# Specular reflection

Diagram of specular reflection
Reflections on still water are an example of specular reflection.

Specular reflection is the mirror-like reflection of light (or of 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 ($\theta _i = \theta _r$ in the figure), and that the incident, normal, and reflected directions are coplanar. This behavior was first discovered through careful observation and measurement by Hero of Alexandria (AD c. 10–70).[1]

## Explanation

Specular reflection is distinct from diffuse reflection, where incoming light is reflected in a broad range of directions. An example of the distinction between specular and diffuse reflection would be glossy and matte paints. Matte paints have almost exclusively diffuse reflection, while glossy paints have both specular and diffuse reflection. A surface built from a non-absorbing powder, such as plaster, can be a nearly perfect diffuser. On the opposite side, polished metallic objects can specularly reflect light very efficiently. The reflecting material of mirrors is usually aluminum or silver.

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 $\theta _i$. 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 compared to the wavelength, the law of reflection no longer holds and the behavior of light is more complicated.

Waves other than visible light can also exhibit specular reflection. This includes other electromagnetic waves, as well as non-electromagnetic waves. Examples include ionospheric reflection of radiowaves, reflection of radio- or microwave radar signals by flying objects, 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 reflectivity of a surface is the ratio of reflected power to incident power. The reflectivity is a material characteristic, depends on the wavelength, and is related to the refractive index of the material through Fresnel's equations. In absorbing materials, like metals, it is related to the electronic absorption spectrum through the imaginary component of the complex refractive index. Measurements of specular reflection are performed with normal or varying incidence reflectometers using a scanning variable-wavelength light source. Lower quality measurements using a glossmeter quantify the glossy appearance of a surface in gloss units.

The image in a flat mirror has these features:

• It is the same distance behind the mirror as the object is in front.
• It is the same size as the object.
• It is the right way up (erect).
• It appears to be laterally inverted, in other words left and right reversed.
• It is virtual, meaning that the image appears to be behind the mirror, and cannot be projected onto a screen.

## Direction of reflection

The direction of a reflected ray is determined by the vector of incidence and the surface normal vector. Given an incident direction $\mathbf{\hat{d}}_\mathrm{i}$ from the surface to the light source and the surface normal direction $\mathbf{\hat{d}}_\mathrm{n},$ the specularly reflected direction $\mathbf{\hat{d}}_\mathrm{s}$ (all unit vectors) is:[2][3]

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

where $\mathbf{\hat{d}}_\mathrm{n} \cdot \mathbf{\hat{d}}_\mathrm{i}$ is a scalar obtained with the dot product. Different authors may define the incident and reflection directions with different signs. Assuming these Euclidean vectors are represented in column form, the equation can be equivalently expressed as a matrix-vector multiplication:

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

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

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

$\mathrm{T}$ denotes transposition and $\mathbf{I}$ is the identity matrix.

## References

1. ^ Sir Thomas Little Heath (1981). A history of Greek mathematics. Volume II: From Aristarchus to Diophantus. ISBN 978-0-486-24074-9.
2. ^ Farin, Gerald; Hansford, Dianne (2005). Practical linear algebra: a geometry toolbox. A K Peters. pp. 191–192. ISBN 978-1-56881-234-2.
3. ^ Comninos, Peter (2006). Mathematical and computer programming techniques for computer graphics. Springer. p. 361. ISBN 978-1-85233-902-9.