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Message 1 of 39 (670 Views)

# Does anyone know the formula (s) used for making a PEllipse?

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05-02-2012 06:10 AM

Hello,

I write programs for large CNC routers, and I'm trying to make a parametric program that uses an ellipse in it.  I need to know the formula or formulas used to produce the arcs that make up a pellipse so that I can get the radii, and end points of each arc.  Any help would be greatly appreciated, Thanks.

-Norm

I've been in the process of writing a vba program and I needed to be able to programmatically draw the arcs to approximate the ellipse the same way that autoCAD does.  I searched and searched and couldn't find anything that says how autoCAD comes up with the approximation that it does.  I found this post in my search.

After a lot of messing with the problem I finally figured out how autoCAD does it.  So if anyone is still interested in this I've attached a drawing file (it's a 2007 version) that shows the math and geometry behind autoCAD's pellipse =1 ellipse approximation.

Hope this helps someone besides me.

Let me know if you have any questions.

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Message 2 of 39 (651 Views)

# Re: Does anyone know the formula (s) used for making a PEllipse?

05-02-2012 08:17 AM in reply to: norm77

In the past I have used the geometry of http://en.wikipedia.org/wiki/Trammel_of_Archimedes in VBA to generate coordinates along eliptical arcs.

A

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Message 3 of 39 (634 Views)

# Re: Does anyone know the formula (s) used for making a PEllipse?

05-02-2012 10:58 AM in reply to: hayalexj

hayalexj wrote:

In the past I have used the geometry of http://en.wikipedia.org/wiki/Trammel_of_Archimedes in VBA to generate coordinates along eliptical arcs.

A

Thanks, I have looked at that already - which is great, but doesn't give enough information.  I have also looked into the "five centered arch" in http://mysite.du.edu/~jcalvert/math/ellipse.htm, which is ok, but not as accurate or usable as the pellipse in AutoCAD.

For programming, the current way a pellipse is created is great.  Each quadrant is made up of 4 arcs.  The end point for each arc is in line with the center of the arcs on either side of it, which flow seamlessly into each other.

For my purposes, I need to be able to calculate the radius, and start & end point of every arc (referenced from the center of the ellipse).  Thanks.

-Norm

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Message 4 of 39 (627 Views)

# Re: Does anyone know the formula (s) used for making a PEllipse?

05-02-2012 11:20 AM in reply to: norm77
I don't know why you would want to create arcs? Surely that's an inaccurate way to do it?

I just generated a series of points along the elliptical arc and fed that directly into the tool path.

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Message 5 of 39 (625 Views)

# Re: Does anyone know the formula (s) used for making a PEllipse?

05-02-2012 11:31 AM in reply to: hayalexj

hayalexj wrote:
I don't know why you would want to create arcs? Surely that's an inaccurate way to do it?

I just generated a series of points along the elliptical arc and fed that directly into the tool path.

Unfortunately I can't create ellipses on the machines I'm programming.  The accuracy of the Pellipse isn't perfect, but good for my application, and easy to check the logic in Excel and AutoCAD.  I really like the Trammel of Archimedes, but I don't know how to apply it.  The basic parametric programming logic I do can be replicated within excel.

-Norm

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Message 6 of 39 (621 Views)

# Re: Does anyone know the formula (s) used for making a PEllipse?

05-02-2012 11:48 AM in reply to: norm77
What I did would not run in Excel without modification. It used the IntersectsWith function to locate the point at which a circle moving along the x axis hit the y axis and then projected arc coordinates through that.
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Message 7 of 39 (613 Views)

# Re: Does anyone know the formula (s) used for making a PEllipse?

05-02-2012 12:40 PM in reply to: hayalexj

This is what I've been able to figure out so far...

Based on the picture shown above:

AF is major axis

AB is minor axis

Line IHD is perpendicular to BF

Line JIC is perpendicular to BD

Line HGE is perpendicular to DF

Line GF = First Arc Radius

Line HE = Second Arc Radius

Line IC = Third Arc Radius

Line JB = Fourth Arc Radius

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Message 8 of 39 (605 Views)

# Re: Does anyone know the formula (s) used for making a PEllipse?

05-02-2012 01:06 PM in reply to: norm77
I think you would have to work out the angle between JC and JA and so on depending on the dims of your arc... Sounds like a lot of effort for an inaccurate way to do it.

The trammel option just requires the ratio of long to short dimension...
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Message 9 of 39 (602 Views)

# Re: Does anyone know the formula (s) used for making a PEllipse?

05-02-2012 01:38 PM in reply to: hayalexj

Yes the trammel option gives X and Y coordinates easily (If only it could be that easy ), but the arcs between can't be calculated which means they would not line up.  Trammel option doesn't give the radii of the arcs.

The key to making this without being able to program ellipses is to have all the arcs have the same start angle as the one that it connects to.

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Message 10 of 39 (598 Views)

# Re: Does anyone know the formula (s) used for making a PEllipse?

05-02-2012 01:43 PM in reply to: norm77
So forget the arcs; they serve no purpose.
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