# Multi-Scale Indoor Scene Geometry Modeling Algorithm Based on Segmentation Results
## 3 Methods
### Recognition and Modeling of Regular Geometric Shapes
#### 3.2.3 Recognition and Modeling of Curved Geometric Objects  
Common curved geometric objects include cylinders, cones, and spheres. Due to factors such as the capturing angle and object occlusion, cylinders and cones exist in two forms in actual capturing scenes, namely planar-surfaces combination and curved-surfaces combination. Due to their different manifestations, the logical processing of recognition and parameter extraction for each curved geometric object is also different.  
When the segmented object belongs to the planar-surfaces combination type, the relationship between the plane and the adjacent curved surfaces' normal vectors is determined by using the plane as the auxiliary matching surface. If the normal vectors of the two are perpendicular to each other, it is a cylinder, corresponding to objects like cups or other cylindrical objects. If there is no perpendicular relationship between the two normal vectors, and the angle with the normal vector of the adjacent surface remains unchanged, it is a cone.  
Due to occlusion or capturing angle, cylinders or cones may also exist with only the curvature of surfaces being captured, in which case, all faces of the object are curved surfaces. In addition to this, spheres are also considered, as they are natural geometric objects consisting only of curved faces. For objects with only curved surfaces, the identification of cylindrical surfaces, conical surfaces, and spherical surfaces is done by analyzing the relative magnitude of the principal curvatures at each point in the object's point cloud data. The values of the principal curvatures can be obtained using the calculation formula, assuming the maximum principal curvature is denoted by \(k_{1}\) and the minimum principal curvature is denoted by \(k_{2}\). The geometric primitive type of the curved surface object can be determined based on the relationship between the extrema of the principal curvatures using the following criteria: (1) Cylinder: \(k_{1}=0,k_{2}>0\), and \(k_{2}\) remains unchanged. (2) Cone: \(k_{1}=0,k_{2}>0\), and \(k_{2}\) varies. (3) Sphere: \(k_{1}=k_{2}\), and both \(k_{1}\) and \(k_{2}\) are positive constants.  
After determining the initial geometric primitive type of the segmented object using the aforementioned method, as prior knowledge for the RANSAC algorithm, the corresponding geometric primitives are used for parameter extraction. For spheres, the modeling can be directly accomplished by using the sphere center and radius. However, for cylinders and cones, after obtaining the parameter equations of the cylindrical surface and conical surface, the point cloud of the objects needs to be projected onto the corresponding axis, and the distance between the two farthest projected points is taken as the height of the cylinder and cone.  
To visually demonstrate the modeling effectiveness of curved geometric objects, point cloud data of the corresponding surfaces of known geometric parameters and dimensions are generated. The reconstruction result of a single curved surface object in the scene is shown in Figure 5, where Figure 5 represents the oriented bounding box established for  
Figure 4: Reconstructed result diagram of planar objects.  
the original point cloud. As can be seen from Figure 5c, for different types of curved surface objects, the reconstructed geometric point cloud fits well with the original point cloud and effectively fills in the missing data based on the obtained 3D model.