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:<math>\max(0, \hat{R}_m \cdot \hat{V})^\alpha = \max(0, 1-\lambda)^{\beta \gamma} = \left(\max(0,1-\lambda)^\beta\right)^\gamma \approx \max(0, 1 - \beta \lambda)^\gamma </math>
where <math>\lambda = 1 - \hat{R}_m \cdot \hat{V}</math>, and <math>\beta = \alpha / \gamma\,</math> is a [[real number]] which doesn't have to be an [[integer]]. If <math>\gamma</math> is chosen to be a power of 2, i.e. <math>\gamma = 2^n</math> where <math>n</math> is an integer, then the expression <math>(1 - \beta\lambda)^\gamma</math> can be more efficiently calculated by squaring <math>(1 - \beta\lambda)\ n</math> times, i.e.
:<math>(1 - \beta\lambda)^\gamma \,=\, (1 - \beta\lambda)^{2^n} \,=\, (1 - \beta\lambda)^{\overbrace{\scriptstyle 2\,\cdot\,2\,\cdot\,\dots\,\cdot\,2}^n} \,=\, (\dots((1 - \beta\lambda)\overbrace{^2)^2\dots)^2}^n.</math>
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:<math>1 = \sqrt{(N_x^2 + N_z^2)}</math>
Because of the powers of two in the equation there are two possible solutions for the normal direction. Thus some prior information of the geometry is needed to define the correct normal direction. The normals are directly related to angles of inclination of the line on the object surface. Thus the normals allow the calculation of the relative surface heights of the line on the object using a [[line integral]], if we assume a continuous surface.
If the object is not cylindrical, we have three unknown normal values <math>N=[N_x, N_y, N_z]</math>. Then the two equations still allow the normal to rotate around the view vector, thus additional constraints are needed from prior geometric information. For instance in [[facial recognition system|face recognition]] those geometric constraints can be obtained using [[principal component analysis]] (PCA) on a database of depth-maps of faces, allowing only surface normals solutions which are found in a normal population.<ref>{{cite book| title=Model-Based Illumination Correction for Face Images in Uncontrolled Scenarios|date=September 2009|last1=Boom |first1= B.J. |last2=Spreeuwers|first2= L.J. |last3=Veldhuis|first3= R.N.J.| volume=5702| pages=33–40| doi=10.1007/978-3-642-03767-2 | issue=2009| series=Lecture Notes in Computer Science| editor1-last=Jiang| editor1-first=Xiaoyi| editor2-last=Petkov| editor-link2= Nicolai Petkov| editor2-first=Nicolai| isbn=978-3-642-03766-5| url=https://halshs.archives-ouvertes.fr/halshs-00420059/document| bibcode=2009LNCS.5702.....J| hdl=11693/26732| hdl-access=free}}</ref>
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* [https://web.archive.org/web/20160525135133/http://michal.is/projects/phong-reflection-model-matlab/ Phong reflection model in Matlab]
* {{usurped|1=[https://web.archive.org/web/20180816064924/http://www.sunandblackcat.com/tipFullView.php?l=eng&topicid=30 Phong reflection model in GLSL]}}
[[Category:Computer graphics algorithms]]
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