# Computer-Aided Design: Development of a Software Tool for Solving Loci Problems

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## Abstract

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## 1. Introduction

## 2. Definitions and Foundations

- Those involving a circle and an external point where the point curve (hyperbola) containing the centres of all possible circles, tangent to the given circle and passing through the point, was to be analytically defined.
- Those involving any two mutually external circles where the point curve (hyperbola branch) containing the centres of a circle tangent to the two given circles was to be analytically defined.

- The implementation of the routines allowing software to depict the correct function in situ and checking that the results were accurate by providing a graphical output as in professional CAD software.
- Defining and solving an exercise to illustrate the process and envisage the potential for solving actual geometries in order to compare the proposed procedures with those used by existing graphical software.

**Case 1**: With a circle and an external point (Figure 2), any circle tangent to the given circle and passing by the given point will fulfil the following conditions:

- Its centre will be a point in the geometric locus of the points lying at the smallest possible distance from the circle coinciding with its distance from the point. This will allow the centres of all circles passing by the point and being external tangents to the given circle to be identified.
- Its centre will be a point in the geometric locus of the points lying at the greatest possible distance from a circle coinciding with its distance from the point. This will allow the centres of all circles passing by the point and being internal tangents to the given circle to be identified.

**Case 2**: In the case of circles tangent to two external ones of radius R1 and R2 (Figure 3), one of the following conditions must be fulfilled:

- The centre will be a point in the geometric locus of the points lying at the maximum distance from the two circles, which will allow the centres of all circles having the two given circles as internal tangents to be determined.
- The centre will be a point in the geometric locus of the points lying at the shortest possible distance from the two given circles, which will allow the centres of all externally tangent circles to be identified.
- The centre will be a point in the geometric locus of the points lying at the shortest possible distance from one given circle and greatest from the other, which will allow the centres of all circles being externally tangent to the first and internally tangent to the second to be identified.
- The centre will be a point in the geometric locus of the points lying at the greatest distance from one of the given circles and shortest from the other, which will allow the centres of all circles being internally tangent to the former and externally tangent to the latter to be identified.

## 3. Practical Example

## 4. Implementation of the Software Tool

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- The ends connecting the segment to the next should have the same tangent.
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- The tangent at its midpoint should have the same direction as the segment connecting the previous and next point.
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- The segment should pass through the three points.

## 5. Conclusions

- The maximum and minimum point-to-circle distance needed to obtain the equations of the curves corresponding to different relative positions.
- The boundary conditions needed to determine the boundaries of the zones where the hyperbolic solutions lie, discard those curve segments not fulfilling the requirements and draw the segments that do fulfil them.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Cummins, C.; Seale, M.; Macente, A.; Certini, D.; Mastropaolo, E.; Viola, I.M.; Nakayama, N. A separated vortex ring underlies the flight of the dandelion. Nature
**2018**, 562, 414–418. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rojas-Sola, J.I. Advances in engineering graphics: Improvements and new proposals. Symmetry
**2021**, 13, 827. [Google Scholar] [CrossRef] - Llach, D.C. Reconstructing “sketchpad” and the “coons patch”: Toward an archaeology of CAD. Leonardo
**2018**, 51, 429–430. [Google Scholar] - Rojas-Sola, J.I.; Del Río-Cidoncha, G.; Ortíz-Marín, R.; López-Pedregal, J.M. Design and development of sheet-metal elbows using programming with visual basic for applications in CATIA. Symmetry
**2020**, 13, 33. [Google Scholar] [CrossRef] - Pan, Z.; Wang, X.; Teng, R.; Cao, X. Computer-aided design-while-engineering technology in top-down modeling of mechanical product. Comput. Ind.
**2016**, 75, 151–161. [Google Scholar] [CrossRef] - Wang, H.; Zeng, Y.; Li, E.; Huang, G.; Gao, G.; Li, G. “Seen Is Solution” a CAD/CAE integrated parallel reanalysis design system. Comput. Methods Appl. Mech. Eng.
**2016**, 299, 187–214. [Google Scholar] [CrossRef] - Bonneau, G.P.; Bartoň, M.; Nelaturi, S. SPM 2021 Editorial. Comput. Aided Des.
**2022**, 143, 103138. [Google Scholar] [CrossRef] - Chang, Y.S.; Chen, M.Y.C.; Chuang, M.J.; Chou, C.H. Improving creative self-efficacy and performance through computer-aided design application. Think. Ski. Creat.
**2019**, 31, 103–111. [Google Scholar] [CrossRef] - Flores, R.L.; Belaud, J.P.; Negny, S.; Le Lann, J.M. Open computer aided innovation to promote innovation in process engineering. Chem. Eng. Res. Des.
**2015**, 103, 90–107. [Google Scholar] [CrossRef] [Green Version] - Husig, S.; Kohn, S. Computer aided innovation−State of the art from a new product development perspective. Comput. Ind.
**2009**, 60, 551–562. [Google Scholar] [CrossRef] [Green Version] - Zanni-Merk, C.; Cavallucci, D.; Rousselot, F. An ontological basis for computer aided innovation. Comput. Ind.
**2009**, 60, 563–574. [Google Scholar] [CrossRef] - Li, J.; Li, Y. Research and application of computer aided design system for product innovation. J. Comput. Methods Sci. Eng.
**2019**, 19, S41–S46. [Google Scholar] [CrossRef] - Bonnici, A.; Akman, A.; Calleja, G.; Camilleri, K.P.; Fehling, P.; Ferreira, A.; Hermuth, F.; Israel, J.H.; Landwehr, T.; Liu, J.; et al. Sketch-based interaction and modeling: Where do we stand? AI EDAM-Artif. Intell. Eng.Des. Anal. Manuf.
**2019**, 33, 370–388. [Google Scholar] [CrossRef] - Bobenrieth, C.; Cordier, F.; Habibi, A.; Seo, H. Descriptive: Interactive 3D shape modeling from a single descriptive sketch. Comput. Aided Des.
**2020**, 128, 102904. [Google Scholar] [CrossRef] - Gryaditskaya, Y.; Sypesteyn, M.; Hoftijzer, J.W.; Pont, S.; Durand, F.; Bousseau, A. OpenSketch: A richly-annotated dataset of product design sketches. ACM Trans. Graph.
**2019**, 38, 232. [Google Scholar] [CrossRef] [Green Version] - Sun, L.; Xiang, W.; Chai, C.; Yang, Z.; Zhang, K. Designers’ perception during sketching: An examination of creative segment theory using eye movements. Des. Stud.
**2014**, 35, 593–613. [Google Scholar] [CrossRef] - Sun, L.; Xiang, W.; Chai, C.; Wang, C.; Huang, Q. Creative Segment: A descriptive theory applied to computer-aided sketching. Des. Stud.
**2014**, 35, 54–79. [Google Scholar] [CrossRef] - Veisz, D.; Namouz, E.Z.; Joshi, S.; Summers, J.D. Computer-aided design versus sketching: An exploratory case study. AI EDAM-Artif. Intell. Eng. Des. Anal. Manuf.
**2012**, 26, 317–335. [Google Scholar] [CrossRef] - Willis, K.D.D.; Jayaraman, P.K.; Lambourne, J.G.; Chu, H.; Pu, Y.W. Engineering sketch generation for computer-aided design. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), Nashville, TN, USA, 19–25 June 2021; pp. 2105–2114. [Google Scholar] [CrossRef]
- Zhong, Y.; Qi, Y.G.; Gryaditskaya, Y.; Zhang, H.G.; Song, Y.Z. Towards practical sketch-based 3D shape generation: The role of professional sketches. IEEE Trans. Circuits Syst. Video Technol.
**2021**, 31, 3518–3528. [Google Scholar] [CrossRef] - Velichová, D. Geometry in engineering education. Eur. J. Eng. Educ.
**2002**, 27, 289–296. [Google Scholar] [CrossRef] - Voronina, M.V.; Tretyakova, Z.O.; Moroz, O.N.; Folomkin, A.I. Descriptive geometry in educational process of technical university in Russia today. Int. J. Environ. Sci. Educ.
**2016**, 11, 10911–10922. [Google Scholar] - Ullah, A.M.M.S.; Harib, K.H. Tutorials for integrating CAD/CAM in engineering curricula. Educ. Sci.
**2018**, 8, 151. [Google Scholar] [CrossRef] [Green Version] - Liapi, K.A. Geometry in architectural engineering education revisited. J. Archit. Eng.
**2002**, 8, 80–88. [Google Scholar] [CrossRef] - Ranger, F.; Vezeau, S.; Lortie, M. Traditional product representations and new digital tools in the dimensioning activity: A designers’ point of view on difficulties and needs. Des. J.
**2018**, 21, 707–730. [Google Scholar] [CrossRef] - Papachristou, E.; Kyratsis, P.; Bilalis, N. A comparative study of open-source and licensed CAD software to support garment development learning. Machines
**2019**, 7, 30. [Google Scholar] [CrossRef] [Green Version] - Blažek, J.; Pech, P. Locus Computation in Dynamic Geometry Environment. Math. Comput. Sci.
**2019**, 13, 31–40. [Google Scholar] [CrossRef] - Kovács, Z. Achievements and Challenges in Automatic Locus and Envelope Animations in Dynamic Geometry. Math. Comput. Sci.
**2019**, 13, 131–141. [Google Scholar] [CrossRef] [Green Version] - Rojas-Sola, J.I.; Hernandez-Diaz, D.; Villar-Ribera, R.; Hernandez-Abad, V.; Hernandez-Abad, F. Computer-Aided Sketching: Incorporating the locus to improve the three-dimensional geometric design. Symmetry
**2020**, 12, 1181. [Google Scholar] [CrossRef] - Hernandez-Abad, F.; Hernandez-Abad, V.; Ochoa-Vives, M. Lugares Geométricos: Su Aplicación a Tangencias; UPC: Barcelona, Spain, 1993. (In Spanish) [Google Scholar]
- Hernández-Abad, F.; Rojas-Sola, J.I.; Hernández-Abad, V.; Ochoa-Vives, M.; Font-Andreu, J.; Hernandez-Diaz, D.; Villar-Ribera, R. Educational software to learn the essentials of engineering Graphics. Comput. Appl. Eng. Educ.
**2012**, 20, 1–18. [Google Scholar] [CrossRef] [Green Version] - Hernández-Abad, F.; Rojas-Sola, J.I.; Hernández-Abad, V.; Ochoa-Vives, M.; Font-Andreu, J.; Hernandez-Diaz, D.; Villar-Ribera, R. Interactive educational software of textile design. Comput. Appl. Eng. Educ.
**2012**, 20, 161–174. [Google Scholar] [CrossRef]

**Figure 1.**Examples of conical curves as loci: (

**a**) equidistance between a straight line and a circle; (

**b**) locus of points at given distance from a fixed point and twice that distance from another.

**Figure 2.**A circle and an external point, together with the curve containing all possible centres of the circles that are tangent interior (

**left**) and exterior (

**right**) to the given circle and pass by the given point (hyperbolic branches).

**Figure 3.**External circles and curve containing all possible centres of the circles that are internally tangent to them (hyperbolic branch).

**Figure 5.**Whole hyperbola (canonical position) and definition of the parameters of the canonical equation of the hyperbola.

**Figure 6.**Hyperbolic branch corresponding to the geometric locus of all possible centres of circles that are external tangents to the given circles.

**Figure 7.**Curve (hyperbolic branch) corresponding to the geometric locus of the centres of circles that are external tangents to that of radius R1 and internal tangents to that of radius R2.

**Figure 8.**Curve (hyperbola) corresponding to the geometric locus of the centres of circles that are external tangents to the circle of radius R1 and internal to that of radius R2, and vice versa (two branches).

**Figure 10.**Graphical depiction of the problem involving the original circles of the two cogwheels in Figure 9.

**Figure 14.**Implementation of the procedures used to represent the hyperbolic branches in graphical form.

**Figure 23.**Geometric locus of the centres of circles that are external tangents to the two given circles.

**Figure 25.**Geometric locus of the centres of circles that are internal tangents to two mutually external circles.

**Figure 26.**Geometric locus of the centres of circles that are external tangents to the first given circle and internal tangents to the second.

**Figure 27.**Geometric locus of the centres of circles that are internal tangents to the first given circle and external tangents to the second.

Type | Geometric Restrictions on Tangency | Case Number |
---|---|---|

A | An external tangent to the circle passing through the point | 1 |

An internal tangent to the circle passing through the point | 2 | |

B | An internal tangent to both circles | 3 |

An external tangent to both circles | 4 | |

An external tangent to the first circle and an internal tangent to the second | 5 | |

An internal tangent to the first circle and an external tangent to the second | 6 |

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## Share and Cite

**MDPI and ACS Style**

Hernández-Díaz, D.; Hernández-Abad, F.; Hernández-Abad, V.; Villar-Ribera, R.; Julián, F.; Rojas-Sola, J.I.
Computer-Aided Design: Development of a Software Tool for Solving Loci Problems. *Symmetry* **2023**, *15*, 10.
https://doi.org/10.3390/sym15010010

**AMA Style**

Hernández-Díaz D, Hernández-Abad F, Hernández-Abad V, Villar-Ribera R, Julián F, Rojas-Sola JI.
Computer-Aided Design: Development of a Software Tool for Solving Loci Problems. *Symmetry*. 2023; 15(1):10.
https://doi.org/10.3390/sym15010010

**Chicago/Turabian Style**

Hernández-Díaz, David, Francisco Hernández-Abad, Vicente Hernández-Abad, Ricardo Villar-Ribera, Fernando Julián, and José Ignacio Rojas-Sola.
2023. "Computer-Aided Design: Development of a Software Tool for Solving Loci Problems" *Symmetry* 15, no. 1: 10.
https://doi.org/10.3390/sym15010010