Lets say I have a XYZ coordinate system, with a secondary coordinate system made with rotations of 15 degrees on the X axis and 5 degrees on the Z axis. On a sketch made on this new coordinate system in the X-Z plane, what kind of eclipse would I need to make in order to see a perfect circle when looking at the sketch from the ORIGINAL X-Z plane? As in, what major and minor radii would I need, as well as any other transformations afterwards?
I've been thinking about this for a while and it seems really simple when there's a rotation on only one axis, lets say the X axis. The major radius would change on the eclipse (in the Z axis direction) and would be equal to the following equation, while the minor radius would just be the original radius of the circle since there's no "shift" in that direction.
major radius = original radius / cosine (angle of rotation along X axis)
I just can't wrap my head around what changes when you rotate the coordinate system around two angles. Originally I thought the second angle would only affect the minor diameter and the two components only affected their respective radii on the eclipse, but the circle doesn't match perfectly when trying this on a CAD system. Is there a subsequent rotation to the eclipse that I need to make after or some combination of trigonometry?
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An ellipse is a section through a cylinder, usually an angled section. The cylinder's diameter never changes, regardless of how you rotate the cutting plane through the cylinder. Therefore, the minor axis value of the ellipse will always be half of the cylinder's diameter (or equal to the cylinder's radius). The major axis will change based on the rotation about the Z axis - the rotation about the X axis will control the angle at which the ellipse is tilted.
See the attached file, change the second Datum CSYS parameters to any angle and you will see the Section update - notice that if you don't change the value of the cylindrical diameter, the minor axis is always the same on the resulting ellipse. The ellipse will rotate and change major axis value but the minor is always the same until the cylinder is changed to have a larger diameter.
Just draw the circle in the original plane, and project onto the tilted plane.
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In your model, I understand that a rotation along the Y axis (rotating the cutting plane through the cylinder) only results in a rotation of the ellipse. However regarding your last sentence "The major axis will change based on the rotation about the Z axis - the rotation about the X axis will control the angle at which the ellipse is tilted.", won't the rotation about the X axis, controlling the tilt angle of the ellipse, affect the projection into the original XZ plane in a similar way to the Z axis rotation?
I guess you do bring up a good point. I am just looking for an alternate solution as I try to learn NX and its capabilities.
I think that the definition of an ellipse, is a planar cut through a cylinder, there are two extremes of that, the perfect circle and two parallel lines. Anything in between will be ellipses.