Christmas Tree Modeling in GeoGebra AR

Key Questions: Notice how the blue surface contains the ceiling (greatest integer) function. If the greatest integer function symbols were removed, what would this surface look like instead? Why is this tree "off center"? That is, why isn't (0,0,0) located on the line (parallel to the zAxis) that passes through the center of the pink cylinder? What point on the plane z = 0 DOES lies on the line that is parallel to the zAxis AND passes through center of the cylinder? How can we mathematically justify this? What is the highest z-value attained by the blue tree part?

Quick Demo