Cylinders and Quadric Surfaces

Definition The trace of a surface $S$ in a plane is the intersection of $S$ with that plane. For a surface given by an equation in $x$, $y$ and $z$, the trace in the horizontal plane $z = k$ is found by substituting $z = k$, which leaves an equation in $x$ and $y$ describing a curve in that plane. Traces in the planes $x = k$ and $y = k$ are found the same way. A trace can be a curve, a point, a pair of lines, or empty.

Definition Let $C$ be a curve in a plane and let $\ell$ be a line not parallel to that plane. The cylinder through $C$ with rulings parallel to $\ell$ is the union of all lines parallel to $\ell$ that pass through a point of $C$. In particular, if an equation in $x$, $y$ and $z$ does not contain $z$, it describes a cylinder with rulings parallel to the $z$ axis. A point $(x, y, z)$ lies on it exactly when $(x, y)$ satisfies the equation, whatever $z$ is. The same holds with the roles of the variables exchanged.

Definition For constants $a, b, c > 0$, the surface

is an ellipsoid centered at the origin. It meets the axes at $(\pm a, 0, 0)$, $(0, \pm b, 0)$ and $(0, 0, \pm c)$. Its trace in $z = k$ is an ellipse for $\lvert k \rvert < c$, a single point for $\lvert k \rvert = c$, and empty for $\lvert k \rvert > c$, and the same pattern holds in the other directions. When $a = b = c$ it is a sphere of radius $a$.

Definition For constants $a, b > 0$ and $c \ne 0$, the elliptic paraboloid and the hyperbolic paraboloid with vertex at the origin are

The elliptic one has horizontal traces that are ellipses on the side where $z/c > 0$, the vertex alone at $z = 0$, nothing on the other side, and parabolas as vertical traces. It opens upward when $c > 0$ and downward when $c < 0$. The hyperbolic one has hyperbolas as traces in $z = k$ for $k \ne 0$, the pair of lines $y = \pm \frac{b}{a} x$ for $k = 0$, and parabolas opening in opposite directions in the planes $x = k$ and $y = k$. Its shape near the origin is a saddle.

Definition For constants $a, b, c > 0$, the elliptic cone with vertex at the origin and axis along the $z$ axis is

Its traces in $z = k$ are ellipses for $k \ne 0$ and the single point at the origin for $k = 0$. Its traces in $x = 0$ and $y = 0$ are pairs of lines through the origin. If a point $(x, y, z)$ lies on the cone, so does $(tx, ty, tz)$ for every real $t$, so the cone is a union of lines through its vertex.

Definition For constants $a, b, c > 0$, the hyperboloid of one sheet and the hyperboloid of two sheets with axis along the $z$ axis are

With the right side equal to $1$, one minus sign on the left gives one sheet and two minus signs give two sheets, and the axis belongs to the variable whose sign differs from the other two. On the one-sheet surface every trace $z = k$ is an ellipse, never empty, so the surface is one connected piece. On the two-sheet surface the trace $z = k$ satisfies $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{k^2}{c^2} - 1$, which is empty for $\lvert k \rvert < c$, so the surface splits into two pieces.

Definition Replacing $x$, $y$ and $z$ in an equation by $x - x_0$, $y - y_0$ and $z - z_0$ moves the surface by the vector $\langle x_0, y_0, z_0 \rangle$ without changing its shape. A center, vertex or axis moves the same way. To recognize a shifted surface, complete the square in each variable that appears both squared and to the first power.

Definition A quadric surface is the set of points satisfying a second degree equation in $x$, $y$ and $z$. This lesson treats equations with no cross terms such as $xy$, $yz$ or $xz$. Cross terms can be removed by rotating the axes, which is not needed here. An equation without cross terms can be identified as follows.

1. If a variable is missing, the surface is a cylinder over the curve in the other two variables.
2. Complete the square in every variable that appears both squared and to the first power.
3. If all three variables are squared, move the constant to the right side, and if it is not zero, divide by it so the right side is $1$. With right side $1$, no minus signs on the left give an ellipsoid, one minus sign a hyperboloid of one sheet, two minus signs a hyperboloid of two sheets, and three minus signs the empty set. With right side $0$, mixed signs give a cone and equal signs give only the center point.
4. If two variables are squared and the third appears only to the first power, the surface is an elliptic paraboloid when the squares have equal signs and a hyperbolic paraboloid when they have opposite signs.
5. If only one variable is squared and another appears to the first power, the surface is a parabolic cylinder, as in $z = x^2$ or $x^2 = y + z$.

Degenerate cases also occur. The equation $x^2 + y^2 + z^2 = -1$ has no points, $x^2 + y^2 + z^2 = 0$ is a single point, $x^2 + y^2 = 0$ is the $z$ axis, $x^2 - y^2 = 0$ is the pair of planes $y = x$ and $y = -x$, and $x^2 = 1$ is the pair of parallel planes $x = 1$ and $x = -1$.