Conical Representation of Sphere

[Leo Authersh]

Is Sphere a more generalized form of Cone i.e. formed by 2 dimensional rotation to 360° of a cone?

Or is Cone a more generalized form of Sphere since sphere can be formed by rotating about Z axis a zero eccentric planar intersection of a cone?

@fresh_42 @FactChecker @WWGD

 

[phinds]

Is Sphere a more generalized form of Cone i.e. formed by 2 dimensional rotation to 360° of a cone?

Or is Cone a more generalized form of Sphere since sphere can be formed by rotating about Z axis a zero eccentric planar intersection of a cone?

@fresh_42 @FactChecker @WWGD

A sphere is a degenerate case of an ellipsoid just as a circle is a degenerate case of an ellipse.

 

[fresh_42]

See https://www.physicsforums.com/threads/quadratic-equation-with-three-variables.925132/#post-5839147

 

[Leo Authersh]

See https://www.physicsforums.com/threads/quadratic-equation-with-three-variables.925132/#post-5839147

This representation is excellent. I remember the term Hyperboloid.

But what confuses me is that, if the hyperboloid is rotated to 90°, we get a cube. How is a cube which is a linear geometric form that has one variable be formed by a Hyperboloid that has three variables?

 

[Leo Authersh]

@Infrared

 

[Mark44]

But what confuses me is that, if the hyperboloid is rotated to 90°, we get a cube.

Why do you think this?

How is a cube which is a linear geometric form that has one variable be formed by a Hyperboloid that has three variables?

This isn't right, either. Let's look at two dimensions first. The unit square in the first quadrant does not have a single equation. Instead, it has four equations, one for each side, along with inequalities that indicate the minimum and maximum values of the variable on each side. For example, the upper horizontal side would be represented by the equation y = 1, and the inequality $0 \le x \le 1$. There would be an equation/inequality pair for each side.

For a cube you would need equation/inequality pairs for each of the six faces.

Back to the hyperboloid. There are actually two kinds of hyperboloids -- hyperboloid of one sheet (or surface) and hyperboloid of two sheets (two distinct surfaces). If you take the hyperbola $x^2 - y^2 = 1$ and rotate it about the x-axis, you get a hyperboloid of two sheets (in three dimensions). If you rotate the same hyperbola about the y-axis, you get a hyperboloid of one sheet.

Do a web search on, say, wikipedia for hyperboloid to see the formulas and graphs of these quadric surfaces.

Mod note: Thread moved to General Mathematics -- the question isn't really about topology or analysis.