What are the equations for conic sections in 4 dimensions?

[MarcL]

Homework Statement

The problem is given in the picture attached. However we are given the equation of a 4 dimensional shape with certain choice given to explain the shape in that dimension.

Homework Equations

Equations of different shape such as:

sphere: √(x^2+y^2+z^2) = r
Ellipse: (x-h)/a^2 + (y-k)/b^2 = K
Hyperbola: (x-h)/a^2 - (y-k)/b^2 = 1

The Attempt at a Solution

Well for question one and two, I said 1 ( which is w=x^2+y^2+z^2) is a "a collection of equally spaced concentric spheres" and 2 ( w=√(x^2+y^2+z^2) was "a collection of unequally spaced concentric spheres ".Though I know the difference between both is the radius where one is w and the other is √w. But why does have a radius of √w make the sphere unequally spaced?

My teacher barely touched the topic so I have a hard time figuring out how to visualize these shapes.

 

[MarcL]

I couldn't see my attachment... here it is ( trying again)

 

[brmath]

You are correct about #1.

 

[brmath]

You are correct about #1. I do know the answers to some of the others, and can guess some, but perhaps it would be best to tell you how I would go about this if I had to do the problem.

Go back to 3 dimensions, and look up the equations of the conic sections. Answer the questions when z is omitted. It has to be similar in 4 dimensions.