# How to find change in angle when vertical Scale of depth axis is modified?

Let Q be the Quaternion that perform a rotation of angle degrees around some axis of rotation. Later I change vertical exaggeration of my Vertical axis (z) to some value VS, e.g. make the vertical (z) scale to 50% or 25% etc without altering the z-values/range? How does it effect angle (w) and orientation(x,y,z) of Quaternion Q. How do I find this change in angle.

• When you scale the vertical axis, do you mean you change the shapes of things, so that a sphere around the origin would become a flattened or elongated spheroid? In that case any rotation about the vertical axis, or $180$ degrees around an axis in the $x,y$ plane, has the same effect as before (in particular, it rotates the spheroid onto itself); any other rotation will not be exactly reproducible, so you have to make some arbitrary choices. One possible choice is to use the exact same quaternion as before. – David K Oct 22 '18 at 12:39
• If "scaling" the vertical axis means that the shapes of all objects remain the same but the basis vector in the $z$ direction gets longer or shorter, then after scaling a sphere is still a sphere but its maximum $z$ measurement might be half of its maximum $x$ or $y$ measurements. In that case, if you blindly apply a quaternion to the new coordinate system as if it were an orthonormal system, in general you won't get a rotation. To get a rotation, you could first transform the coordinates to an orthonormal system, then rotate, then reverse the first transformation. – David K Oct 22 '18 at 12:43
• So the answer depends on what you really mean to do: how are you interpreting the scaling, how do you interpret a rotation. The words "without altering the z-values/range" suggest that you mean to flatten or elongate spheres by this change of vertical scale. Then in general you cannot reproduce the effects of most rotations on objects, so the question then is how should we decide what new rotation the quaternion should represent? – David K Oct 22 '18 at 12:53
• Its like this if the point was {0,0,1} after changing the vertical scale e.g.to 10 it would be render on {0,0,10} position. its effect same like if sphere would get elongate, or if i have a curved line would gets straighten. so my question is around that Quaternion, Is it possible to get change in rotation angle and update Quaternion Q as well? – Megamind Oct 23 '18 at 4:34
• In general you don't "update" any rotations after a transformation like this. It is unclear why you think this would be possible, or what you expect it to do. For example, let's take the square with coordinates $(0,\pm1,\pm1)$ (in the $y,z$ plane, centered around the origin) and rotate it $45$ degrees around the $x$ axis. You can work out where the square's vertices are after rotation. Now apply your scaling factor $10$ so the square becomes the rectangle $(0,\pm1,\pm10).$ How should it be rotated? – David K Oct 23 '18 at 12:24