Ibn Al Haytham et les équations algébriques.

Pablo_de_retour · March 2019

Bonjour,

On parle sur le lien suivant : https://en.wikipedia.org/wiki/Alhazen's_problem du problème d'Alhazen célèbre en optique d'après certaines sources.
J'aimerais vous demander des clarifications autour du paragraphe suivant qui se trouve sur ce même lien ci-dessus :

Geometric formulation :

The problem comprises drawing lines from two points in a circle meeting at a third point on its circumference and making equal angles with the normal at that point. This is equivalent to finding the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in order to carom off the edge of the table and hit another ball at a second given point. Thus, its main application in optics is to solve the problem, "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 an equation of the fourth degree.

Ce paragraphe évoque la trouvaille d'une équation algébrique de $4$-ème degré dû à Alhazen, et résultant des calculs faits autour du problème d'Alhazen. De quelle équation il s'agit plus exactement, et comment il l'a obtenu ?

Merci d'avance.

Pablo_de_retour · March 2019

Sur le meme lien que j'ai évoqué plus haut, ( dans un autre paragraphe ), on parle d'une autre équation issue du problème d'Alhazen, mais cette fois çi de $10$- ème degré. est ce que vous connaissez cette équation, ainsi que sa méthode de formulation ?

Voici de quel paragraphe il s'agit :

Influence :

Ibn al-Haytham solved the problem using conic sections and a geometric proof, but later mathematicians such as Christiaan Huygens, James Gregory, Guillaume de l'Hôpital, Isaac Barrow, and many others, attempted to find an algebraic solution to the problem, using various methods, including analytic methods of geometry and derivation by complex numbers.[5] An algebraic solution to the problem was finally found in 1997 by the Oxford mathematician Peter M. Neumann.[6] Recently, Mitsubishi Electric Research Labs researchers solved the extension of Alhazen's problem to general rotationally symmetric quadric mirrors including hyperbolic, parabolic and elliptical mirrors.[7] They showed that the mirror reflection point can be computed by solving an eighth degree equation in the most general case. If the camera (eye) is placed on the axis of the mirror, the degree of the equation reduces to six.[8] Alhazen's problem can also be extended to multiple refractions from a spherical ball. Given a light source and a spherical ball of certain refractive index, the closest point on the spherical ball where the light is refracted to the eye of the observer can be obtained by solving a tenth degree equation.

Merci d'avance.

Ibn Al Haytham et les équations algébriques.

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