what is the right circular of cone and what is the right circular of cylinder

I have some questions.

1)what is the parametrization of cone and what is the parametrization of cylinder?

2) what is the right circular of cone and what is the right circular of cylinder?

I hope someone can answer these question

thanks

Solution 1:

You shouldn't say "the" parameterization, because there are several of them.

For a cylinder, here is one parameterization:

$$S(u,v) = (a\cos u, a\sin u, v)$$

This is a right circular cylinder with radius $a$ whose centerline is the $z$-axis. Its equation is $x^2 + y^2 = a^2$.

For a cone, here is one parameterization:

$$S(u,v) = (kv\cos u, kv\sin u, v)$$

This is a circular cone with its tip at the origin and whose centerline is the $z$-axis. The half-angle of this cone is $\alpha = \tan^{-1}k$, as shown in the picture below. It's equation is $x^2 + y^2 = kz^2$.

There is a common special case: if $k=1$, then the equation of the cone becomes simply $x^2 + y^2 = z^2$. In this case, the half-angle of the cone is $\alpha = \tan^{-1}(1)$, which is $\pi/4$, or 45 degrees. So the total "spread" angle of the cone is 90 degrees.

If you want cones and cylinders that are in more general positions, the equations are more complicated. But, it's usually best to work with shapes that are in "standard" position, as given above, so that the equations are nice and simple.

Question 2: The word "right" in the phrase "right circular cylinder" is a bit similar to the word "upright". It just means that the centerline of the cone or cylinder is perpendicular to its base circle. So, if the base circle is in the $xy$ plane, this means that the centerline is parallel to the $z$ axis. In other words, these are the commonly-seen types of cones and cylinders, not strange tilted ones.