Applications
Polynomials of high degrees often appear in problems involving optimization, and sometimes these polynomials happen to be quartics, but this is a coincidence.
Quartics often arise in computer graphics and during ray-tracing against surfaces such as quadric or tori surfaces, which are the next level beyond the sphere and developable surfaces.
Another frequent generator of quartics is the intersection of two ellipses.
In computer-aided manufacturing, the torus is a common shape 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. Over 10% of the computational time in a CAM system can be consumed simply calculating the solution to millions of quartic equations.
A program demonstrating various analytic solutions to the quartic was provided in Graphics Gems Book V. However, none of the three algorithms implemented are unconditionally stable. In an updated version of the paper, which compares the 3 algorithms from the original paper and 2 others, it is demonstrated that computationally stable solutions exist only for 4 of the possible 16 sign combinations of the quartic coefficients.
Read more about this topic: Quartic Function