963833843
-2.81718576806369,48.6345745656318
-2.45580675,49.6693527478793
-2.07573755971215,50.7041309301268
-1.67987075323029,51.7389091123742
-1.27121911598265,52.7736872946217
-0.852892733679289,53.8084654768692
-0.428075322681925,54.8432436591167
-9.02618353594998E-16,55.8780218413642
0.428075322681928,56.9128000236117
0.852892733679296,57.9475782058592
1.27121911598265,58.9823563881066
1.67987075323029,60.0171345703541
2.07573755971215,61.0519127526016
2.45580675,62.0866909348491
2.81718576806369,63.1214691170966
3.15712430136914,64.1562472993441
3.4730352124174,65.1910254815916
3.76251422842316,66.225803663839
4.02335823928241,67.2605818460865
4.2535820645706,68.295360028334
4.45143356196434,69.3301382105815
4.61540696210245,70.364916392829
4.74425432840004,71.3996945750765
4.83699505459943,72.4344727573239
4.89292332777584,73.4692509395714
4.9116135,74.5040291218189

Ensure that you have a spline with 73 Fit-Points ... It may duplicate the last one depending if there is a CR/LF carried over or not.

=> 72 intervals of 5 degrees.
Fit-Point 36 would match with your arc center.

Your developed contour fairly matches with these coordinates for a certain span but there are issues ...
... Rounded values issues and more.

Remark that your CW arc does not start at 180° ... It starts just below that at 180.36603569°
For some reason your arc ends at 107.88673406° making it an arc of -72.47930163° with an arc length of exactly 15.000"
That relates to somewhat halfway between Fit-Point 50-51 (36 + (72.479... / 5) = 50.49...)

How you come to a developed length of >18< or measured value 17.9 or marked as '17 7/8' is a mystery for me.
Advice: Work in decimal notation, rounded values to 1/4, 1/2 or 1/8 have no meaning here, re-entering them as exact is faulty.

We could limit the function to between 180.36603569° and 107.88673406° so that the endpoints of our Fit-Point spline will correlate with the arc.
But splines end 'normal' ... You must overdo it at least 2-3 points to have a correct curvature near the point of interest.
When you trim a Fit-Point spline at some position, you get a Control-Point spline with a proper end tangent to preserve the curvature.
Store a copy of the original because there is no easy way back except undoing in the same QCAD instance.

Regards,
CVH

[franksbedford]

I want the CNC cut panels to line up in a straight line.

Means what exactly? Metal? Wood? Styrofoam? Stripes welded / clued together for each of the 8 sites? What is the thickness of the material? Final shape? A real curve or a semi "curve" built out of straight segments?

Single wall cardboard. 0.15". Taped with fiber reinforced Kraft tape. The panels follow the curve of the arch support on each of the eight hexagon sides. The edges of the eight panel meet, where I need to tape them. The final mold is packed with castable refractory material. After the refractory cures, the mold is removed, with the casting taking the shape of the mold. I hope this is more clear.

d36_dome_arch copy.pdf

(120.2 KiB) Downloaded 11 times

I am now going to read CVH's posts at least a couple of times. I've got some learning to do.

Thanks!

[franksbedford]

Here is the piece I need to design. The top and bottom measurements are known because the piece connects (tape) to a hexagon at the top and bottom. The curve (or series of tangents) is the unknown.

I need to do this four times -- for four different size domes. The top hexagon is fixed. A 10" hexagon with 4 1/4" sides. The bottom hexagon varies with the size of the dome. 28", 32", 36" and 40".

I hope this is clear.

dome_cover.pdf

(333.83 KiB) Downloaded 16 times

[franksbedford]

Ensure that you have a spline with 73 Fit-Points ... It may duplicate the last one depending if there is a CR/LF carried over or not.

=> 72 intervals of 5 degrees.
Fit-Point 36 would match with your arc center.

Your developed contour fairly matches with these coordinates for a certain span but there are issues ...
... Rounded values issues and more.

Remark that your CW arc does not start at 180° ... It starts just below that at 180.36603569°
For some reason your arc ends at 107.88673406° making it an arc of -72.47930163° with an arc length of exactly 15.000"
That relates to somewhat halfway between Fit-Point 50-51 (36 + (72.479... / 5) = 50.49...)

How you come to a developed length of >18< or measured value 17.9 or marked as '17 7/8' is a mystery for me.
Advice: Work in decimal notation, rounded values to 1/4, 1/2 or 1/8 have no meaning here, re-entering them as exact is faulty.

We could limit the function to between 180.36603569° and 107.88673406° so that the endpoints of our Fit-Point spline will correlate with the arc.
But splines end 'normal' ... You must overdo it at least 2-3 points to have a correct curvature near the point of interest.
When you trim a Fit-Point spline at some position, you get a Control-Point spline with a proper end tangent to preserve the curvature.
Store a copy of the original because there is no easy way back except undoing in the same QCAD instance.

Regards,
CVH

Thanks for this. I have changed the drawings to decimal. The reason for the funny angles is because I used an arc with two point and height to connect two known points. There is a 2" high vertical piece at the bottom of the mold that is necessary for structural reasons and a 10" wide hexagon at the top (5" on each side) for structural and design reasons. The curve that connects those two points is arbitrary. It's a 300 lb. cast pizza oven , so precision isn't important. For example, three tangents would be fine.

The length of the cover (17.82") comes from the length of the arc. Screenshot.

I'm back to reading about splines. And open to ideas!

Screenshot 2024-03-24 at 10.16.06 AM.png (1.82 MiB) Viewed 804 times

[franksbedford]

One other thing. I can see that I've been "designing" these pieces without accounting for the thickness of the material. 0.15" cardboard. I'll start fixing that on all the Qcad files.

The pieces will be cut by a CNC laser.

[CVH]

The top hexagon is fixed. A 10" hexagon with 4 1/4" sides. The bottom hexagon varies with the size of the dome. 28", 32", 36" and 40".

NOT a hexagon ... It is a regular octagon.

A regular octagon with opposite sides 31.5 apart has a side length of 13.047727... and NOT 13 1/4 as in your dome_cover pdf.
-> The math is easy: side = span/(1+sqrt(2)) = 13.047727214752494037253194812605

One with opposite sides 10 apart has a side length of 4.142136... and NOT 4 1/4.

Speaking of a dome of 32" ... Thus inside = 31 1/2 + 2 x 1/4 ... 1/4 thick material, am I right?

Now I know what the unconnected 5.0" long line piece represents ...
But the height above the circle center would be wrong.
Why wasn't the arc shape connected to that?

As remarked, your arc center does not match with the slot at the left.
It is drawn 0.075753 higher in the DXF posted on Sat Mar 23, 2024 10:00 am

So ... Reconstructing your arc over here is a bit ambiguous.
Don't know where it should start exactly ...
... It was 15.000" long, trimmed to a rectangle of 11.3603024 by 8.2155174 ...
... It ended in the middle of nowhere ...
... It won't connect to the open endpoint of the line piece that is 5.0 long.


But the math for the cosine/sine function is sound ... You only have to put the right numbers in it.
Perhaps you can provide a drawing with a proper layout without using rounded/truncated values.

Regards,
CVH

[CVH]

so precision isn't important. For example, three tangents would be fine.

The pieces will be cut by a CNC laser.

A laser won't care if it is reduced to 3, 5, 10 or more segments.
It will cut the cosine/sin curve approximated with the same ease.

The length of the cover (17.82") comes from the length of the arc. Screenshot.

Can't measure on a screenshot, nor on a pdf, the arc length in the initial provided drawing was truncated to be 15.000 long.
Now knowing that it must be expanded it is 17.89864163 long when I trim it and not 17.82....
The main problems: Where is your center located? Are this the same arcs? Did the design change over the last hours/days?

There is a 2" high vertical piece at the bottom

It measures 1.90625, the slot itself is 0.1875 wide ... Top is at 2.09375

The curve that connects those two points is arbitrary.

Arbitrary or not, at some point you should settle for a certain arc radius:
R11.85768452 or R11.9509759 or yet another.

Remark also that your inner supporting shapes can not be 16" wide to span 32".
You need to account for A) 2 times the thickness that covers it and B) that you can not merge them in the middle to a single point.
If all your material is 0.1875 thick, they will form an octagonal hole with a span of 0.452665.
So it should be 16" - cover - 0.226333" wide.

Regards,
CVH

[franksbedford]

This is all very helpful. Thanks! I've used Qcad in the past for conceptual design, but not for accurate production and manufacturing design. The input is very helpful.

Octagon it is.

Here is the updated .dxf for the 32" model. The panel is 16" wide because two are held together with an interlocking octagon (notched) to make 32". Though of course it should be 31.7" less 0.3" (two thicknesses of 0.15" cardboard covers). There at 0.1875" notches cut into many of the pieces where two mold panels slide together. That explains the x.91" measurements which are now fixed. I am measuring to the center of the notch.

I've left the curved section between the two fixed points blank.

What is the best method for drawing that curve to get a clean radius, so I can calculate the fixed-point spline measurements? In the past I used the arc two points and height tool, and simply dragged a curve from one end point to the other, and set the height to look right.

If three tangents make it easier to calculate the fixed point spline than an arc, that would be fine.

Last thing. I was under the impression that Qcad calculates the "length" of an arc and provides that measurement in the property editor. In the screenshot below, I selected Arc 1, and the property editor shows a length of 17.820". Is that wrong?

Screenshot 2024-03-24 at 10.16.06 AM.png (1.82 MiB) Viewed 736 times

[franksbedford]

This is what the assembled mold looks like.

IMG_2404.jpeg (3.44 MiB) Viewed 731 times

[CVH]

FB,

From the given equation it should be clear that we need the radius or the diameter of the cylinders joined at 45°.

As said: Reconstructing your arc over here is a bit ambiguous.
A circle, an arc is defined with 3 points that are not co-linear or with 2 points and a radius or its center and a radius or by tangents.

The arc is drawn CW.
Let us call the starting point S and the ending point E.

If we mirror the support insert then we have 4 points: S - E - E' -S'.
But an arc drawn from 3 of these points is far more shallow as what you presented so far.

In all drawings d32_dome_cover.dxf & d32_dome_arch(1).dxf & d32_dome_arch(2).dxf I discover a half supporting insert of 16 wide and 14 high.
You agreed that 16 wide is not yet correct but the last file without arc is not yet adapted.

For the first 2: At the left, the top of the slot is 2+(0.1875/2)=2.09375 above the base, let the base be our Y-Zero.
Let the middle of the dome be X-Zero ... Only for now that coincides with the bottom right corner of the insert.
(X-Zero, Y-Zero) is our Relative Zero position for all files.

The arc in d32_dome_cover.dxf started at S but did NOT end in E.
Center was @(-4.14255746, 2.16950263), radius was 11.85768452.
The center is 0.07575263 above the top of the slot.
The arc CANNOT end in E, at best it will end in @(-4.94506237, 14) instead of @(-5, 14)

The arc in d32_dome_arch(1).dxf ended in E but did NOT start in S.
Center was @(-4.04902605, 2.08692006), radius was 11.9509759.
The center is 0.00682994 below the top of the slot.
Because the arc starts at 180 degree that doesn't coincides with S.

I see a third constraint emerging: Being tangent to vertical at the slot near S.
Then your center should be located at @(-4.05641868, 2.09375) and the radius is 11.94358132 for d32_dome_cover.dxf & d32_dome_arch(1).dxf.

Don't type in the numbers although they are precise enough.
They won't hold for d32_dome_arch(2).dxf because there the top of the slot is 2.1000 above the base

Draw a chord from S to E and an orthogonal through the middle of the chord.
Trim the orthogonal to the top horizontal of the slot at S ... The trimmed ending is your center position.
(Circle from Tangent and 2 points (CT1) works too if the vertical is extended somewhat, otherwise: No solution )

If you adapt that in a newer file then it is just a matter of doing the math for the developing of the cover.
Awaiting the newer design ...

Regards,
CVH

[CVH]

Last thing. I was under the impression that Qcad calculates the "length" of an arc and provides that measurement in the property editor. In the screenshot below, I selected Arc 1, and the property editor shows a length of 17.820". Is that wrong?

That is correct for that arc ...
But 17.82056866 itself is not correct because it overshoots the slot by about 0.00683.

And the arc in d32_dome_cover.dxf was drawn truncated to be 15.000 long.
If I trim that arc higher up it is 17.89864163 long.
All because it are different arcs.

Regards,
CVH

[CVH]

I used d32_dome_arch (2).dxf as as a starting point.
Where S is 2.1000 above the base ...

In a few simple steps I get the development of the arc cover.

d32_dome_arch (3).dxf

(155.47 KiB) Downloaded 13 times

But you are not out the woods yet.
First: The support insert should not be 16 wide, then it is unsure if E is at (-5,14).

Cardboard is corrugated to make it stiff.
When we roll that then the exterior can not elongate, can not be stretched like with most plate material.
What will happen is that the exterior becomes segmented and the interior will wrinkle.
This wrinkling is far superior than the minute compression seen with other materials.

The presented development is the interior face regarding the spline shapes left - right.
But that needs to be longer to account for wrinkling.
Doing the same for the exterior face won't fit because the edges need to beveled.
And that would be slightly to long because of segmentation.

The truth will lie somewhere in between.
We could estimate that when we know the thickness (0.15" but slots are 0.1875") and the corrugated nature: Flutes/length.
Double wall would be to stiff (3 sheets).
Also see: https://en.wikipedia.org/wiki/Corrugated_fiberboard

Regards,
CVH