Math 2.0

Geometry Problem

Alhazen's problem

Builds on Law of reflectionCircleConstructible number

Problem

Let Γ\Gamma be a circle with centre OO in the plane. Let AA and BB be two points of the plane that do not lie on Γ\Gamma. Find the points PP on Γ\Gamma at which a ray from AA is reflected to BB by the law of reflection. These are the points PP such that the line OPOP bisects the angle APBAPB.

Remarks

The circle is the cross-section of a spherical mirror by the plane through AA, BB and the centre of the sphere. So the problem asks where to look in a spherical mirror to see a given point. Ibn al-Haytham (Alhazen) solved it geometrically with the help of conic sections. In coordinates, the condition on PP leads to a polynomial equation of degree 44. Peter M. Neumann proved in 1997 that the points PP cannot in general be constructed with ruler and compass, so the problem is not a problem of constructible numbers.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Calculus of variations, concept, no evidence levelCalculus of variationsconceptCircle, concept, no evidence levelCircleconceptConstructible number, concept, no evidence levelConstructible numberconceptEuler–Lagrange equation, concept, no evidence levelEuler–Lagrange equati…conceptGeodesic, concept, no evidence levelGeodesicconceptNormal vector, concept, no evidence levelNormal vectorconceptReflection, concept, no evidence levelReflectionconceptFermat's principle, statement, no evidence levelFermat's principlestatementLaw of reflection, statement, no evidence levelLaw of reflectionstatementAlhazen's problem, problem, no evidence levelAlhazen's problemproblem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Calculus of variations concept
  2. Circle concept
  3. Constructible number concept
  4. Euler–Lagrange equation concept
  5. Geodesic concept
  6. Normal vector concept
  7. Reflection concept
  8. Fermat's principle statement
  9. Law of reflection statement
  10. Alhazen's problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Alhazen%27s_problem
About this page
TagAZ2NV4EBAY
URIm2://AZ2NV4EBAY
Kindproblem
WritingW0 stub
MSC51M15, 78A05, 51M04
StatusThis copy of the site has no build data. Run m2 status AZ2NV4EBAY.
AboutLaw of reflection
UsesLaw of reflection, Circle, Constructible number
Same as
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.