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Water mathematics

The mathematics of water

You have seen many different kinds of water in your life already and hopefully they seemed realistic enough. Different natural water types have several properties in common but they are absolutely different in others. The key ingredients of the realistic water are:

• Reflecting the objects visible above the surface

• Refracting the objects under the surface

• Multiple reflections and refraction

• Appropriate ratio between reflection and refraction: the Fresnel term

• Some small modification in the colors (dirtier water), water-fog

• Moving surface

• Specular reflection

• Deep water phenomena

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Reflection

The definition according to wikipedia: “Reflection is the change in direction of a wave front at an interface between two different media so that the wave front returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves.” In some aspects the surface of the water acts like a perfect mirror. The electromagnetic waves of the light are reflected on the surface. The physical law for this: the angle of reflection equals to the angle of incidence, and we measure these angles to the normal-vector of the surface, namely the two angles α = β:

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Taking into consideration only the reflection behavior of the water it is easy to calculate the color of the pixels on the water surface. Mirroring the position of the camera to the plane of the surface gives the exact location of the virtual view: just determine which color has the object which is visible through every pixel of the water from the virtual view. This idea is shown on the next figure:

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If the camera is in point A, the perceived color on the water surface will be the color of the object visible from point B through the same intersection point. The point B is exactly the same far from the plane of the water as point A - the two distances are marked with letter “k” on the figure.

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Refraction

The speed of electromagnetic waves is different in different media. The change happening when it passes from one medium to another causes the phenomenon of refraction. The Snell’s law (named after the Dutch mathematician Willebrord Snellius) describes the relationship between the angle of incidence and refraction: the ratio between them is a constant depending on the media or more exactly the ratio of the sines of the angles equals the ratio of velocities in the two media:

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or

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These angles are measured in respect to the normal-vector of the boundary between the two media.

According to this law the direction of the wave can be refracted towards or from the normal line, depending on the relative refraction-indices of the media. The next figure shows an example:

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In this case the velocity is lower in the second medium (V2 ................
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