## 3 Method
### Gaussian-Voronoi Map  
The primary challenge in generating terrain data using large language models is the lack of stability when outputting large amounts of text, making it difficult to directly output complex terrain data. To tackle this challenge, we introduce the Gaussian-Voronoi map, an innovative data structure. This method combines the spatial partitioning capability of Voronoi diagrams with the smoothing characteristics of Gaussian functions to simulate diverse terrain features. Figure 2 explains the working principle of the Gaussian-Voronoi map. Specifically, we first use a Gaussian function to roughly express the height of a certain area:  
\[G=h\cdot e^{-\frac{\rho^{2}}{\pi}} \tag{1}\]  
Figure 2: Significant differences exist between the cells of the original Voronoi diagram. To ensure that terrain generation is roughly equivalent across the map, we apply [PERSON]’s relaxation to the Voronoi diagram. This maintains a basic uniformity in cell sizes. where \(h\) represents the elevation height represented by the Gaussian function, \(w\) is the influence weight of the Gaussian function on the map, and \(d\) is the distance from any point on the map to the center of the Gaussian function. Diverging from traditional Gaussian applications, we avoid using a covariance matrix for scaling and rotation control. Instead, we add multiple Gaussian functions to simulate these effects. This simplification helps the large language model process terrain features more efficiently, reducing potential errors. Instead, we achieve a similar effect by adding more Gaussian functions on the map. The advantage of this method is that, since the parameters of the Gaussian functions are generated by a large language model, omitting control over rotation and scaling simplifies the dimensions that the model must handle when analyzing terrain features, thereby effectively reducing the potential error rate. This strategy aims to leverage the capabilities of large language models while alleviating their burden in handling complex terrain generation tasks. After generating Gaussian data, we integrate them into a Voronoi diagram. This step divides the space into regions, each influenced by a Gaussian peak, creating a more varied terrain. We use Voronoi diagrams to subdivide the plane into several approximately equalized areas. For any pixel point on the plane, we find the nearest Voronoi point \(p_{i}\), thus determining the Voronoi area to which the pixel point belongs; next, for each pixel point corresponding to \(p_{i}\), we shape the terrain details by overlapping Gaussians. To capture the terrain's randomness and diversity, we go beyond letting each Gaussian impact the terrain uniformly. We employ a probability function, \(c_{i,j}\), to decide if a Gaussian \(G_{j}\) affects a specific point, enhancing the model's realism and flexibility. The probability function is determined by two Gaussian distribution functions:  
\[c_{i,j}=e^{-\frac{d_{j}^{2}}{m_{j}}} \tag{2}\]  
where \(d_{j}\), \(w_{j}\), and \(n\) have the same meanings as in the previous formula, with \(n\) being the L2 norm of the map size. For all \(G_{j}\) determined to be superimposed on \(p_{i}\), we simply sum their values to obtain the value of that block:  
\[H(p_{i})=\sum_{j=1}^{k}G_{j} \tag{3}\]  
Finally, rendering each block according to its data into a grayscale image yields the 3D terrain heightmap. By inputting data to the large language model in the form of Gaussians and decoding them through the Gaussian-Voronoi map structure, we can convert simple summaries of terrain features into complex terrains; by adjusting the number of Voronoi blocks, we can obtain 3D terrain heightmaps of specified size and complexity. We chose Gaussian as the container for simple terrain features mainly because Gaussian can summarize the general terrain height with just four parameters, as mentioned above: the two-dimensional coordinates, elevation height, and influence weight, allowing the large language model to describe and adjust 3D terrains with high accuracy.