# Drawing an approximation to a circle in isometric projection

A circle viewed from from the side is an ellipse.

A common approximation can be found on the web (eg do a google image search for isometric circle). This produces something like (with arc centers T,U,A and N): Alternatively, make a 4 Bezier Curves with control points of each arc being the mid points of a side of the parallelogram and the included corner.

Here is the ellipse which is the image of circle: $x^2+3y^2=3/2$.

Here is a picture of the ellipse (black), the composite Bezier curve(red/green) and the 4 circular curves(blue/orange): How is this justified? Why do the arcs join up smoothly?

EDIT: I would like to:

• Predict the error (for a bezier approximation to a circle here is an error estimate: wikipedia).

• Generalise either of these methods, say, include more Bezier control points, include more composite arcs, find more circular arc centers ...

• Is that not just an ellipse? – muaddib Jun 12 '15 at 1:00
• True, the true isometric projection is an ellipse. But circular arcs are easier to draw. Hence, the common approximation. – pdmclean Jun 12 '15 at 1:01
• Ah, I see. 4321 Comment is now long enough :) – muaddib Jun 12 '15 at 1:06

3. An arc of the ellipse "near" the major axis or minor axis is "approximately circular", since it's the image of a circular arc under a linear transformation $T$ and the axes of the ellipse are the eigenspaces of $T$.