Accuracy of Isocircles

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Given a cube with inscribe circles on each of the faces, prove the isocircle function of AutoCAD produces a true presentation if the cube is viewed an isometric view.

I tried to prove this by rotating a square with a inscribe circle along a hinge (segment bc) which would be the right vertical edge of the square. The rotation was 45 degrees and the result is View B.

Then I tried to rotate the result as viewed in View B along hinge (segment cd). I first tried to do this at 30 degrees but I could not reproduce to the same aspect ratio of the major and minor axises of the ellipse within the given cube I am trying to prove true. However I had to do this rotation at 35 degrees to get close but it still was not the results I expected. My estimated result is View C.


In theory when a circle is rotated it produces an ellipse. A circle is just an ellipse with the major and minor axis’s being equal. I tried to rotate View A appropriately but I could not reproduce the ellipse in exact proportion.

I know how to use AutoCAD to produce the ellipse using isoplane and isocircle. That is how the cube was created functions but that is not the scope of my question. Basically asking to produce one side of the cube from VIEW A as a Isometric view.

Please provide reasoning and support of any proof or this proof you provide.

CAD drawing and JPED is attached.

Very Respectfully,


Raymond Di Leo, Jr.

 

[chicopee]

Well, measure a few points along the isocircle from a reference point such as its centers o that you have "x" and "y" values for the points, then knowing the lengths of the major and minor axes of the isocircle develop the elliptical equation in terms of "x" and "y" and see if these points fall on the isocircle. By moving the cursor to the center of the isocircle, the "x" and "y" positions of the points mentioned above will be easily identified for you.

 

[SEANT]

This probably does not qualify as a proof but:

If you analyze an Isometric direction it will show 1, -1, 1 (for SouthEast Iso). So, rotating the 2D geometry by that amount about the hinge bc by 45 degrees makes sense because ACOS(1/SQRT(2)) = 45.

For the next step, though, the angle off of the XY plane for a Viewpoint of 1, -1, 1 (or about the segment cd) would need to be ACOS(SQRT(2)/SQRT(3)), about 35.26439 degrees.

 

[chicopee]

I have reviewed several of my references as you brought up an interesting point. I have concluded that an isometric ellipse is not a true ellipse as you would get if you plotted one on a normal view. Although a projected isometric ellipse is correct on 30d inclined plane, any measurement not along the directions of the principal axes would not be true measurement and that includes angles. Note to remember that drawn isometric ellipse have slightly longer dimensions along the directions of the principal axis than that of projected isometric ellipse.