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Refractive Texture Mapping, Part One
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# Refractive Texture Mapping, Part One

November 10, 2000 Page 2 of 4

Perturbations

The water mesh is implemented as a flat polygonal mesh using the left-hand coordinate system, where the Y-axis represents the up dimension and the Z-axis points away from the viewer. This mesh is divided in n slices in either dimension specified by the user. The figure below shows the top view of the mesh as if it's been divided in three slices.

Sphere mapping or refractive texture mapping is not too interesting over a flat mesh, especially for simulating water. Taking this fact into account, it's necessary to perturb the mesh.

There are several ways of perturbing the mesh to look like water, some faster and some slower. A very efficient way is by using the array approach described by Hugo Elias and Jeff Lander ("A Clean Start: Washing Away the Millennium," Game Developer, December 1999). However, the array approach only simulates approximations of ripples on the mesh. If the application needs to perturb the mesh with a constant directional wave, or a combination of directional waves, the array approach cannot be used. Another method is to use a spring model, modeling the flat surface as a set of two dimensional springs, which can produce very realistic effects. Lastly, we can simply use sums of sine or cosine functions which creates very good results. For the sake of simplicity, the sample program explained here implements perturbations using sums of sine wave functions.

 Figure 4. Mesh divided in three slices, top view.

Sine Waves

In order to understand how the sample program perturbs the polygonal mesh using sine wave sums, let's quickly review some properties of sine wave functions.

In 2D space (X, Y) as function of X, sine waves can be written as:

y(x) = a * sin(kx - p), where

x = X-axis value
y = Y-axis value
a = wave amplitude
k = wave number (frequency)
p = phase shift

(Equation 3)

The amplitude (a) is how big the wave is because it multiplies the whole function. The wave number (k) is related to the wave frequency: the smaller this number is, the greater the frequency and vice versa. The phase shift shifts the wave to either a positive or negative direction of the X-axis. In terms of animation, we can use the time as a phase shift to make the waves move each frame.

In 3D space, the sine wave equation can be extended to (using the left-hand coordinate system):

y(x,z) = a * sin(k(xi + zj) - p) (Equation 4)

where
x = X-axis value
y = Y-axis value (in this particular case, the height of the wave)
z = Z-axis value
a = wave amplitude
k = wave number (frequency)
p = phase shift
(i, j) = wave directional vector components

Equation 4 is a straightforward extension of Equation 3 for 3D space. The only part that needs special attention is the wave directional vector components (i, j). In this case, they define in which direction the wave is traveling on the XZ plane. As an example, let's write the wave equation as if the wave is traveling 45 degrees (p/4 radians) in respect to the X-axis.

 Figure 5. Top view of a wave traveling at 45 degrees.

If we project the wave directional vector (unit vector) on the X- and Z-axis, by simple trigonometry the components can be computed as:

and

So the equation for this example can be written like:

Page 2 of 4

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