Cylinders and Quadric Surfaces

Cylinder

A cylinder (from Greek "roller" or "tumbler") is one of the most basic curvilinear geometric shapes. The surface is formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since antiquity. A cylinder can be seen as a polyhedral limiting case of an $n$-gonal prism where $n$ approaches infinity.

Cylinder

A right circular cylinder with radius $r$ and height $h$.

In common use, a cylinder is taken to mean a finite section of a right circular cylinder, i.e. the cylinder with the generating lines perpendicular to the bases, with its ends closed to form two circular surfaces. If the cylinder has a radius $r$ and length (height) $h$, then its volume is given by $V = \pi r^2h$, and its surface area is $A = 2\pi rh$ without the top and bottom, and $2\pi r(r + h)$ with them.

Quadric Surface

A quadric, or quadric surface, is any $D$-dimensional hypersurface in $(D+1)$-dimensional space defined as the locus of zeros of a quadratic polynomial. In coordinates $\{x_1, x_2, \cdots, x_{D+1} \}$, the general quadric is defined by the algebraic equation: 

$\displaystyle{\sum_{i,j=1}^{D+1} x_i Q_{ij} x_j + \sum_{i=1}^{D+1} P_i x_i + R = 0}$

Cylinders, spheres, ellipsoids, etc. are special cases of quadric surfaces.

Examples

Ellipsoid: 

$\displaystyle{{x^2 \over a^2} + {y^2 \over b^2} + {z^2 \over c^2} = 1}$

Sphere: 

$\displaystyle{{x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over a^2} = 1}$

Elliptic paraboloid: 

$\displaystyle{{x^2 \over a^2} + {y^2 \over b^2} - z = 0}$

Cone: 

$\displaystyle{{x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 0}$

Parabolic cylinder: 

$x^2 + 2ay = 0$

Ellipsoid

An ellipsoid given as ${x^2 \over a^2} + {y^2 \over b^2} + {z^2 \over c^2} = 1$.