Scaling a sphere to give identical result, regardless of rotation

Question

how can I maintain a consistent scaling of a sphere regardless of the rotation of the model? For example, take the following (scaled) sphere:

In this image, we start with a sphere, looking at it directly through the Z-azis. The sphere is scaled by 0.7x on the Y (up/down) axis, giving it a 'squashed' appearance.

A 3D vector that describes the scaling of this sphere is [1.0, 0.7, 1.0]. The red dot indicates an arbitrarily chosen point on the surface of the sphere that we will mark as the 'top'.

Now, after the sphere rotates 90 degrees through the Z-axis, the same scaling coefficients [1.0, 0.7, 1.0] when applied will produce a result like this:

My question is as follows:

Given a rotation quaternion Q that represents the rotation of the sphere, how can I use Q to correctly alter the scaling coefficients vector in order to produce a result like this instead:

Notice that the eventual 'oriented shape' of the sphere is the same as in the first image, even though a 90-degree-through-Z rotation has been applied. In this instance, a corrected scaling coefficient vector would be something like [0.7, 1.0, 1.0].

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I've tried simply multiplying the scaling coefficients vector by Q but this gives an incorrect result. This is actually trivial to see if you imagine rotating [1.0, 1.0, 1.0] by 45 degrees through any angle. The answer should always remain as [1.0, 1.0, 1.0] but it does not.

Answer 1

We can construct a matrix representing this transformation by simply reversing the order in which we normally apply scale and rotation:

 Matrix4x4 RotationInvariantScale(Vector3 axisAlignedScaleCoeffs, Quaternion orientation)
 {
     Matrix4x4 scaleMatrix = Matrix4x4.Scale(axisAlignedScaleCoeffs);
     Matrix4x4 rotationMatrix = Matrix4x4.TRS(Vector3.zero, orientation, Vector3.one);
     return scaleMatrix * rotationMatrix;
 }

Here I draw a model with this matrix as I change the input rotation:

(Made in Unity, using Graphics.DrawMesh to render using my custom matrix, since Unity won't allow us to set the matrix on a Transform component directly)

Depending on your full setup, you may find this is easier to execute using parenting: place the sphere's visual as the child of an otherwise invisible parent. Apply scaling on the parent object, and rotation on the visible child object. This can avoid hacking in a skewed matrix manually.