Discrete Variable Conditional 3D Models in Computer Extended Descriptive Geometry (CeDG): Preliminary Study on Cylindrical Polygonal Sheet-Metal Elbow

Abstract

Computer extended Descriptive Geometry (CeDG) technique has demonstrated its ability to build parametric computational models of 3D systems supported by Descriptive Geometry procedures in previous studies. When a parameter causes a non-continuous and qualitative change in the 3D model, it requires an additional logic. The treatment of visibility is a well-studied example of qualitative changes that are associated with continuous parameters. We are interested in qualitative changes associated with discrete parameters (discrete variable conditional 3D models). This work deals with the preliminary study of a methodology for the construction of such 3D models. A cylindrical polygonal sheet-metal elbow model of n ferrules (neutral surface) with the ability to cater for the change of variable n (n > 2, discrete) is used in this work as a target system to study the limits and accuracy of CeDG based on GeoGebra in the construction of discrete variable conditional models. We successfully achieved a conditional CeDG model (3D), including the flat pattern, for any number of ferrules. This model overpasses the scope of the polygonal elbow model from LogiTRACE v14, which is restricted to $n\le 12$. In addition, flat patterns in CeDG are exact in opposition to that of LogiTRACE, which are prismatic approximation with a given number of generatrices.

Appendix

The code that implements the DVCM model of the cylindrical elbow is divided into two blocks. The first one obtains the vertical projection of the elbow, defined by planes that are perpendicular to the projection plane, as indicated in the methodology section. If the system were not initially in this position, the model would include the necessary changes of projection direction.

The same list is divided into sub-blocks that define each of the tasks that are performed sequentially. The input data are the ends of the projective segments of the openings. These are assigned to the PortA1, PortA2, PortB1 and PortB2 points handled by the code (blue lines).

Elbow vertical projection

The following code block obtains the flat pattern of the ferrules on a reference cylinder. Figure 5 shows the main objects on which the calculation is based. Thus, baseCyl is the real magnitude of the opening, on which the OmegaCyl free point is defined, that identifies the gomegaCyl generatrix and the ArcomegaCyl arc. This figure also shows the vertical projection of the cylinder’s axis, axisCil. A list of points (highlighted in green), ListPlanesD, is calculated on the axis. These points represent the intersection with the axis of each plane which, in turn, contains the intersection between two adjacent ferrules.

The angle formed between these planes and the axis (is sign-alternating, which is treated as (−1)^iFerrules in the code, DO list) is stored in the PhiPlaneAxis variable (it is shown with the 83º value in Fig. 2) and allows the planes’ projections to be defined, their projection points to be calculated and their flat patterns to be finally computed.

Flat pattern