ization of base mesh under the assumption that surface with larger area can catch more surface details of original mesh, so that we give it more samples by enlarge its parametric area.
3.2.3 Displacment Map Generation
With base mash and it’s parameterization, we are going to build the displacement map at last.
We sample rays from the base mesh surface along the direction of inverse interpolated vertex-normal, computing the distance and the original surface normal of the point at which the ray intersects the surface of original mesh. Then store the original normal vector with the displace-ment value as a 4-channels luminance to the pixel.
3.3 Wind Erosion Simulation Equations
In this section, we will show how to simulate such phenomena by our wind erosion equation.
Before we show our wind erosion simulation equation, the Wind Erosion eQuation(WEQ) have to be introduced first. In the environment and soil science field, W. S. Chepil et. al[11] use wind tunnels and field studies to develop the first wind erosion prediction equation. The equation expressed in function form is:
E = f (I, K, C, L, V ), (3.1)
where E is the potential average annual soil loss, I is the soil erodibility index, K is the soil ridge roughness factor, C is the climate factor, L is unsheltered distance across a field, and V is the equivalent vegetative cover. This equation has been used for a long time to predict the soil loss caused by wind erosion in agricultural fields.
Observing the WEQ, we know this equation is designed for predicting the soil loss in geo-morphology. Inspired by the WEQ, we design our wind erosion equation which suited for our modeling and rendering wind erosion phenomena on meshes. Our equation is based on these observations:
• The soil loss, E, is proportional to wind force(C factor and L factor in WEQ)
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• The soil loss, E, is propositional to objects appearance roughness(K factor in WEQ)
• The soil loss, E, is propositional to objects intrinsic composition(I factor in WEQ)
• The soil loss make mesh surface lost the fine details in height field
• The appearance roughness is actually the degree of surface variance, which we can refer to the curvature over mesh
To sum up, the lost of surface detail is propositional to the wind and the mesh’s intrinsic composition combine with it’s curvature. Hence we design our wind erosion simulation equa-tion on mesh:
H(x) = W t(x)
R(x) , (3.2)
where x is the sample point on mesh which we are compute; H(x) is the eroded height for this point; W t(x) is the wind force factor of x; R(x) is a function reflect the relationships between eroded height and material of x.
And for simplex, here we assume the wind force has a constant effect over the entire mesh.
For each sample point on mesh surface, wind force affect erosion quantity more on the upwind side of mesh, but less on the downwind side of mesh. Then we get the W t(x) equation:
W t(x) = ~Wd· ~Nd(x) + basicW, (3.3) where x is the sample point on mesh which we are compute; ~Wd is the wind direction in the simulation environment; ~Nd(x) is the average normal direction of the sample point x; basicW is a constant denote the basic wind effect of erosion, and it is the first control parameter in our simulation equation.
As we mentioned material of each sample point x in Equation3.2. Material is the property that shows under the variance of curvature, how much soil will lost on the sample point. For example, when sample point x has higher curvature on mesh surface, it means that the mesh structure have less strength to support this sample and this sample would easily lost it’s soil by wind force. For lower curvature sample point, it would be harder for wind to blow away the
3.3 Wind Erosion Simulation Equations 15
soil of that sample. But how much soil lost is caused by this kind of material? If the material is stronger, no matter the curvature is high or low, soil lost of the sample is small. But when the material is weaker, a tiny curvature raise would cause high soil lost of the sample. R(x) is designed to represent it. As a consequence, when the material changes, function R(x) changes, too.
To define one R(x), which means to define a new material to curvature mapping relation-ships, we need three more control parameters: Emin, Emedium, and Emax. Emin, Emedium, Emax denotes respectively the minimum, the medium, and the maximum soil lost in one time step ero-sion simulation. In our setup, the minimum soil lost occurs when this material has only basic wind effect and its curvature is 0. The maximum soil lost occurs when this material direct faces the wind force and its curvature is 1. The medium soil lost occurs when curvature of this sample point is 0.5. Hence we get that R(x) will pass three point: (0,basicWE
min ) as (x0, y0), (0.5, Emedium) where x is the curvature of the sample point. R(x) return the soil lost ratio of this kind of material under the curvature and wind condition of the sample point.Fig3.2 and Fig3.3 show different R(x).
Note that we consider curvature is signed and the range is [−1, 1]. When curvature of sample point is below zero, this sample is on concave part of this mesh. We treat this as an exception, and will discuses in the later section. On the other hand, in order to make R(x) be a monotonic decrease function, once Emin and Emax are given, Emedium should be bound in [y0, y2], and basicW should be bound in (E Emin
max−Emin, ∞). Either Emediumor basicW are not satisfied their bounding condition will be corrected in system. By well designed R(x), we can satisfied the needs of H(x), and a stable simulation system.
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Figure 3.2: Emin=42.0, Emax=100.0, Emedium=0.08, basicW =3.95
Figure 3.3: Emin=20.0, Emax=55.0, Emedium=0.117, basicW =3.80