Editing Quartic function (section)

==Applications==
Each [[coordinate]] of the intersection points of two [[conic section]]s is a solution of a quartic equation. The same is true for the intersection of a line and a [[torus]]. It follows that quartic equations often arise in [[computational geometry]] and all related fields such as [[computer graphics]], [[computer-aided design]], [[computer-aided manufacturing]] and [[optics]]. Here are examples of other geometric problems whose solution involves solving a quartic equation.

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 {{math|''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.<ref>{{Cite web|url=http://people.math.gatech.edu/~etnyre/class/4441Fall16/ShifrinDiffGeo.pdf|title=DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces, p. 36|website=math.gatech.edu}}</ref>

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.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Crossed Ladders Problem|url=https://mathworld.wolfram.com/CrossedLaddersProblem.html|access-date=2020-07-27|website=mathworld.wolfram.com|language=en}}</ref>

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.<ref name=MacTutor>{{MacTutor|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham}}</ref><ref>{{citation|title=Scientific Method, Statistical Method and the Speed of Light|first1=R. J.|last1=MacKay|first2=R. W.|last2=Oldford|journal=Statistical Science|volume=15|issue=3|date=August 2000|pages=254–78|doi=10.1214/ss/1009212817|mr=1847825|doi-access=free}}</ref><ref name=Weiss>{{Citation|last = Neumann|first = Peter M.|author-link = Peter M. Neumann|journal = [[American Mathematical Monthly]]|title = Reflections on Reflection in a Spherical Mirror|year = 1998|volume = 105|issue = 6|pages = 523–528|doi = 10.2307/2589403|jstor = 2589403}}</ref>

Finding the [[distance of closest approach of ellipses and ellipsoids#Distance of closest approach of two ellipses|distance of closest approach of two ellipses]] involves solving a quartic equation.

The [[eigenvalue]]s of a 4×4 [[matrix (mathematics)|matrix]] are the roots of a quartic polynomial which is the [[characteristic polynomial]] of the matrix.

The characteristic equation of a fourth-order linear [[difference equation]] or [[differential equation]] is a quartic equation. An example arises in the [[Bending#Timoshenko-Rayleigh theory|Timoshenko-Rayleigh theory]] of beam bending.<ref>{{Cite book|last=Shabana|first=A. A.|url=https://books.google.com/books?id=G2UyBTji18oC&q=Timoshenko-Rayleigh+theory&pg=PA2|title=Theory of Vibration: An Introduction|date=1995-12-08|publisher=Springer Science & Business Media|isbn=978-0-387-94524-8|language=en}}</ref>

[[Intersection (Euclidean geometry)|Intersections]] between spheres, cylinders, or other [[quadric]]s can be found using quartic equations.