An Overview of Quadric Surfaces

The General Equation of a Quadric Surface

The Features of the Equation

1. The highest degree: the highest degree of the variables in the equation is 2
2. The types of terms: it contains quadratic terms, linear terms and a constant term
3. Symmetry: the equation may have symmetry about the coordinate axes

The Classification of Quadric Surfaces

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

The Ellipsoid Family (three eigenvalues of the same sign)

• The sphere: the three eigenvalues are equal and of the same sign
• The ellipsoid: the three eigenvalues are of the same sign but not equal

The Hyperboloid and Cone Family (two of one sign, one of another)

• The hyperboloid of one sheet / of two sheets: two eigenvalues of the same sign and one of another (with a nonzero constant term; which of the two it is depends on the sign of the constant term)
• The elliptic cone: two eigenvalues of the same sign and one of another, and the surface passes through the origin (zero constant term)

The Paraboloid Family (exactly one zero eigenvalue)

• The elliptic paraboloid: one eigenvalue is zero and the other two are of the same sign
• The hyperbolic paraboloid (the saddle surface): one eigenvalue is zero and the other two are of different signs

The Cylinder Family (degenerate surfaces)

• The elliptic cylinder: one eigenvalue is zero and the other two are of the same sign, with no linear term in the corresponding direction
• The hyperbolic cylinder: one eigenvalue is zero and the other two are of different signs, with no linear term in the corresponding direction
• The parabolic cylinder: two eigenvalues are zero and one is nonzero, and there is a linear term

The Geometric Properties of Quadric Surfaces

Symmetry

Most quadric surfaces have symmetry about the coordinate axes or the coordinate planes.

The Nature of the Sections

• the ellipsoid family: the sections by planes are ellipses
• the paraboloid family: the sections by planes are parabolas
• the hyperboloid family: the sections by planes are hyperbolas

The Behaviour at Infinity

• the ellipsoid: bounded, with its asymptote behaviour at infinity
• the paraboloid: opens out towards infinity
• the hyperboloid: has asymptotic surfaces

Coordinate Transformations

Why Standardisation Matters

Through a coordinate transformation, the equation of a general quadric surface can be turned into a standard form, which makes it easy to identify and to study.

The Principal Axis Transformation

Through an orthogonal transformation, the quadratic form can be turned into its standard form, which then determines the type of the surface.

---

Summary

Symbols Used in This Article