How to define a solid cyclinder or any solid objects parametrically?

[DorumonSg]

How to define a solid cyclinder or any solid objects parametrically?

I can't figure out what do I do with the z axis for example a Cylinder :

x = 0.5*cos(theta)
y = sin(theta)

0*pi <= theta <= 2*pi

This will make an eclipse.

But wad about z?

I know we have to stretch z to the height we want. But how do we do that?

Lets take it I want the height to be 2.

So :

z = t

0 <= t <= 2

But it won't work. Becuz' it only stretches the z axis, infact it not only just stretches the z axis, it stretches the eclipse I defined using x y to the point z = 2 on the z axis.

I can't figure out how to stretch the eclipse properly on the z axis so it becomes a proper solid cyclinder?

 

[DorumonSg]

Oh and another question.

According to the notes my teacher gave me,

The equation of a circle is r^2 - x^2 - y^2 = 0 where r is the radius.

While the equation for a circle disk(half space) is

r^2 - x^2 - y^2 >= 0

Why is this so? And why does half space mean in 2D?