Quadric Surfaces

Definition: A quadric surface is a second-degree surface $Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz = J$ of the three variables $x, y, z$.

We will now look at some examples of quadric surfaces.

Spheres

Figure	Surface
	Spheres: Equations that can be reduced into the form by completing the square to $(x - x_0)^2 + (y - y_0)^2 + (z-z_0)^2 = k^2$ form a sphere (a hollow ball) centered at the point $(x_0, y_0, z_0)$ and that has radius $k$. If $A = B = C \neq 0$ from the general equation given above, then if the equation represents a quadric surface then it will also be a sphere.

Ellipsoids

Figure	Surface
	Ellipsoids: Equations in the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$, $a, b, c \neq 0$ represents ellipsoids with semi-axes of lengths $a$, $b$, and $c$.

Cylinders

Figure	Surface
	Right Circular Cylinders: Equations in the form $x^2 + y^2 = k^2$, $x^2 + z^2 = k^2$, or $y^2 + z^2 = k^2$, $k \neq 0$ represent right circular cylinders with circular cross sections to the planes parallel to $z = k$, $y = k$, and $x = k$ respectively.
	Elliptic Cylinders: Equations in the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $\frac{x^2}{a^2} + \frac{z^2}{c^2} = 1$, and $\frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$, $a, b, c \neq 0$ represent cylinders with elliptic cross sections to the planes parallel to $z = k$, $y = k$ and $z = k$ respectively.
	Parabolic Cylinders: Equations in the form $x = y^2$, $x = z^2$, $y = x^2$, $y = z^2$, $z = x^2$, and $z = y^2$, $a, b, c \neq 0$ represent cylinders with parabolic cross sections to the planes parallel to $z = k$, $y = k$, and $x = k$ respectively.
	Hyperbolic Cylinders: Equations in the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, $\frac{x^2}{a^2} - \frac{z^2}{c^2} = 1$, $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$, $\frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$, $\frac{z^2}{c^2} - \frac{x^2}{a^2} = 1$, and $\frac{z^2}{c^2} - \frac{y^2}{b^2} = 1$, $a, b, c \neq 0$ represent cylinders with hyperbolic cross sections to the planes parallel to $z = k$, $y = k$, and $x = k$ respectively.

Cones

Figure	Surface
	Right-Circular Cone: Equations in the form $x^2 + y^2 = z^2$, $x^2 + z^2 = y^2$, and $y^2 + z^2 = x^2$ represent right-circular cones with circular cross sections to the planes parallel to $z = k$, $y = k$, and $x = k$ respectively.
	Elliptic Cone: Equations in the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}$, $\frac{x^2}{a^2} + \frac{z^2}{c^2} = \frac{y^2}{c^2}$ and $\frac{y^2}{b^2} + \frac{z^2}{c^2} = \frac{x^2}{a^2}$ represent elliptic cones with elliptic cross sections to the planes parallel to $z = k$, $y = k$, and $x = k$ respectively.

Paraboloids

Figure	Surface
	Elliptic Paraboloids: Equations in the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z}{c}$, $\frac{x^2}{a^2} + \frac{z^2}{c^2} = \frac{y}{b}$, and $\frac{y^2}{b^2} + \frac{z^2}{c^2} = \frac{x}{a}$, $a, b, c \neq 0$ represent elliptic paraboloids with elliptic cross sections to the planes parallel to $z = k$, $y = k$ and $x = k$ respectively.
	Hyperbolic Paraboloids: Equations in the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = \frac{z}{c}$, $\frac{x^2}{a^2} - \frac{z^2}{c^2} = \frac{y}{b}$, and $\frac{y^2}{b^2} - \frac{z^2}{c^2} = \frac{x}{a}$, $a, b, c \neq 0$ represent hyperbolic paraboloids with hyperbolic cross sections to the planes parallel to $z = k$, $y = k$, and $x = k$ respectively.

Hyperboloids

Figure	Surface
	Hyperboloids of One Sheet: Equations in the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$, $\frac{x^2}{a^2} + \frac{z^2}{c^2} - \frac{y^2}{b^2} = 1$, and $\frac{y^2}{b^2} + \frac{z^2}{c^2} - \frac{x^2}{a^2} = 1$, $a, b, c \neq 0$ represent hyperboloids of one sheet.
	Hyperboloids of Two Sheets: Equations in the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1$, $\frac{x^2}{a^2} + \frac{z^2}{c^2} - \frac{y^2}{b^2} = -1$, and $\frac{y^2}{b^2} + \frac{z^2}{c^2} - \frac{x^2}{a^2} = -1$, $a, b, c \neq 0$ represent hyperboloids of two sheets.

Identifying Quadric Surfaces

It is a good practice to remember some of the general equations for the quadric surfaces noted above, however, it is tedious to remember them all. We will now look at a method of identifying quadric surfaces, but before we do so, we will look at the following definition that will allow us to identify these surfaces.

Definition: Let $S$ be a surface. A Trace or Cross-Section to $S$ is a curve that intersects $S$.

Often times, looking at traces/cross-sections on planes that are on the $xy$, $yz$, and $xz$ planes will be the most useful in determining what surface an equation represents, provided the surface is centered around one of these axes.

Example 1

Determine the quadric surface $S$ represented by the equation $\frac{x^2}{4} + \frac{y^2}{16} + \frac{z^2}{81} = 1$.

Let $x = 0$. Then the trace of $S$ on the $yz$-plane is the equation $\frac{y^2}{16} + \frac{z^2}{81} = 1$ which is an ellipse with semi-minor axis $4$ and semi-major axis $9$.

Let $y = 0$. Then the trace of $S$ on the $xz$-plane is the equation $\frac{x^2}{4} + \frac{z^2}{81} = 1$ which is an ellipse with semi-minor axis $2$ and semi-major axis $9$.

Let $z = 0$. Then the trace of $S$ on the $xy$-plane is the equation $\frac{x^2}{4} + \frac{y^2}{16} = 1$ which is an ellipse with semi-minor axis $2$ and semi-major axis $4$.

Putting these all together, we can see that $S$ represents an ellipsoid.

Example 2

Determine the quadric surface $S$ represented by the equation $x = y^2 + z^2$.

Let $x = 0$. Then the trace of $S$ on the $yz$-plane is the equation $0 = y^2 + z^2$ which is just the point $(0, 0, 0)$.

Let $y = 0$. Then the trace of $S$ on the $xz$-plane is the equation $x = z^2$ which is a parabola.

Let $z = 0$. Then the trace of $S$ on the $xy$-plane is the equation $x = y^2$ which is a parabola.

Putting these all together, we can see that $S$ represents a paraboloid.