Intersection Between Surfaces Using Computer Extended Descriptive Geometry (CeDG): Application to the Focal Illumination of a Sphere

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Springer Nature in Advances in Design Engineering III February 2023, available at: https://doi.org/10.1007/978-3-031-20325-1_55 © 2023 Springer Nature Switzerland AG Intersection Between Surfaces Using Computer Extended Descriptive Geometry (CeDG): Application to the Focal Illumination of a Sphere Manuel Prado-Velasco and Laura García-Ruesgas Abstract Computer Extended Descriptive Geometry (CeDG) is a new approach to computer modelling of 3D geometric systems that tries to overcome several limitations of current CAD systems. A preliminary version of CeDG for GeoGebra has demonstrated advantages in sheet metal and mechanisms field. This paper develops the theoretical basis of the Locus-based Surfaces’ Intersection Method (LSIM) of CeDG and compares it against the standard Descriptive Geometry technique, through the calculation of the illumination of a sphere by a focal light beam. Results showed that, in opposition to standard Descriptive Geometry technique, the LSIM CeDG model combines less complexity (geometric procedure requires only one iteration) with the capability to be extended to other parameters’ values and projections, keeping the compliance with geometrical requirements. Accuracy metrics have demonstrated that LSIM can generate the exact (true) surface intersection curve through its projections, thanks to the geometric integrity of the underlying dynamic geometry software (GeoGebra). Keywords Descriptive geometry · CAD · Computer parametric graphic modelling · Dynamic geometry software · CeDG 1 Introduction Descriptive Geometry (DG) and its derived representation systems, such as dihedral system, have defined the fundamentals of graphical representation in engineering since the second half of the 19th century, establishing an important scientific technical corpus [1]. This one includes many texts that continue to provide a solid basis for university teaching [2–7], even though their use in the professional field has been practically replaced by computer aided design software (CAD) [8]. The geometry of CAD model is mainly represented by fit curves and surfaces, such as B-splines, which provide high control and accuracy [9, 10]. Computer Extended Descriptive Geometry (CeDG) is a new approach to computer modelling of 3D geometric systems [11], which tries to address the limitations of current CAD systems pointed out by some authors [11–13]. Briefly, CAD tools allow the construction of virtual prototypes of 3D systems that can be manipulated in space and easily projected according to the chosen representation system. However, they do not facilitate the creation of the model when it depends on some implicit parameter, in addition to presenting shortcomings in the calculation of flat patterns of sheet metal surfaces. The CeDG approach combines the ability of DG to solve spatial geometric problems with the ability of dynamic geometry in the process of building geometricalgebraic models [14– 16]. Unlike CAD systems, CeDG parametric models preserve the integrity of curves and surfaces. A preliminary version of the CeDG approach implemented on the dynamic geometry software GeoGebra™ [17] has demonstrated its ability to overcome some of the above limitations [11, 18]. The objective of this study is to present and analyze the method used in CeDG models to obtain the intersection curve between two surfaces. The usual procedure in DG consists of defining it by means of its projections, which are calculated in turn by interpolation on a set of points belonging to them. The points are obtained by means of DG techniques, applied iteratively. Accordingly, the accuracy and complexity of the resulting curve is proportional to the number of points obtained [7]. The CeDG approach uses an extension of the DG procedures, which allows the generation of a specific locus function for each projection of the intersection curve sought. The locus function is automatically determined from the sequence of geometric-algebraic instructions associated with the calculation of a single generic point of the intersection [11]. This paper develops the theory that defines the foundation of the Locus-based Surfaces’ Intersection Method (LSIM) and compares this one with respect to the standard DG procedures through a case study. 2 Methods The study is developed according to the following workflow: i. The theory of LSIM is developed using the 2D locus function of the dynamic geometry software as algebraic support. Several examples are succinctly presented to describe two variants of this novel method. ii. A case study has been defined to evaluate the goodness of LSIM. This is a sphere illuminated by a focal (spot) light beam. The problem is solved using both standard descriptive geometry procedures and the LSIM-based CeDG approach. iii. Solutions to the illuminated surface reached in the previous point are compared to give the methodological advantages and the improvement in accuracy of CeDG against the standard descriptive geometry procedures. The novel LSIM requires the development of a theory that is succinctly presented in the following Section. The examples selected to support the theoretical description include a cone-cylinder intersection and the focal illumination of the sphere that is subsequently executed as case study. The case study is defined with detail in this Section with the aim of facilitating the definition of metrics for the comparison stage. The spatial system that defines the case study appears in Fig. 1 through the projections. The focal spot that illuminates the sphere is above the horizontal plane that contains the circular directrix of the light cone (distance DV), which in turn has a diameter of 2·rBCono. The diameter of the sphere is 2·rEsf and its center o-o’ is above the previous horizontal plane at the distance Dsph. These parameters control completely the focal illumination of the sphere through a conical light beam with solid angle equal to Ω. Those main parameters are defined in Table 1, together with the default value that was used during the model building process. The value of the solid angle of the conical light beam may be written as follows: from which the default value of Ω is 0.27 π sr. Fig. 1 Sphere to be illuminated from a focal light beam with spot in v-v’ and cone surface defined by the spot as vertex, vertical axis and circular directrix Table 1 Main parameters of the sphere illuminated from a focal light beam The illumination of the sphere is computed through the warped curve in the sphere that encloses the lightened zone. Concerning the metrics, we use the next qualitative properties to compare the CeDG methodology against that of the standard descriptive geometry procedures: 1. Complexity of the method and number of iterations required to get the proper solution. 2. Additional procedures required to guarantee that the solution is compliant with geometrical and physical requirements. 3. Capability of the methodology to extend the solution to other parameters’ values and projections The sought warped curve that defines the boundary of the lightened zone in the sphere is composed by a piece of the cone-sphere intersection and an arc of the separatrix circle associated with the light spot. We evaluate the accuracy of the cone-sphere intersection, taken advantage that the projection of this curve in the vertical plane of Fig. 1 is an arc of parabola. The reason is that both surfaces are revolution quadrics with axes defining a plane parallel to vertical plane [6]. The metrics that quantify the comparison of accuracies are extracted as follows: 1. The conic associated to the true (exact) cone-sphere intersection projection is computed using five true points of this conic, in the dynamic geometry software (GeoGebra). 2. We calculate the dispersion of the conics identified by GeoGebra when their definition points are moved along the cone-sphere intersection projection, from the true point towards its extreme points, defined as equidistant points from their adjacent true points. The procedure is clarified in Fig. 2, which shows a true point and their two extreme points M1 and M2. The true points and projection curve have been extracted and enlarged from Fig. 9. 3. The horizontal distance between any point in the calculated projection curve and the true projection curve (absolute error curve), Err(y), is calculated as a function of the vertical distance, y. 4. The influence of dimensional parameters of the system in Err(y) is obtained and discussed. Fig. 2 True point (p’1p’2)in the projection of cone-sphere intersection and points M1 and M2, which are equidistant to the adjacent true points (p’3p’4,p’5p’6) 3 Locus Based Surface Intersection Theory The intersection between two surfaces produces a curve in space that in general will not be flat. The calculation of these curves is a ubiquitous problem in science and engineering, including the 3D definition of any industrial part or system, the spatial analysis of biomechanical systems, or the study of a focal lighting system, to name a few examples. The general technique for calculating the intersection curve between surfaces in descriptive geometry is based on the use of a set of auxiliary surfaces, defined in such a way that the encounter between any one of them and the two data surfaces produces a pair of simple curves that will be cut by belonging to the same auxiliary surface, to provide points of the intersection curve sought. This procedure must be repeated until enough points is obtained to define the curve with the required accuracy. The details concerning the typology of auxiliary surfaces to be used depending on the datum surfaces are part of the large body of existing knowledge in descriptive geometry. One feature to note in this general technique is that the points that meet each auxiliary surface are exact (true). This property provides some control over the accuracy of the resulting 3D curve. This procedure on paper is iterative and time-consuming and requires obtaining notable points of the curve, such as those belonging to contour lines and auxiliary boundary surfaces, which ensure that it meets a set of minimum quality criteria. As the contour lines depend on the direction and type of projection, the points of contact with the desired curve also depend on the direction and type of projection, which is an additional shortcoming of the technique. The above limitations are inherent to the manual technique of performing the procedures, but not to the descriptive geometry itself. Thus, the general technique of calculating the intersection curve can be formulated as follows. Calling σ the plane curve resulting from the projection of the intersection curve C onto a plane of interest, σ will be defined by the set of all its points pσ. On the other hand, if the auxiliary surfaces that give the points pσ of σ are expressed as S(ω), where the parameter ω in a set Ω identifies each of the possible surfaces, then the curve σ can be expressed as the locus of the points pσ (ω) for all ω. That is: where L (ω) is a function that represents the curve parametrically, providing points in flat space (plane on which C is projected) and taking ω ∈ Ω as a parameter. The function L exists and will be continuous under certain usual continuity assumptions on S(ω) and on data surfaces. Under these conditions, CeDG allows to construct the functions L i associated with the projections of the intersection curve C onto the planes of interest denoted by the index i (σi), using the Geogebra locus command. This command is mainly employed in two forms: with a reference point, locus (pσ, pref) and with a parameter, locus (pσ, t). In the first form, the set Ω is defined by a reference locus with points pref, while the second form sets Ω as an interval in R defining the values of t. To clarify the constructive procedure in each case, two examples are discussed below. 3.1 Ω Defined as a Reference Locus The first example uses the cylinder-cone bite of Fig. 3. The intersection between these surfaces is a warped curve that can be obtained on paper by the general technique of intersection between surfaces of descriptive geometry, but the process is very sensitive to drawing errors due to the nature of the curve. Figure 3 shows three projections of such a curve, obtained using the LSIM-based CeDG model. Auxiliary surfaces used for this system are planes defined by the vertex of the cone (V) and one of the generatrixes of the cylinder. Since the cylinder is perpendicular to the profile plane, its projection in this plane coincides with its circular directrix. Figure 3 shows a generic auxiliary plane perpendicular to the profile plane and defined by the line passing through the points v" and 3"4" in the profile. This auxiliary plane allows obtaining two points of the intersection curve sought, defined by 3 and 4 in the horizontal projection, and 3’ and 4’ in the vertical one. To build the LSIM-based CeDG model of this system, we start by calculating the points 3 and 4 in 3D starting from the auxiliary plane mentioned above, defined in terms of a free point on the reference locus. The free point is 3"4" (pref) and the reference locus is the arc of circle inside the cone between the boundary points 1" and 2" in the profile, as shown in Fig. 3. This point can be created graphically by clicking on the arc after selecting the point tool (Point(Arc) command). Once created, it can be dragged with the mouse along the arc. Once the chosen plane is placed in a comfortable situation to work (preferably an intermediate one), the general intersection technique is applied, which will provide the points 3 and 4 belonging to the intersection curve sought, symmetrically placed on the left and right side of the plan and elevation views. According to the previous analysis, the arc of circle is theΩlocus that defines the projections of the intersecting curve according to Eq. (2). Thus, the horizontal projection of the curve, σ1, will be: Fig. 3 LSIM based CeDG model of cylinder-cone bite where 3 and 4 denote the horizontal projections of the points obtained through the auxiliary plane. In the same way the vertical projection will be: where 3’ and 4’ are the vertical projections of the points obtained through the auxiliary plane. Equations (3) and (4) define the cylinder-cone intersection curve by means of algebraic entities created in the process of determining the pair of points associated with the generic auxiliary plane, thus defining an exact (true) and complete solution. The intersection curve computation process in CeDG only requires the use of a generic auxiliary surface and does not need to calculate points related to boundaries or others to obtain the complete solution of the system. The L function that defines each projection is specific of the modeled system and can be composed of several leaves, as in this example. Each leaf has been represented in a diLerent color in Fig. 3. It is remarkable to note that, in agreement with the small distance between points 1” and 2”, almost the entire cylinder lies inside the cone. A slight shift of the cylin-drical directrix to the left is enough to convert the bite in a penetration characterized by two unconnected warped curves. Each curve will be defined in such a case for a single leaf. The LSIM-based CeDG model may automatically addresses this change in the nature of the intersection, although the analysis exceeds the scope of this paper. 3.2 Ω Defined as Interval in R The second example shows the intersection of a conical surface with a sphere, a situation that can appear in diLerent situations of technical interest. One of them is the illumination of a sphere from a focal light beam, which was defined as case study in Methods section, and it is shown in Fig. 1. When solving this system, it is necessary to calculate the intersection of the light cone with the illuminating sphere, to obtain a portion of the curve that limits the illuminated area. The curve is completed by a portion of the separatrix circle defined from the light source, which distinguishes the spherical area that can be illuminated from that which cannot. The spherical illuminated area is therefore defined as the area that can be illuminated (internal to the separatrix circumference) and that which is eLectively illuminated by the eLective light cone (intersection of light cone sphere). The CeDG solution of this system is presented in Fig. 4. The boundary curve of the illuminated surface, given by its projections, is given as the sum of a portion of the separatrix curve (flat) connected by points P1 and P2 with the cone-sphere intersection curve (warped). From the complete construction process, we are interested in the LSIM-based solution for the cone-sphere intersection curve. Choosing horizontal planes as auxil-iary surfaces, S(ω), where ω is some real value parameter that allows to identify them, each of them generates a circle in the cone and another in the sphere. These circles may encounter each other to produce one (tangent circles) or two (secant circles) points of the intersection curve sought. Setting as ω parameter the vertical distance of the horizontal plane in relation to the point I1 (see Fig. 4), the intersection curve will be given by its projections σ1 (horizontal) and σ2 (vertical), derived from Eq. (2)for t = ω ∈Ω, beingΩan interval in R given by [0, Lim2] with Lim2 equal to the vertical distance of point I2 with respect to I1. The horizontal projection consists of two leaves: Fig. 9 Conic dispersion in cone-sphere intersection from interpolation spline-based solution Fig. 10 Conic associated to the cone-sphere intersection obtained by the LSIM As expected, the errors of the true five points pertaining to the vertical projection of the cone-sphere intersection are null. Their vertical distances are marked in the abscissa axis of Fig. 11. The maximum absolute error is 0.35 mm (0.71 % with respect to the horizontal distance between I’1 and I’2). Although the errors are not large, in agreement with projections presented in Fig. 8 (right), they could increase in other projections. Previous shortcomings in the geometrical and physical compliance, together with this dimensional error, reduce the capability of the standard descriptive geometry technique to extend the solution to other parameters’ values and projections, in opposition to LSIM-based CeDG. Fig. 11 Horizontal absolute error(cm)ofthe vertical projection of the interpolation spline-based cone-sphere solution, as a function of the vertical dimension We have finally evaluated the influence of the model’s parameters on the Err(y) dimensional error of the interpolation spline-based cone-sphere solution for the same projection. The Fig. 12 shows how the light beam focus radius, defined by rBCono parameter, aLects to Err(y) when it goes down from 70 mm to 60 mm (red trajecto-ries), and when it goes up from 70 mm to 80 mm (blue trajectories). The maximum absolute errors were 0.32 mm (0.72 %) for rBCono = 60 mm and 0.38 mm (0.74 %) for rBCono = 80 mm. The values of the vertical distances marked in the abscissa axis refer to rBCono = 70 mm. The dimensional error behavior is similar for the other parameters. Therefore, although the interpolation spline-based cone-sphere intersection solution has limita-tions that diLicult the extension to other parameters’ values and projections when the compliance with geometrical and physical requirements is kept, the model is robust against perturbations in these parameters. The diLerence between the conic eccentricity (c/a) of the reference parabola (1.00028) and the true parabola (1) is associated to the accuracy of GeoGebra to solve the 5-equations system defined by the 5 points that define the parabola coeLicients. The relative error is 0.028 %. Although it is a small error, it aLects to the capability of CeDG to control the integrity of geometry (c/a > 1 is a hyperbola) and the accuracy and thus more research is required to improve this issue. Fig. 12 Influence of the light beam focus radius on the horizontal absolute error of the interpolation splinebased cone-sphere intersection (60–70 mm red; 70–80 mm blue, increments of 1mm) 5 Conclusions CeDG is a novel approach to computer modelling of 3D geometric systems based on descriptive geometry, which overcomes several limitations of current CAD systems. This study has presented first the theory underlying the Locus-based Surfaces’ Intersection Method (LSIM), and second a case study about the focal illumination of a sphere that allowed comparing LSIM against standard descriptive geometry procedures. The outcomes demonstrate that the complexity of the LSIM CeDG solution, in terms of the number of iterations required, is lesser than that of the standard descrip-tive geometry procedures. In addition, the accuracy of LSIM CeDG was much higher than that of standard procedures. In fact, the accuracy was only limited by the numer-ical precision of GeoGebra, since the computed intersection curve is formulated through algebraic objects that keep the geometric integrity of the curve projections. We may conclude that CeDG provides the true cone-sphere intersection curve. Finally, the LSIM CeDG solution kept the compliance with geometrical require-ments when the projection and parameters’ values of the model were changed, in opposition with the standard technique that requires additional geometrical restrictions. CeDG is currently in a starting version that will evolve to include the automa-tion and improvement of many geometric procedures to facilitate its diLusion to professional and academic domains. References 1. Monge G (1851) An elementary treatise on descriptive geometry, with a theory of shadows and of perspective. Oxford University, London, p 137 2. Izquierdo Asensi F (1996) Geometrı.a Descriptiva Superior y Aplicada, cuarta ed. Paraninfo, p 642 3. Asensi FI (2000) Geometrı.a Descriptiva [Descriptive Geometry], 24th edn. Editorial Paraninfo, p 287 4. Gonza.lez Monsalve M, Palencia Cortes J (2006) Geometrı.a descriptiva 5. de Abajo FJR (2007) Geometrı.a descriptiva.Tomo I. Sistema Die.drico, Editorial Donostiarra 6. Taibo A= (2009) Geometrı.a descriptiva y sus aplicaciones II: Curvas y super?icies. Editorial Tebar S.L. Madrid, vol II, p 446 7. Leighton Wellman B (1987) Geometrı.a Descriptiva [Technical Descriptive Geometry]. Editorial Reverte., S.A., Sevilla, p 615 8. Weisberg DE (2008) Computer-aided design strong roots at MIT. In: The engineering design revolution: the people, companies and computer systems that changed Forever the Practice of Engineering, p 650. http://cadhistory.net/ 9. Marsh D (2005) Applied Geometry For Computer Graphics and CAD, 2nd edn. Springer-Verlag, London, United States of America, p 350 10. Boissonnat J-D, Teillaud M (eds) (2006) Effective computational geometry for curves and surfaces. Springer 11. Prado-Velasco M, Ortı.z Marı.n R, Garcı.a L, Rio-Cidoncha MGD (2021) Graphical modelling with computer extended descriptive geometry (CeDG): description and comparison with CAD. Comput Aided Des Appl 18:272– 284. https://doi.org/10.14733/cadaps.2021.272-284 12. Migliari R (2012) Descriptive geometry: from its past to its future. Nexus Netw J 14:555–571. https://doi.org/10.1007/s00004-012-0127-3 13. Stachel H (2007) The status of todays descriptive geometry related education (Cad/Cg/Dg) in Europe. J Graph Sci Japan 41:15–20. https://doi.org/10.5989/jsgs.41.Supplement1_15 14. Hildenbrand D, Oldenburg R (2007) Geometric algebra: a foundation of elementary geometry with possible applications in computer algebra based dynamic geometry systems. Electron J Math Technol 1:1–19 15. Kova.cs Z, Recio T, Ve.lez MP (2021) GeoGebra discovery in context. Electron Proc Theoret Comput Sci 352:141– 147. https://doi.org/10.4204/eptcs.352.16 16. Kllogjeri Q, Kllogjer P (2017) Geogebra: a vital bridge linking mathematics with biology and other sciences. SM J Bio 3 17. Hohenwarter J, Hohenwarter M (2019) Geogebra classic manual. https://wiki.geogebra.org/ en/Manual 18. Prado-Velasco M, Ortiz-Marı.n R (2021) Comparison of computer extended descriptive geometry (CeDG) with CAD in the modeling of sheet metal patterns. Symmetry 13:685