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Flexible terrain erosion

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Abstract

In this paper, we present a novel particle-based method for simulating erosion on various terrain representations, including height fields, voxel grids, material layers, and implicit terrains. Our approach breaks down erosion into two key processes—terrain alteration and material transport—allowing for flexibility in simulation. We utilize independent particles governed by basic particle physics principles, enabling efficient parallel computation. For increased precision, a vector field can adjust particle speed, adaptable for realistic fluid simulations or user-defined control. We address material alteration in 3D terrains with a set of equations applicable across diverse models, requiring only per-particle specifications for size, density, coefficient of restitution, and sediment capacity. Our modular algorithm is versatile for real-time and offline use, suitable for both 2.5D and 3D terrains.

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A Computation of a metaball

A Computation of a metaball

We use the following formula to evaluate a metaball in space with a center c and of radius R:

$$\begin{aligned} g(p) = 1 - \frac{||p - c||}{R} \end{aligned}$$

using the Euclidean distance.

We have a total amount \( Q \) to define in this space, so the final metaball function f needs to satisfy Eqs. (15) and (16):

$$\begin{aligned} f(p)&= \lambda g(p) \end{aligned}$$
(15)
$$\begin{aligned} \int _{p \in V_{3D}}{f \, dp}&= Q \end{aligned}$$
(16)

First, let’s exploit the radial symmetry of the metaball and rewrite \(g(p) = 1 - r\) by using the polar coordinates of the point \(p - c\).

We can then integrate g over the volume \(V_{3D}\) as

$$\begin{aligned}&\int _{0}^{1}{ \int _{0}^{\pi }{ \int _{0}^{2\pi }{ g(r) r^2 \sin (\theta )\, {\text {d}}r} \, {\textbf {d}}\theta } \, {\text {d}}\phi } \nonumber \\&\quad = \int _{0}^{1}{ \int _{0}^{\pi }{ \int _{0}^{2\pi }{ (1 - r) r^2 \sin (\theta )\, {\text {d}}r} \, {\text {d}}\theta } \, {\text {d}}\phi } \nonumber \\&\quad = \int _{0}^{1}{ (1 - r)r^2 \, {\text {d}}r} \times \int _{0}^{\pi }{ \sin {\theta } \, {\text {d}}\theta } \times \int _{0}^{2\pi }{ 1 \, {\text {d}}\phi } \end{aligned}$$

We then break down the integrals one by one such as

$$\begin{aligned}{} & {} \int _{0}^{1}{ (1 - r)r^2 \, {\text {d}}r} = \frac{1}{12} \nonumber \\{} & {} \int _{0}^{\pi }{ \sin {\theta } \, {\text {d}}\theta } = 2 \nonumber \\{} & {} \int _{0}^{2\pi }{ 1 \, {\text {d}}\phi } = 2 \pi \nonumber \end{aligned}$$

By combining all these integrals, we get \(\int {g} = \frac{1}{12} \times 2 \times 2\pi = \frac{\pi }{3}\).

So given \(\int {f} = q_\textrm{detachment} \) and \(\int {f} = \lambda \int {g}\), we can deduce that \(\lambda = \frac{ Q }{\int {g}} = \frac{3}{\pi } Q \).

From (15) we finally get

$$\begin{aligned} f(p) = \frac{3 Q }{\pi } \left( 1 - \frac{||p - c||}{R} \right) \end{aligned}$$
(17)

, representing the rate of change on the evaluation function of the terrain surface.

The integration in the voxel space is out of the scope of this paper and a numerical solution is instead proposed in Sect. 4.4.

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Hartley, M., Mellado, N., Fiorio, C. et al. Flexible terrain erosion. Vis Comput 40, 4593–4607 (2024). https://doi.org/10.1007/s00371-024-03444-w

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