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Diagram of specular reflection

Reflections on still water are an example of specular reflection.

Specular reflection is the mirror-like reflection of light (or sometimes other kinds of wave) from a surface, in which light from a single incoming direction (a ray) is reflected into a single outgoing direction. Such behavior is described by the law of reflection, which states that the direction of incoming light (the incident ray), and the direction of outgoing light reflected (the reflected ray) make the same angle with respect to the surface normal, thus the angle of incidence equals the angle of reflection; this is commonly stated as $θ i = θ r$ . A second defining characteristic of specular reflection is that incident, normal, and reflected directions are coplanar. This behavior was first discovered through careful observation and measurement by Hero of Alexandria (c. 10–70 AD).^[1]

This is in contrast to diffuse reflection, where incoming light is reflected in a broad range of directions. The most familiar example of the distinction between specular and diffuse reflection would be glossy and matte paints. While both exhibit a combination of specular and diffuse reflection, matte paints have a higher proportion of diffuse reflection and glossy paints have a greater proportion of specular reflection. Very highly polished surfaces, such as high quality mirrors, can exhibit almost complete specular reflection.

Even when a surface exhibits only specular reflection with no diffuse reflection, not all of the light is necessarily reflected. Some of the light may be absorbed by the materials. Additionally, depending on the type of material behind the surface, some of the light may be transmitted through the surface. For most interfaces between materials, the fraction of the light that is reflected increases with increasing angle of incidence $θ i$ . If the light is propagating in a material with a higher index of refraction than the material whose surface it strikes, then total internal reflection may occur (if the angle of incidence is greater than a certain critical angle). Specular reflection from a dielectric such as water can affect polarization and at Brewster's angle reflected light is completely linearly polarized parallel to the interface.

The law of reflection arises from diffraction of a plane wave (with small wavelength) on a flat boundary: when the boundary size is much larger than the wavelength then electrons of the boundary are seen oscillating exactly in phase only from one direction—the specular direction. If a mirror becomes very small (comparable to the wavelength), the law of reflection no longer holds and the behaviour of light is more complicated.

Usually, the term specular reflection refers to visible light; however the term is also widely used for other electromagnetic waves. The specular reflection of non-electromagnetic waves follows basically the same laws. Examples include acoustic mirrors, which reflect sound, and atomic mirrors, which reflect neutral atoms. For the efficient reflection of atoms from a solid-state mirror, very cold atoms and/or grazing incidence are used in order to provide significant quantum reflection; ridged mirrors are used to enhance the specular reflection of atoms.

The inclusion of specular reflection in dentistry helps improve the aesthetic quality of an inlay, onlay or filling, allowing the appearance of the material 'flowing' in with the natural dentition.

Specular reflection can be most accurately measured using a glossmeter. The measurement is based on the refractive index of an object. The standard units for measurement are "gloss units".

1 Calculation
2 Notes
3 References
4 See also

Calculation

Given an incident direction $\mathbf{\hat{d}}_\mathrm{i}$ and the surface normal direction $\mathbf{\hat{d}}_\mathrm{n}$ , the specularly reflected direction $\mathbf{\hat{d}}_\mathrm{s}$ (all unit vectors) can be calculated as:^[note
1]^[note
2]

$\mathbf{\hat{d}}_\mathrm{n} = 2 (\mathbf{\hat{d}}_\mathrm{n} \cdot \mathbf{\hat{d}}_\mathrm{i}) \mathbf{\hat{d}}_\mathrm{n} - \mathbf{\hat{d}}_\mathrm{i},$

where $\mathbf{\hat{d}}_\mathrm{n} \cdot \mathbf{\hat{d}}_\mathrm{i}$ is a scalar obtained with the dot product, $\cdot$ . (Be careful in that different authors may define the incident and reflection directions with different sign than above, i.e., into vs. out of the surface.) Assuming these Euclidean vectors are represented in column form, we may express the equation equivalently as a matrix-vector multiplication:

$\mathbf{\hat{d}}_\mathrm{n} = \mathbf{R} \; \mathbf{\hat{d}}_\mathrm{i},$

where $\mathbf{R}$ is the so-called Householder transformation matrix, defined as:

$\mathbf{R} = 2 \mathbf{\hat{d}}_\mathrm{n} \mathbf{\hat{d}}_\mathrm{n}^\mathrm{T} - \mathbf{I};$

$T$ denotes transposition and $\mathbf{I}$ is the identity matrix. If only unnormalized surface normal vector $\mathbf{d}_\mathrm{n} = \left| \mathbf{d}_\mathrm{n} \right| \mathbf{\hat{d}}_\mathrm{n}$ is available, a square root otherwise required to obtain the normal length $\left| \mathbf{d}_\mathrm{n} \right| = \sqrt{(\mathbf{d}_\mathrm{n} \cdot \mathbf{d}_\mathrm{n})}$ can be avoided as follows:

$\mathbf{\hat{d}}_\mathrm{n} = 2 \left( (\mathbf{d}_\mathrm{n} \cdot \mathbf{\hat{d}}_\mathrm{i}) / (\mathbf{d}_\mathrm{n} \cdot \mathbf{d}_\mathrm{n}) \right) \mathbf{d}_\mathrm{n} - \mathbf{\hat{d}}_\mathrm{i},$

or, in terms of $\mathbf{R}$ :

$\mathbf{R} = 2 (\mathbf{d}_\mathrm{n}^\mathrm{T} \mathbf{d}_\mathrm{n})^{-1} (\mathbf{d}_\mathrm{n} \mathbf{d}_\mathrm{n}^\mathrm{T}) - \mathbf{I},$

where $\mathbf{d}_\mathrm{n}^\mathrm{T} \mathbf{d}_\mathrm{n} = \mathbf{d}_\mathrm{n} \cdot \mathbf{d}_\mathrm{n}$ is again a scalar and $\mathbf{d}_\mathrm{n} \mathbf{d}_\mathrm{n}^\mathrm{T}$ is a matrix.

Notes

^ ^[2], p.191-192.
^ ^[3], p.361.

References

^ Sir Thomas Little Heath (1981). A history of Greek mathematics. 2. ISBN 0486240746, 9780486240749.
^ Farin, Gerald; Hansford, Dianne (2005). Practical linear algebra: a geometry toolbox. A K Peters. pp. 394. ISBN 978-1-56881-234-2. http://www.farinhansford.com/books/pla/.
^ Comninos, Peter (2006). Mathematical and computer programming techniques for computer graphics. Springer. pp. 547. ISBN 978-1-85233-902-9.

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