Editing Specular reflection (section)

== Law of reflection ==
[[File:Heart of the City water feature Sheffield - geograph.org.uk - 618552.jpg|thumb|Specular reflection from a wet metal sphere]]
[[File:Marble ball - Kongens Have - Copenhagen - DSC07898.JPG|thumb|Diffuse reflection from a marble ball]]
When light encounters a boundary of a material, it is affected by the optical and electronic response functions of the material to electromagnetic waves. Optical processes, which comprise [[Reflection (physics)|reflection]] and [[refraction]], are expressed by the difference of the refractive index on both sides of the boundary, whereas [[reflectance]] and [[Absorption (optics)|absorption]] are the real and imaginary parts of the response due to the [[electronic structure]] of the material.<ref>{{cite book|last1=Fox|first1=Mark|title=Optical properties of solids|url={{google books |plainurl=y |id=5WkVDAAAQBAJ}}|date=2010|publisher=Oxford University Press|location=Oxford|isbn=978-0-19-957336-3|page=1|edition=2nd}}</ref>
The degree of participation of each of these processes in the transmission is a function of the frequency, or wavelength, of the light, its polarization, and its angle of incidence. In general, reflection increases with increasing angle of incidence, and with increasing absorptivity at the boundary. The [[Fresnel equations]] describe the physics at the optical boundary.

Reflection may occur as specular, or mirror-like, reflection and [[diffuse reflection]]. Specular reflection reflects all light which arrives from a given direction at the same angle, whereas diffuse reflection reflects light in a broad range of directions. The distinction may be illustrated with surfaces coated with [[glossy]] [[paint]] and [[Matte (surface)|matte]] paint. Matte paints exhibit essentially complete diffuse reflection, while glossy paints show a larger component of specular behavior. A surface built from a non-absorbing powder, such as plaster, can be a nearly perfect diffuser, whereas polished metallic objects can specularly reflect light very efficiently. The reflecting material of mirrors is usually aluminum or silver.

Light propagates in space as a wave front of electromagnetic fields. A ray of light is characterized by the direction normal to the wave front (''wave normal''). When a ray encounters a surface, the angle that the wave normal makes with respect to the [[surface normal]] is called the [[angle of incidence (optics)|angle of incidence]] and the plane defined by both directions is the [[plane of incidence]]. Reflection of the incident ray also occurs in the plane of incidence.

The law of reflection states that the angle of reflection of a ray equals the angle of incidence, and that the incident direction, the surface normal, and the reflected direction are [[coplanar]].

When the light is incident perpendicularly to the surface, it is reflected straight back in the source direction.

The phenomenon of reflection arises from the [[diffraction]] of a [[plane wave]] on a flat boundary. When the boundary size is much larger than the [[wavelength]], then the electromagnetic fields at the boundary are oscillating exactly in phase only for the specular direction.

=== Vector formulation ===
{{see also|Snell's law#Vector form}}

The law of reflection can also be equivalently expressed using [[linear algebra]]. The direction of a reflected ray is determined by the vector of incidence and the [[surface normal]] vector. Given an incident direction <math>\mathbf{\hat{d}}_\mathrm{i}</math> from the light source to the surface and the surface normal direction <math>\mathbf{\hat{d}}_\mathrm{n},</math> the specularly reflected direction <math>\mathbf{\hat{d}}_\mathrm{s}</math> (all [[unit vector]]s) is:<ref>{{cite book |last=Haines |first=Eric |year=2021 |editor-last1=Marrs |editor-first1=Adam |editor-last2=Shirley |editor-first2=Peter |editor-last3=Wald |editor-first3=Ingo |title=Ray Tracing Gems II |publisher=Apress |pages=105–108 |chapter=Chapter 8: Reflection and Refraction Formulas |doi=10.1007/978-1-4842-7185-8_8 |isbn=978-1-4842-7185-8|s2cid=238899623 }}</ref><ref>{{cite book | last = Comninos | first = Peter | title = Mathematical and computer programming techniques for computer graphics | publisher = Springer | year = 2006 | url = {{google books |plainurl=y |id=Kdb7-YnnOVwC|page=361}}| isbn = 978-1-85233-902-9 | page = 361 | url-status = live | archive-url = https://web.archive.org/web/20180114234526/https://books.google.com/books?id=Kdb7-YnnOVwC&lpg=PR3&dq=isbn%3A9781852339029&pg=PA361 | archive-date = 2018-01-14 }}</ref>
: <math>\mathbf{\hat{d}}_\mathrm{s} = \mathbf{\hat{d}}_\mathrm{i} - 2 \mathbf{\hat{d}}_\mathrm{n} \left(\mathbf{\hat{d}}_\mathrm{n} \cdot \mathbf{\hat{d}}_\mathrm{i}\right),
</math>
where <math>\mathbf{\hat{d}}_\mathrm{n} \cdot \mathbf{\hat{d}}_\mathrm{i}</math> is a scalar obtained with the [[dot product]]. Different authors may define the incident and reflection directions with [[sign convention|different signs]].
Assuming these [[Euclidean vector]]s are represented in [[column vector|column form]], the equation can be equivalently expressed as a matrix-vector multiplication:<ref>{{cite book | last1 = Farin | first1 = Gerald | last2 = Hansford | first2 = Dianne | author2-link = Dianne Hansford | title = Practical linear algebra: a geometry toolbox | url=http://www.farinhansford.com/books/pla/
| publisher = A K Peters | year = 2005 | isbn = 978-1-56881-234-2 | pages = 191–192 | url-status = live | archive-url = https://web.archive.org/web/20100307145056/http://www.farinhansford.com/books/pla/ | archive-date = 2010-03-07 }} {{google books|id=Iq6qVt22RZUC|title=Practical linear algebra: a geometry toolbox}}</ref>
: <math>\mathbf{\hat{d}}_\mathrm{s} = \mathbf{R} \; \mathbf{\hat{d}}_\mathrm{i},</math>
where <math>\mathbf{R}</math> is the so-called [[Householder matrix|Householder transformation matrix]], defined as:
: <math>\mathbf{R} =  \mathbf{I} - 2 \mathbf{\hat{d}}_\mathrm{n} \mathbf{\hat{d}}_\mathrm{n}^\mathrm{T};</math>
in terms of the [[identity matrix]] <math>\mathbf{I}</math> and twice the [[outer product]] of <math>\mathbf{\hat{d}}_\mathrm{n}</math>.