Quadric Surfaces

The General Equation of a Quadric Surface

Characteristics

1. The highest degree of any variable is 2.
2. Terms include quadratic, linear, and constant terms.
3. The surface often has symmetry about a coordinate axis or plane.

Classification of Quadric Surfaces

The type of a quadric surface is largely determined by the signs of the three eigenvalues $\lambda_1, \lambda_2, \lambda_3$ of its quadratic-form matrix; degenerate cases such as cylinders also involve the linear terms.

Ellipsoid Family (three eigenvalues of the same sign)

• Sphere: all three eigenvalues equal and of the same sign.
• Ellipsoid: all three of the same sign but not equal.

Hyperboloid and Cone Family (two equal-sign, one different)

• Hyperboloid of one / two sheets: two eigenvalues share a sign and the third differs (nonzero constant term; the sheet count depends on the sign of the constant).
• Elliptic cone: two share a sign and the third differs, and the surface passes through the origin (zero constant term).

Paraboloid Family (exactly one zero eigenvalue)

• Elliptic paraboloid: one zero eigenvalue, the other two share a sign.
• Hyperbolic paraboloid (saddle surface): one zero eigenvalue, the other two differ in sign.

Cylinder Family (degenerate)

• Elliptic cylinder: one zero eigenvalue, the other two share a sign, with no linear term in the zero direction.
• Hyperbolic cylinder: one zero eigenvalue, the other two differ in sign, no linear term in the zero direction.
• Parabolic cylinder: two zero eigenvalues, one nonzero, with a linear term present.

Standard Forms

Sphere

$$(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 = R^2$$

Ellipsoid

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

Elliptic Cone

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}$$

Elliptic Paraboloid

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z}{c}$$

Hyperbolic Paraboloid (Saddle)

$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = \frac{z}{c}$$

Hyperboloid of One Sheet

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$$

Hyperboloid of Two Sheets

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1$$

Geometric Properties

Symmetry

Most quadric surfaces are symmetric about a coordinate axis or coordinate plane.

Sections

• Ellipsoid family: sections by planes are ellipses.
• Paraboloid family: sections are parabolas.
• Hyperboloid family: sections are hyperbolas.

Behavior at Infinity

• Ellipsoid: bounded.
• Paraboloid: opens to infinity.
• Hyperboloid: has asymptotic surfaces (the cone).

Why Standardization Matters

By a coordinate transformation (principal-axis / orthogonal transformation), the general quadric equation can be reduced to a standard form, making the type and properties of the surface easy to identify and study.

Worked Example

Identify the surface $\frac{x^2}{4} + \frac{y^2}{9} - \frac{z^2}{16} = 1$.

Solution: two positive squared terms and one negative, equal to 1—this is a hyperboloid of one sheet. Its trace in the plane $z=0$ is the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$.

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Summary

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