[dnautics]

1) this doesn't work for cases where your data are positive-definite.

However, let's set that aside. I apologize for being a bit obfuscatory. My point is: If this is the case, then the explanation in the OP is totally misleading, because your data shouldn't look like an ellipsoid, but rather a circle. PCA should only be used in situations where there is a reason to believe there is a mechanistically justifiable "hidden value" that underlies otherwise uncontrolled "independent variables", thus making a dimensional reduction reasonable.

This is not at all the situation that the OP goes over in the first part of the post.

[twstws]

The example was a little clunky, but I don't find it misleading. A biplot of two normalized variables is elliptical, if the variables are correlated. This particular hand-drawn example does indeed look a bit weird, but that doesn't detract from the main point. It clearly shows the relationship between the original data and the ordination; it's a rigid rotation.

This is easily grasped with a 2d example, despite the fact that PCA makes no sense with only two variables.