# Multiellipsoidal Mapping Algorithm
## 1 Introduction  
Many algorithms have been developed for the robotic mapping problem [1], which arises when a robot is provided with a spatial model of the environment. Unlike many 3D reconstruction algorithms that use interpolation to obtain the best object rendering, the principal aim of robotic mapping is to create maps that are rich in information and low in memory cost. The environment and objects could be acquired through 3D sensor like Laser Rangefinder or multiples camera view with algorithms like Structure from Motion [2]. With these techniques we obtain dense point clouds. On the one hand, there exist 3D mapping algorithms, e.g., Simultaneous Localization and Mapping (SLAM), that model key point maps. The principal features of such algorithms are representations with low memory cost and ease of use. However, the maps are not information-rich; consequently, such algorithms cannot be used in many applications, such as object manipulation and aerial navigation. On the other hand, many algorithms create extensive object representations as 3D reconstructions based on the Radial Basis Function (RBF) [3] or other functions [4], resulting in richer information maps, albeit at a high memory cost; additionally, such algorithms are difficult to use. Finding a good balance between representation and memory cost is an important current research topic. In the last decade, algorithms based on object mapping with geometric entities have become very popular for many applications [1], such as urban and office environment mapping. Object mapping refers to algorithms that approximate the volume shape of a point cloud (it is to say approximate a point cloud) with multiply geometric entities, this is analogous to use curve fitting in a regressionproblem, but we use geometric entities to describe and preserve the volume shapes instead of functions because there is not dependent variables. The quality of the approximation could be measured using the mean distance of the points to the geometric entity. Geometric entities have been used before to approximate point clouds, for instance, we observe the use of multiplanar representations [5] and spherical and linear representations [6; 7; 8; 9]. Each of these approaches fits the parameters of their geometric entities to obtain the best approximation of sensor data. However, if more complex entities are used, we can obtain approximations of point clouds with better accuracy, and using fewer entities than those obtained by using mulitiplanar, spherical or linear representations. In this paper, a new object-mapping algorithm is presented. This algorithm is capable of approximating the point cloud with multiple ellipsoids and other quadratic surfaces such as spheres, pairs of planes, and pseudocylinders, which allows us to reduce the number of geometric entities used to represent the objects and complete scenes of the environment, this feature is shown in a controlled experiment, but to automate this process a hierarchical clustering must be implemented, as discussed in the future work section. This algorithm can also be related to Hyperellipsoidal Neurons (HNs) [10] based on Geometric Algebra (GA) [11], where every neuron is trained with k-means++ [12] and Differential Evolution (DE) [13] for adapting the point cloud implicit surface. In the following sections, we show that the multiellipsoidal mapping algorithm is capable of creating information-rich maps with low memory cost and that the entities obtained with the algorithm are even capable of deforming into other quadratic surfaces due to their representation in GA as multivectors. The entities are described in GA; then, the map is suitable for developing new algorithms that work in this algebra, such as path planning [14], pose estimation [15; 16], and other tasks [17]. The paper is organized as follows. In Section 2, we introduce several mathematical and algorithmic tools used in this paper. Section 3 presents the adaptation strategy of the ellipsoidal surfaces to a point cloud.