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
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 .
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 , arrangements of
sets of quadrics [2,10,18,22,29], and boundary representations of quadric-based solid models [13,21].