Morphing Surfaces in GeoGebra AR

KEY QUESTIONS FOR THE FIRST TWO SCREENCASTS:
After the first transformation (in each screencast), cross sections parallel to the plane [i]z[/i] = 0 are SQUARES. How can we mathematically justify this for each screencast? [br][br]For each transformation after the first (in each screencast), are there any regions over the xy-plane for which the surface is UNDEFINED? If so, what are these regions? How can we tell?
MORPHING A PARABOLOID
MORPHING A CONE
KEY QUESTIONS FOR THE NEXT SCREENCAST (BELOW):
After the transformations are completed (for both pyramids), how would you describe regions (over the xy-plane) for which both surfaces are undefined? How do we know this to be true?
MORPHING A SQUARE PYRAMID
– Tim Brzezinski

Information: Morphing Surfaces in GeoGebra AR