Cylinders and Quadric Surfaces

Cylinders

Definition 1. A cylinder is a surface that is generated by moving a straight line along a given planar curve while holding the line parallel to a given fixed line. The curve is called a generating curve for the cylinder. In solid geometry, where cylinder means circular cylinder, the generating curves are circles, but now we allow generating curves of any kind.

Remark

Remark. any curve $f(x, y) = c$ in the $xy$-plane defines a cylinder parallel to the $z$-axis whose equation is also $f(x, y) = c$. For instance, the equation $x^2 + y^2 = 1$ defines the circular cylinder made by the lines parallel to the $z$-axis that pass through the circle $x^2 + y^2 = 1$ in the $xy$-plane. In a similar way, any curve $g(x, z) = c$ in the $xz$-plane defines a cylinder parallel to the $y$-axis whose space equation is also $g(x, z) = c$. Any curve $h(y, z) = c$ defines a cylinder parallel to the $x$-axis whose space equation is also $h(y, z) = c$. The axis of a cylinder need not be parallel to a coordinate axis, however.

Quadric Surfaces

Definition 2. A quadric surface is the graph in space of a second-degree equation $$Ax^2 + By^2 + Cz^2 +Dxy + Exz + Fyz + Gz + Hy + Iz + J= 0,$$ where $A, B, C, D, E, F, G, H, I$ and $J$ are constants. Quadric Sufaces can be rewritten as ellipsoids, elliptical paraboloids, elliptical cones, hyperboloid of one sheet, hyperboloid of two sheets, and hyperbolic paraboloid. (Spheres are special cases of ellipsoids).

The equations of quadric surfaces being symmetric with respect to the zzz-axis

(1) Ellipsoid: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$$
(2) Elliptical Paraboloid: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z}{c}$$
(3) Ellpitcal Cone: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}$$
(4) Hyperboloid of One Sheet: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$$
(5) Hyperboloid of One Sheet: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1$$
(6) Hyperbolic Paraboloid: $$\frac{y^2}{b^2} - \frac{x^2}{a^2} = \frac{z}{c}, \text{ } c > 0$$