Technicalities when graphing in 3-space

[sc0tt]

Background info for technical question:
I just started a summer Calculus III course and the introductory topic is graphing in 3-space (just a warning I tend to over think everything). Our assignment was to visualize the graph of

Z = X$$^{2}$$ + y$$^{2}$$​

I plotted points in a table from -3, 3 for both x and y. I visualized this as a topographic table with the Z-values reaching out of my paper. This gives a shape that is similar to a bowl, but the corners stretch upwards (at 18 units) and the center at 0. That's what I drew.

My math teacher and everyone else in the class drew a paraboloid, meaning the top was a flat circle not pointed. The teacher argued that there is a point where all the Z values would be the same and that's what gives the circular shape. I argued that keeping to scale, the shape has corners and that choosing an arbitrary Z-value is creating the picture you want, not the actual graph.

Technicalities:
Does this function have a circular top? I believe if you were to expand the shape, say to -4, 4 for x and y, a section with a delta Z would be added to the previous graph. If we were to approach infinity we would be adding more and more pieces that were shaped like before creating the corners. At the same time I realize this is adding sort of a 4th dimension, because all the points, whether you go to infinity or not, exist simultaneously and so the thought of this delta-Z shape wouldn't exist. Right?!

Also, the thought of this delta Z shape led me to think of what the 3d cross section of this shape would be. Would it be sort of like a frustum from a cone, or would it be the before mentioned shape i.e. the difference between a graph with x and y bounds of -3 and 3 vs. -4 and 4?

My Conclusion:
My conclusion is that everything is arbitrary, it doesn't matter whether you think of it as a circular top or corners because the shape goes to infinity so you could end it however you want.

For the 3d cross sections, its also arbitrary. I would think sticking to the essence of the shape it would be a weird shaped 3d cross section with the delta Z, not a frustum-like shape, it also depends on what you view this 3d cross section as.

Hopefully some doctor out there has some insight on my problem and that everything isn't arbitrary.
Thanks if you read this looonnng post,
-Scott

 

[slider142]

You probably drew something like this:

Your teacher decided to arbitrarily highlight the circular cross-section of the graph, while you decided to draw as much of the graph as you had domain, highlighting the parabolic cross-sections. There is not really any reason to argue against either representation, although your teacher's representation is more native to cylindrical coordinates than Cartesian.

 

[sc0tt]

That makes sense, but is there no definitive shape for this graph? Because scales vary, any graph could be manipulated to look like anything. Isn't there standards for this kind of stuff?

And I'm starting to grasp my 3D cross section question. You would have to define which planes were intersecting this shape, for instance Z=1 and Z=3. This would create the frustum shape, but you can have any type of cross section really. I have no idea what my Delta Z shape is though, it sure does look funny.

 

[slider142]

You can cut the graph off using any shape you want, as different views of the graph will be useful for different objectives. The graph itself does not change, as the points (x, y, z) that satisfy the equation do not change. All you're changing is the location, orientation, and size of your bounding box that looks at some finite region of the graph.

 

[sc0tt]

Yea, that's the problem I'm running into I guess. Everyone has different manipulated versions of this graph, but when I try to visualize this graph, I never know how I should picture it. That is the most frustrating. It seems like there is no definitive answer to quench my visual problem, so I should just arbitrarily pick a shape and roll with it.