Beyond cubic polynomials: Applications?

Question

Cubic polynomials are crucially important in computer graphics: for example, cubic Bézier curves/surfaces, and cubic splines, which have many practical applications. Essentially visual continuity constraints force the polynomials to be cubic; quadratic would not suffice.

My question is:

Q. Are there practical applications that lead to a need to study polynomials of degree 4 or higher?

I am seeking justification for exploring polynomials of higher degree, justification that is external to pure mathematics (as is graphics).

---

         
          ^{(Image from Adobe bezier-pen-tool.)}

---

Answer 1

Shamir's secret sharing is a cryptographic application of polynomial interpolation where the order of the polynomial depends on the number of shares which you want to be required to access the secret. E.g. if you want 12 stars worth of generals to be required to launch your missiles, you take a random polynomial of degree 11 which passes through $(0, \text{launch code})$ and give four points on the polynomial to each 4-star general, three to each 3-star general, etc.

Answer 2

Quartics and higher degree polynomials frequently arise when intersecting lower degree curves, e,g, the intersection of two conics, or a line and a torus. Such curve intersections often occur in physical problems. For example, Wikipedia lists the following examples.

• In computer-aided manufacturing, the torus is a shape that is commonly associated with the endmill cutter. To calculate its location relative to a triangulated surface, the position of a horizontal torus on the z-axis must be found where it is tangent to a fixed line, and this requires the solution of a general quartic equation to be calculated.

• A quartic equation arises also in the process of solving the crossed ladders problem, in which the lengths of two crossed ladders, each based against one wall and leaning against another, are given along with the height at which they cross, and the distance between the walls is to be found.

• In optics, Alhazen's problem is "Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer." This leads to a quartic equation.

Similarly for surfaces and higher dimensions, e.g. intersections of such are ubiquitous in geometric modelling etc, e.g. see the excerpt below from a paper on quadric intersection algorithms.

The simplest of all the curved surfaces, quadrics (i.e., algebraic surfaces of degree two), are fundamental geometric objects, arising in such diverse contexts as geometric modeling, statistical classification, pattern recognition, and computational geometry. In geometric modeling, for instance, they play an important role in the design of mechanical parts; patches of natural quadrics (planes, cones, spheres and cylinders) and tori make up to 95% of all mechanical pieces according to Requicha and Voelcker [19].

Computing the intersection of two general quadrics is a fundamental problem and an explicit (i.e., parametric) representation of the intersection is desirable for most applications. Indeed, computing intersections is at the basis of many more complex geometric operations such as computing convex hulls of quadric patches [12], arrangements of sets of quadrics [2,10,18,22,29], and boundary representations of quadric-based solid models [13,21].

Answer 3

BCH and Reed-Solomon codes are classes of error correcting codes constructed from polynomials over finite fields. The polynomials occurring in such constructions have higher degrees. These examples also provide motivation for studying vector spaces over finite fields.

Answer 4

One important example is the Chebfun software package, which uses higher order polynomial interpolants on carefully chosen points to accurately represent a variety of functions.