Villarceau Circles as a Path to Machine Toroid Surfaces in 5-axis

Villarceau Circles as a Path to Machine Toroid Surfaces in 5-axis | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Villarceau Circles as a Path to Machine Toroid Surfaces in 5-axis Javier Castellote, Manuel Estrems Amestoy, Óscar de Francisco, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7048862/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 5 You are reading this latest preprint version Abstract Toroid surfaces are frequently found in mechanical components with high functional performance. In this work, a way of machining this type of surface has been developed using Villarceau circles which are the result of the intersection of a toroid with an inclined plane at a particular angle. These circles are neither the directrix nor the generatrix circles of the toroid. Its use as a cutting path has advantages such as the possibilities to get different surfaces textures orientation. As an example of application, the 5-axes machining with a cylindrical mill of a toroidal propeller that belongs to a centrifugal impeller compressor is presented. This includes the geometrical description, the planning of the machining process, the description of the machine tool and fixtures, the definition of trajectories to maintain a constant thickness of blades, and the machined part. Several defects and inconveniences will be discussed. Villarceau Circles Toroid Surfaces 5-Axis Machining Centrifugal Impeller Compressor Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 1 Introduction Toroidal surfaces are widely used in mechanical components such as valves, junctions, and propellers due to their geometric and functional properties. Achieving high-precision manufacturing of these surfaces presents significant challenges, particularly in finishing operations, where surface texture and geometric accuracy directly impact component performance. Traditional approaches rely on axial and radial toolpaths for machining toroidal geometries. However, these conventional strategies often suffer from geometric deviations, uneven surface quality, and limitations in tool accessibility. A novel approach for machining toroidal surfaces is based on the Villarceau circles, first described by Yvon Villarceau in 1848 [ 1 ]. These circles arise when a torus is intersected by a plane at a specific oblique angle, forming two circular sections distinct from the equatorial and meridional sections. While Villarceau circles have been extensively studied in the field of differential geometry [ 2 ], their application in advanced manufacturing remains underexplored. Recent developments in multi-axis machining suggest that integrating Villarceau circles into toolpath generation can optimize machining performance by improving surface finish, reducing tool deflection, and enhancing process efficiency [ 3 ]. The emergence of five-axis machining has revolutionized the manufacturing of freeform surfaces, including toroidal components. Five-axis strategies enable better control over tool orientation, minimizing abrupt changes in tool engagement and eliminating undesirable cutting conditions such as zero-velocity points [ 4 ]. Studies on trajectory generation for freeform surfaces indicate that curvature-adaptive toolpaths enhance machining accuracy and surface quality, particularly in the context of circular interpolation techniques [ 5 ]. The use of Villarceau circles as a machining strategy aligns with these advancements by enabling continuous tool movement along natural circular trajectories, thereby reducing sudden accelerations and decelerations that degrade machining precision. Recent research has also emphasized the role of advanced interpolation techniques in optimizing CNC programming for complex geometries. Dong et al. [ 6 ] highlighted the significance of multi-scale tool orientation methods in freeform surface machining, demonstrating that advanced interpolation strategies, such as NURBS-based toolpath planning and circular arc approximation, contribute to reducing computational overhead and improving the continuity of tool trajectories, thereby enhancing machining efficiency and accuracy. These advancements facilitate the practical implementation of Villarceau-based machining strategies in industrial applications. Additionally, optimized toolpath interpolation has been demonstrated to improve surface roughness and overall machining efficiency in five-axis milling processes. Langeron et al. [ 7 ] introduced a B-spline-based toolpath computation method that significantly improves tool trajectory continuity, leading to reduced machining time and enhanced surface quality. Their findings confirm the benefits of polynomial interpolation in achieving smoother tool movements and minimizing sudden directional changes, which are critical for achieving high-precision results in five-axis milling operations. The application of Villarceau circles in machining toroidal surfaces offers several key advantages: Optimized Surface Roughness: By avoiding purely radial or axial tool orientations, machining trajectories based on Villarceau circles improve surface texture control, which is critical in components subject to friction and aerodynamic constraints. Simplified CNC Programming: The inherent circular nature of Villarceau-based toolpaths reduces the complexity of CNC code by leveraging circular interpolation techniques, thus improving computational efficiency and program readability. Enhanced accessibility in Five-Axis Machining: Villarceau trajectories provide continuous tool access to the entire toroidal surface, optimizing cutting conditions and preventing low tangential velocity zones that typically arise in conventional machining strategies. This study aims to validate the feasibility of using Villarceau circles for machining toroidal surfaces by applying them to the production of a centrifugal impeller air compressor rotor as example. The methodology includes the geometrical description, the planning of the machining process, the description of the machine tool, the definition of trajectories to maintain a constant thickness of blades, the CNC program and, finally, the machining of the part. The results will demonstrate their potential to enhance precision, efficiency, and overall machining quality compared to conventional strategies. 2 Geometrical description The parametric equation of a toroid is generated by the revolution around Z axis of a circumference C 1 with radius r in the plane XZ with center in O 1 at a distance R of the Z axis, as shown in Fig. 1 . Its parametric equations are: $$\:\varvec{P}\left(u,v\right)=\left(\begin{array}{c}x\\\:y\\\:z\end{array}\right)=\left(\begin{array}{c}\left(R+r*\text{cos}\left(v\right)\right)*\text{c}\text{o}\text{s}\left(u\right)\\\:\left(R+r*\text{cos}\left(v\right)\right)*\text{s}\text{i}\text{n}\left(u\right)\\\:r*\text{s}\text{i}\text{n}\left(v\right)\end{array}\right)$$ 1 Villarceau’s Circles, in geometry, are a pair of circles generated in a toroid by obliquely sectioning a plane that passes through the center, according to Eq. 2 . $$\:\text{cos}\left(\alpha\:\right)*z+\text{s}\text{i}\text{n}\left(\alpha\:\right)*x=0$$ 2 The intersection is represented in Fig. 2 and the parametric expression of the curve is determined by substituting the value of u in Eq. 1 the expression of Eq. 3 . $$\:u=\text{a}\text{c}\text{o}\text{s}\left[\frac{\sqrt{{R}^{2}-{r}^{2}}*sen\left(v\right)}{R+r*cos\left(v\right)}\right]$$ 3 3 Description of the component to be machined as an example. For the machining of a circle of Villarceau on a toroid, a rotor of centrifugal propeller of gases has been designed. The primitive of the component is a toroid that has been previously machined in a CNC lathe. The blades of the impeller trace the trajectory of one of Villarceau’s circle depending on the direction of rotation. The thickness of the blade has been established to give some rigidity to the blade. The part is machined by a cylindrical mill of 5 mm in diameter. The faces of the blade are parallel to the circle with a direction normal to the surface in the same circle. A Depth of cutting is evolving linearly from the entrance at the neck of the toroid until the exit at the base. The numbers of blades are limited due to the diameter of the mill and its relation with the diameter of the toroid in the place of cutting, in such a way that at certain distances from the neck a doble of blade could emerge. In Fig. 3 a drawing of the workpiece with its dimensions is depicted as well as the solid model of the design component. 4 Description of the machine tool and planning of the machining process The machine tool is a CNC milling machine LAGUN 650 with a rotary table Spirsin Divisores Electrónicos S.L. model 8210.2F/125 that allows the interpolation of trajectories in the B and C axes. Axis B is limited from − 10 o to 100 o . The configuration is shown in Fig. 4 . The CNC program uses the ISO G-code for programing the coordinates X, Y and Z position of the tool with length compensation, and the angular coordinates B and C of the rotary table. For the fixture of the preform to the rotary table, an utensil has been designed requiring a drilling at its axis for the screw fixation as shown in Fig. 5 . The surface of the toroid has been painted to make clear the surface that has not be machined in the final part. Definition of trajectories to maintain a constant thickness of blades The trajectory of the tool is defined from the Villarceau circle over the toroid surface following a parallel curve at a constant distance of the semi-width of the blade as shown in Fig. 7 . The cutting of one face begins from the neck with a table position B = 90 o as shown inf Fig. 6 and it ends in a vertical position. The mill is always in a position perpendicular to the surface and with a depth cut that varies from a maximum at the beginning and a minimum at the end. The position and orientation of all points that must be followed by the center of the mill at the tip has been calculated in a worksheet for the two faces of the blade. The inverse kinematics is made by the methodology described in [ 2 ]. The code is executed as many times as the number of blades N has the part after the preselection of C an angle 2π/N. An important issue is that a security distance is required to avoid impacts between tool and the workpiece, and to complete the machining of the surface at the beginning and at the end of the trajectory parallel to Villarceau’s circle. Figure 7 represents the initial position of the tool before the machining of each face of the blade. Each trajectory of the tool for the machining of one face has been discretized in 40 points in which linear interpolation G1 has been applied. Intermediate blades have been machined between the complete blades that will reinforce the fluid flow during its performance. 5 Analysis of the final part The final part is represented in Fig. 8 . The mill is a cylindrical tool of 5 mm in diameter. The surface between blades does not coincides with a toroid due to the intersection with the trajectories to generate the intermediate blades. This creates steps between the two surfaces that are more accused in the exit of the tool. This effect could be avoided through a machined trajectory between initial trajectories or by using mills with smaller diameter after the faces of the blades were machined. 6 Conclusions This article has demonstrated the use of Villarceau circles as a basis for machining toroids, facilitating programming cutting paths and generating textures other than circumferential or radial ones. The machining of a centrifugal gas compressor rotor which blades follow the geometry of these circles has been machined as an example of application. Declarations Funding We acknowledge the Department of Mechanical, Materials and Manufacturing Engineering of the Universidad Politécnica de Cartagena for partially funded the elaboration and presentation of the present work. Conflicts of interest Not applicable Author’s Contribution All authors contributed to the conceptualization and design of the study. Concept and Methodology Manuel Estrems and Javier Castellote; formal analysis, Wilmer E. Cumbicus, Manuel Estrems and Javier Castellote; CNC work: Javier Castellote, Manuel Estrems and Oscar de Francisco; writing-original draft preparation, Javier Castellote and Manuel Estrems; writing-review and editing, Wilmer Cumbicus and Óscar de Francisco; supervision, Manuel Estrems and Wilmer E. Cumbicus. All authors have read and approved the final manuscript. References Villarceau, Y. (1848). Théorème sur le tore. Nouvelles Annales de Mathématiques, 7, 345-347. http://www.numdam.org/item/NAM_1848_1_7__345_1.pdf (visited 15/02/2025) Struik, D. J. (1988). Lectures on classical differential geometry. Dover Publication. ISBN: 0486656098 Dong, J., He, J., Liu, S., Wan, N., & Chang, Z. (2023). "A Multi-Scale Tool Orientation Generation Method for Freeform Surface Machining with Bull-Nose Tool." Micromachines, 14(6), 1199. https://doi.org/10.3390/mi14061199 Estrems, M., Castellote, J., Cumbicus, W. E., Sánchez, H., Carrero-Blanco, J., de Francisco, Ó., & Arizmendi, M. (2019). Trajectory generation in 5-axis milling of freeform surfaces using circular arc approximation and its influence in surface roughness. Procedia Manufacturing, 41, 208-215. https://doi.org/10.1016/j.promfg.2019.07.048 Altintas, Y. (2012). Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design. Cambridge University Press. ISBN: 978-1107001480 Dong, J., He, J., Liu, S., Wan, N., & Chang, Z. (2023). A Multi-Scale Tool Orientation Generation Method for Freeform Surface Machining with Bull-Nose Tool. Micromachines, 14(6), 1199. https://doi.org/10.3390/mi14061199 Langeron, J.-M., Duc, E., Lartigue, C., & Bourdet, P. (2004). A new format for 5-axis tool path computation, using B-spline curves. Computer-Aided Design, 36(12), 1219-1229. https://doi.org/10.1016/j.cad.2003.11.007 Cite Share Download PDF Status: Under Revision Version 1 posted Editorial decision: Major Revisions Needed 10 Aug, 2025 Reviewers agreed at journal 10 Jul, 2025 Reviewers invited by journal 10 Jul, 2025 Editor assigned by journal 09 Jul, 2025 First submitted to journal 08 Jul, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7048862","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":483714296,"identity":"3154eb04-4d71-461c-a06f-f570462dcc70","order_by":0,"name":"Javier Castellote","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Javier","middleName":"","lastName":"Castellote","suffix":""},{"id":483714297,"identity":"1d0f1156-0c71-4f6e-89b3-e06687e8b1df","order_by":1,"name":"Manuel Estrems 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references.\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-7048862/v1/b0a7af3dbbd0ca9de7765cc3.png"},{"id":86664266,"identity":"818bee03-35eb-4017-af27-9f58b55da0cd","added_by":"auto","created_at":"2025-07-14 10:53:52","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":87049,"visible":true,"origin":"","legend":"\u003cp\u003eGeneration of Circles of Villarceau by intersection of a toroid by a plane.\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-7048862/v1/fb906bbbc4d5a9d1ca35b7a6.png"},{"id":86664267,"identity":"211ab718-00ed-4752-807d-67f56ef9c458","added_by":"auto","created_at":"2025-07-14 10:53:52","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":74198,"visible":true,"origin":"","legend":"\u003cp\u003eA CAD solid design of the component and a drawing of the workpiece.\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-7048862/v1/25bddae44feb693f179bd120.png"},{"id":86662837,"identity":"c3c5c3d5-8458-4533-87a4-847bc17ce0e0","added_by":"auto","created_at":"2025-07-14 10:45:52","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":106381,"visible":true,"origin":"","legend":"\u003cp\u003eMachining Center LAGUN 650 with a rotary table for the 5-axes machining.\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-7048862/v1/5ab6c2320921b540529c013a.png"},{"id":86662849,"identity":"b6b7ac32-3bc9-4123-b939-291872300797","added_by":"auto","created_at":"2025-07-14 10:45:52","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":49124,"visible":true,"origin":"","legend":"\u003cp\u003eToroid and utensil for the fixture in machine tool.\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-7048862/v1/9b43b8fa268e84dd8b307f13.png"},{"id":86662847,"identity":"7d4fe962-8613-426b-81f0-15b1a8284755","added_by":"auto","created_at":"2025-07-14 10:45:52","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":147041,"visible":true,"origin":"","legend":"\u003cp\u003eViews of the machining during the process.\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-7048862/v1/49e99d19d6a0c99ae7670745.png"},{"id":86665178,"identity":"a7bb6860-199f-4b23-95f9-71c2f62cda88","added_by":"auto","created_at":"2025-07-14 11:01:52","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":106423,"visible":true,"origin":"","legend":"\u003cp\u003eSecurity distances for the entrance of the trajectories generated for each face of the blade.\u003c/p\u003e","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-7048862/v1/9520064303a5a5b8512d782b.png"},{"id":86662850,"identity":"2858164e-bfd3-4222-bc82-4b176e4f8510","added_by":"auto","created_at":"2025-07-14 10:45:52","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":185270,"visible":true,"origin":"","legend":"\u003cp\u003eViews of the machined part.\u003c/p\u003e","description":"","filename":"floatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-7048862/v1/1c2a60afd8fee84d327bbaa0.png"},{"id":86666805,"identity":"e1a40b5c-de13-454d-b75e-b612921ca3ec","added_by":"auto","created_at":"2025-07-14 11:09:53","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1250596,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7048862/v1/be9bb75f-7e94-497c-b263-6a40bf670f09.pdf"}],"financialInterests":"","formattedTitle":"Villarceau Circles as a Path to Machine Toroid Surfaces in 5-axis","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eToroidal surfaces are widely used in mechanical components such as valves, junctions, and propellers due to their geometric and functional properties. Achieving high-precision manufacturing of these surfaces presents significant challenges, particularly in finishing operations, where surface texture and geometric accuracy directly impact component performance. Traditional approaches rely on axial and radial toolpaths for machining toroidal geometries. However, these conventional strategies often suffer from geometric deviations, uneven surface quality, and limitations in tool accessibility. A novel approach for machining toroidal surfaces is based on the Villarceau circles, first described by Yvon Villarceau in 1848 [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. These circles arise when a torus is intersected by a plane at a specific oblique angle, forming two circular sections distinct from the equatorial and meridional sections. While Villarceau circles have been extensively studied in the field of differential geometry [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e], their application in advanced manufacturing remains underexplored. Recent developments in multi-axis machining suggest that integrating Villarceau circles into toolpath generation can optimize machining performance by improving surface finish, reducing tool deflection, and enhancing process efficiency [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eThe emergence of five-axis machining has revolutionized the manufacturing of freeform surfaces, including toroidal components. Five-axis strategies enable better control over tool orientation, minimizing abrupt changes in tool engagement and eliminating undesirable cutting conditions such as zero-velocity points [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. Studies on trajectory generation for freeform surfaces indicate that curvature-adaptive toolpaths enhance machining accuracy and surface quality, particularly in the context of circular interpolation techniques [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. The use of Villarceau circles as a machining strategy aligns with these advancements by enabling continuous tool movement along natural circular trajectories, thereby reducing sudden accelerations and decelerations that degrade machining precision.\u003c/p\u003e\u003cp\u003eRecent research has also emphasized the role of advanced interpolation techniques in optimizing CNC programming for complex geometries. Dong et al. [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e] highlighted the significance of multi-scale tool orientation methods in freeform surface machining, demonstrating that advanced interpolation strategies, such as NURBS-based toolpath planning and circular arc approximation, contribute to reducing computational overhead and improving the continuity of tool trajectories, thereby enhancing machining efficiency and accuracy. These advancements facilitate the practical implementation of Villarceau-based machining strategies in industrial applications. Additionally, optimized toolpath interpolation has been demonstrated to improve surface roughness and overall machining efficiency in five-axis milling processes. Langeron et al. [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] introduced a B-spline-based toolpath computation method that significantly improves tool trajectory continuity, leading to reduced machining time and enhanced surface quality. Their findings confirm the benefits of polynomial interpolation in achieving smoother tool movements and minimizing sudden directional changes, which are critical for achieving high-precision results in five-axis milling operations.\u003c/p\u003e\u003cp\u003eThe application of Villarceau circles in machining toroidal surfaces offers several key advantages:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003eOptimized Surface Roughness: By avoiding purely radial or axial tool orientations, machining trajectories based on Villarceau circles improve surface texture control, which is critical in components subject to friction and aerodynamic constraints.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eSimplified CNC Programming: The inherent circular nature of Villarceau-based toolpaths reduces the complexity of CNC code by leveraging circular interpolation techniques, thus improving computational efficiency and program readability.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eEnhanced accessibility in Five-Axis Machining: Villarceau trajectories provide continuous tool access to the entire toroidal surface, optimizing cutting conditions and preventing low tangential velocity zones that typically arise in conventional machining strategies.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eThis study aims to validate the feasibility of using Villarceau circles for machining toroidal surfaces by applying them to the production of a centrifugal impeller air compressor rotor as example. The methodology includes the geometrical description, the planning of the machining process, the description of the machine tool, the definition of trajectories to maintain a constant thickness of blades, the CNC program and, finally, the machining of the part. The results will demonstrate their potential to enhance precision, efficiency, and overall machining quality compared to conventional strategies.\u003c/p\u003e"},{"header":"2 Geometrical description","content":"\u003cp\u003eThe parametric equation of a toroid is generated by the revolution around \u003cem\u003eZ\u003c/em\u003e axis of a circumference \u003cem\u003eC\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e with radius \u003cem\u003er\u003c/em\u003e in the plane \u003cem\u003eXZ\u003c/em\u003e with center in \u003cem\u003eO\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e at a distance \u003cem\u003eR\u003c/em\u003e of the \u003cem\u003eZ\u003c/em\u003e axis, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Its parametric equations are:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:\\varvec{P}\\left(u,v\\right)=\\left(\\begin{array}{c}x\\\\\\:y\\\\\\:z\\end{array}\\right)=\\left(\\begin{array}{c}\\left(R+r*\\text{cos}\\left(v\\right)\\right)*\\text{c}\\text{o}\\text{s}\\left(u\\right)\\\\\\:\\left(R+r*\\text{cos}\\left(v\\right)\\right)*\\text{s}\\text{i}\\text{n}\\left(u\\right)\\\\\\:r*\\text{s}\\text{i}\\text{n}\\left(v\\right)\\end{array}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eVillarceau\u0026rsquo;s Circles, in geometry, are a pair of circles generated in a toroid by obliquely sectioning a plane that passes through the center, according to Eq.\u0026nbsp;\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:\\text{cos}\\left(\\alpha\\:\\right)*z+\\text{s}\\text{i}\\text{n}\\left(\\alpha\\:\\right)*x=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe intersection is represented in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e and the parametric expression of the curve is determined by substituting the value of \u003cem\u003eu\u003c/em\u003e in Eq.\u0026nbsp;\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e the expression of Eq.\u0026nbsp;\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:u=\\text{a}\\text{c}\\text{o}\\text{s}\\left[\\frac{\\sqrt{{R}^{2}-{r}^{2}}*sen\\left(v\\right)}{R+r*cos\\left(v\\right)}\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e"},{"header":"3 Description of the component to be machined as an example.","content":"\u003cp\u003eFor the machining of a circle of Villarceau on a toroid, a rotor of centrifugal propeller of gases has been designed. The primitive of the component is a toroid that has been previously machined in a CNC lathe. The blades of the impeller trace the trajectory of one of Villarceau\u0026rsquo;s circle depending on the direction of rotation. The thickness of the blade has been established to give some rigidity to the blade. The part is machined by a cylindrical mill of 5 mm in diameter. The faces of the blade are parallel to the circle with a direction normal to the surface in the same circle. A Depth of cutting is evolving linearly from the entrance at the neck of the toroid until the exit at the base. The numbers of blades are limited due to the diameter of the mill and its relation with the diameter of the toroid in the place of cutting, in such a way that at certain distances from the neck a doble of blade could emerge. In Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ea drawing of the workpiece with its dimensions is depicted as well as the solid model of the design component.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"4 Description of the machine tool and planning of the machining process","content":"\u003cp\u003eThe machine tool is a CNC milling machine LAGUN 650 with a rotary table Spirsin Divisores Electr\u0026oacute;nicos S.L. model 8210.2F/125 that allows the interpolation of trajectories in the B and C axes. Axis \u003cem\u003eB\u003c/em\u003e is limited from \u0026minus;\u0026thinsp;10\u003csup\u003eo\u003c/sup\u003e to 100\u003csup\u003eo\u003c/sup\u003e. The configuration is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe CNC program uses the ISO G-code for programing the coordinates X, Y and Z position of the tool with length compensation, and the angular coordinates \u003cem\u003eB\u003c/em\u003e and \u003cem\u003eC\u003c/em\u003e of the rotary table.\u003c/p\u003e\u003cp\u003eFor the fixture of the preform to the rotary table, an utensil has been designed requiring a drilling at its axis for the screw fixation as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. The surface of the toroid has been painted to make clear the surface that has not be machined in the final part.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eDefinition of trajectories to maintain a constant thickness of blades\u003c/b\u003e\u003c/p\u003e\u003cp\u003eThe trajectory of the tool is defined from the Villarceau circle over the toroid surface following a parallel curve at a constant distance of the semi-width of the blade as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e. The cutting of one face begins from the neck with a table position \u003cem\u003eB\u003c/em\u003e\u0026thinsp;=\u0026thinsp;90\u003csup\u003eo\u003c/sup\u003e as shown inf Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e and it ends in a vertical position. The mill is always in a position perpendicular to the surface and with a depth cut that varies from a maximum at the beginning and a minimum at the end. The position and orientation of all points that must be followed by the center of the mill at the tip has been calculated in a worksheet for the two faces of the blade. The inverse kinematics is made by the methodology described in [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. The code is executed as many times as the number of blades \u003cem\u003eN\u003c/em\u003e has the part after the preselection of \u003cem\u003eC\u003c/em\u003e an angle 2π/N.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eAn important issue is that a security distance is required to avoid impacts between tool and the workpiece, and to complete the machining of the surface at the beginning and at the end of the trajectory parallel to Villarceau\u0026rsquo;s circle. Figure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e represents the initial position of the tool before the machining of each face of the blade.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eEach trajectory of the tool for the machining of one face has been discretized in 40 points in which linear interpolation G1 has been applied. Intermediate blades have been machined between the complete blades that will reinforce the fluid flow during its performance.\u003c/p\u003e"},{"header":"5 Analysis of the final part","content":"\u003cp\u003eThe final part is represented in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e. The mill is a cylindrical tool of 5 mm in diameter. The surface between blades does not coincides with a toroid due to the intersection with the trajectories to generate the intermediate blades. This creates steps between the two surfaces that are more accused in the exit of the tool. This effect could be avoided through a machined trajectory between initial trajectories or by using mills with smaller diameter after the faces of the blades were machined.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"6 Conclusions","content":"\u003cp\u003eThis article has demonstrated the use of Villarceau circles as a basis for machining toroids, facilitating programming cutting paths and generating textures other than circumferential or radial ones. The machining of a centrifugal gas compressor rotor which blades follow the geometry of these circles has been machined as an example of application.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe acknowledge the Department of Mechanical, Materials and Manufacturing Engineering of the Universidad Polit\u0026eacute;cnica de Cartagena for partially funded the elaboration and presentation of the present work.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflicts of interest\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor\u0026rsquo;s Contribution\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll authors contributed to the conceptualization and design of the study. Concept and Methodology Manuel Estrems and Javier Castellote; formal analysis, Wilmer E. Cumbicus, Manuel Estrems and Javier Castellote; CNC work: Javier Castellote, Manuel Estrems and Oscar de Francisco; writing-original draft preparation, Javier Castellote and Manuel Estrems; writing-review and editing, Wilmer Cumbicus and \u0026Oacute;scar de Francisco; supervision, Manuel Estrems and Wilmer E. Cumbicus. \u0026nbsp;All authors have read and approved the final manuscript.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eVillarceau, Y. (1848). Th\u0026eacute;or\u0026egrave;me sur le tore. Nouvelles Annales de Math\u0026eacute;matiques, 7, 345-347. http://www.numdam.org/item/NAM_1848_1_7__345_1.pdf (visited 15/02/2025)\u003c/li\u003e\n\u003cli\u003eStruik, D. J. (1988). Lectures on classical differential geometry. Dover Publication. ISBN: 0486656098\u003c/li\u003e\n\u003cli\u003eDong, J., He, J., Liu, S., Wan, N., \u0026amp; Chang, Z. (2023). \u0026quot;A Multi-Scale Tool Orientation Generation Method for Freeform Surface Machining with Bull-Nose Tool.\u0026quot; Micromachines, 14(6), 1199. https://doi.org/10.3390/mi14061199 \u003c/li\u003e\n\u003cli\u003eEstrems, M., Castellote, J., Cumbicus, W. E., S\u0026aacute;nchez, H., Carrero-Blanco, J., de Francisco, \u0026Oacute;., \u0026amp; Arizmendi, M. (2019). Trajectory generation in 5-axis milling of freeform surfaces using circular arc approximation and its influence in surface roughness. Procedia Manufacturing, 41, 208-215. https://doi.org/10.1016/j.promfg.2019.07.048 \u003c/li\u003e\n\u003cli\u003eAltintas, Y. (2012). Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design. Cambridge University Press. ISBN: 978-1107001480\u003c/li\u003e\n\u003cli\u003eDong, J., He, J., Liu, S., Wan, N., \u0026amp; Chang, Z. (2023). A Multi-Scale Tool Orientation Generation Method for Freeform Surface Machining with Bull-Nose Tool. Micromachines, 14(6), 1199. https://doi.org/10.3390/mi14061199 \u003c/li\u003e\n\u003cli\u003eLangeron, J.-M., Duc, E., Lartigue, C., \u0026amp; Bourdet, P. (2004). A new format for 5-axis tool path computation, using B-spline curves. Computer-Aided Design, 36(12), 1219-1229. https://doi.org/10.1016/j.cad.2003.11.007\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"the-international-journal-of-advanced-manufacturing-technology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jamt","sideBox":"Learn more about [The International Journal of Advanced Manufacturing Technology](https://www.springer.com/journal/170)","snPcode":"170","submissionUrl":"https://submission.nature.com/new-submission/170/3","title":"The International Journal of Advanced Manufacturing Technology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Villarceau Circles, Toroid Surfaces, 5-Axis Machining, Centrifugal Impeller Compressor","lastPublishedDoi":"10.21203/rs.3.rs-7048862/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7048862/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eToroid surfaces are frequently found in mechanical components with high functional performance. In this work, a way of machining this type of surface has been developed using Villarceau circles which are the result of the intersection of a toroid with an inclined plane at a particular angle. These circles are neither the directrix nor the generatrix circles of the toroid. Its use as a cutting path has advantages such as the possibilities to get different surfaces textures orientation. As an example of application, the 5-axes machining with a cylindrical mill of a toroidal propeller that belongs to a centrifugal impeller compressor is presented. This includes the geometrical description, the planning of the machining process, the description of the machine tool and fixtures, the definition of trajectories to maintain a constant thickness of blades, and the machined part. Several defects and inconveniences will be discussed.\u003c/p\u003e","manuscriptTitle":"Villarceau Circles as a Path to Machine Toroid Surfaces in 5-axis","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-07-14 10:45:47","doi":"10.21203/rs.3.rs-7048862/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Major Revisions Needed","date":"2025-08-10T08:46:31+00:00","index":"","fulltext":""},{"type":"reviewerAgreed","content":"","date":"2025-07-11T02:40:15+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-07-10T21:46:20+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-07-09T04:41:29+00:00","index":"","fulltext":""},{"type":"submitted","content":"The International Journal of Advanced Manufacturing Technology","date":"2025-07-08T05:20:33+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"the-international-journal-of-advanced-manufacturing-technology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jamt","sideBox":"Learn more about [The International Journal of Advanced Manufacturing Technology](https://www.springer.com/journal/170)","snPcode":"170","submissionUrl":"https://submission.nature.com/new-submission/170/3","title":"The International Journal of Advanced Manufacturing Technology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"488c922e-9162-43fd-af16-e93a60d906a9","owner":[],"postedDate":"July 14th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"in-revision","subjectAreas":[],"tags":[],"updatedAt":"2025-10-06T20:36:46+00:00","versionOfRecord":[],"versionCreatedAt":"2025-07-14 10:45:47","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7048862","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7048862","identity":"rs-7048862","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}