Quadric Surfaces

Introduction

A "quadric surface" is an algebraic surface, defined by a quadratic (order 2) polynomial. Non-degenerate quadrics in $\mathbb{R}^3$ (familiar 3-dimensional Euclidean space) are categorised as either ellipsoids, paraboloids, or hyperboloids. Our collection contains most of the different types of quadric, including degenerate cases. The links on the left take you to the models.

 

Degree 2 Curves

Let us start by looking at the two-dimensional analogue to quadric surfaces: degree 2 curves in a plane. These are defined by a quadratic form in two-dimensional Euclidean space, $\mathbb{R}^2$ (that is, they are defined on the zero set of a polynomial in $x, y$ where the highest power involved is two, or the $xy$ term has a non-zero coefficient). For example, the familiar unit circle is given by $x^2 + y^2 - 1 =0$.

Degree two curves fall into four (non-degenerate) categories, and are known as "conic sections", since each of a circle, ellipse, parabola, and hyperbola can be realised as the intersection of a cone with a plane. Model VI 5 demonstrates how an ellipse, parabola, or hyperbola can be found by intersecting planes with a cone. Sadly it is missing two pieces, so only shows a hyperbola and parabola. To learn more about conic sections and how they arise in everyday life, see Jill Britton's interesting article.

 

Model VI 5: Complete Model; Showing Hyperbola; Showing Parabola

Model VI 5Model VI 5 - Piece RemovedModel VI 5

 

Defining Quadric Surfaces

 

Before we look at the details of our models, let us first give some concrete definitions for ellipsoids, hyperboloids, and paraboloids.

Ellipsoids are defined by $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$$Imposing $a = b$ gives a "spheroid", and imposing $a = b = c$ gives a sphere. Ellipsoid shapes can be used in architecture, yielding impressive buildings.

Paraboloids may be split into two further cases: elliptic and hyperbolic. An "elliptic paraboloid" is defined by $$\frac{x^2}{a^2} + \frac{y^2}{a^2} - z = 0$$A "circular paraboloid" is the special case of this where $a = b$. A "hyperbolic paraboloid" is given by: $$\frac{x^2}{a^2} - \frac{y^2}{a^2} - z = 0$$Elliptic paraboloids can be attained from ellipsoids by letting the surface stretch out to infinity along one axis.

The general form of a hyperboloid is an "elliptic hyperboloid", given by: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = \pm 1$$If the right hand side is positive we get a hyperboloid "of one sheet", and for the negative case we get a hyperboloid "of two sheets". As before, the special case where $a = b$ gives a "circular hyperboloid". Hyperboloids of one sheet are ruled surfaces, as illustrated by this exhibit. It is interesting to note that while some quadrics are not ruled surfaces in this context, all quadric surfaces are ruled surfaces when considered in projective space.

Degenerate quadric surfaces include (elliptic and circular) cones, (elliptic, circular, parabolic, and hyperbolic) cylinders. Cones can be attained by applying a limiting process to hyperboloids, taking $a, b, c$ to infinity, and are also ruled surfaces.

 

Determining Conics and Ellipsoids

Any conic section is uniquely determined by knowing the position of five distinct points lying on it. Strictly speaking, the points must be in general position - no three of them can be collinear (all lying on one line). So, given five such points in $\mathbb{R}^2$, there is a unique conic passing through them. However, stating five points is not a very practical means of determining a conic (the following construction is much more difficult than solving a degree 2 polynomial), however elegant such constructions may look. An ellipse is uniquely determined by its major and minor axes, and can be constructed by hand using pins and string.

An elllipsoid, like an ellipse, is determined by its axes. There are three principal axes for an ellipsoid though, shown in red here. Knowing the intersection between an ellipsoid and any three planes uniquely determines the ellipsoid, and in fact any such intersection with a plane is an ellipse (or a circle, which is a special case of an ellipse). Again, the construction involved here is far too complicated for this characterisation to provide a useful way of describing ellipsoids [1].

There is a much more elegant way to determine the type of a conic or quadric, using linear algebra. The defining equation of any ellipsoid or hyperboloid can be written in the form $\mathbf{x}^\top\mathbf{A}\mathbf{x}=1$, where $\mathbf{x}=(x,y,z)$ and $\mathbf{A}$ is a real symmetric matrix (equal to its transpose). A famous theorem in linear algebra says that we can find a rotation matrix $\mathbf{P}$, satisfying $\mathbf{PP}^\top=\mathbf{P}^\top\mathbf{P}=\mathbf{I}$, the identity matrix,  such that $\mathbf{PAP}^\top =\mathbf{D}$, where the matrix $\mathbf{D}$ is diagonal with the eigenvalues of $\mathbf{A}$ as its diagonal entries. The eigenvectors of $\mathbf{A}$ are then the principal axes of the quadric. We can write $$\mathbf{x}^\top\mathbf{Ax}=\mathbf{x}^\top \mathbf{P}^\top \mathbf{PAP}^\top\mathbf{Px} = \mathbf{X}^\top\mathbf{D}\mathbf{X}=1,$$ where $\mathbf{X}=\mathbf{Px}$. We see that, in the rotated coordinates $\mathbf{X}$, the coordinate axes are the principal axes of the quadric and the signs of the eigenvalues (diagonal elements of $\mathbf{D}$) determine the type. If they are all positive, it is an ellipsoid; two positive and one negative gives a hyperboloid of one sheet; one positive and two negative gives a hyperboloid of two sheets. If one or more eiegnvalues vanish, we have a degenerate case such as a paraboloid,or a cylinder or even a pair of planes.

 

References

[1] P.P. Klein, On the Ellipsoid and Plane Intersection Equation (2012), http://www.scirp.org/journal/PaperInformation.aspx?PaperID=24506#.VbeCmv...