Editing Specular reflection (section)

=== 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>.