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REVIEW 4 major objections 5 minor 54 references

Exploration of offsets of Cayley ovals and their singularities

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Smooth Cayley ovals still produce cusped, self-intersecting offsets whose topology changes with distance.

desk verdict A genuine first exploration of Cayley oval offsets, but the cusp evidence is computed on the wrong curve and the analysis only covers the external loop. read the letter →

arxiv 2505.21028 v1 pith:WITKAF7L submitted 2025-05-27 math.AG

Thierry Dana-Picard , Daniel Tsirkin This is my paper
classification math.AG MSC 97G4014H5014-04
keywords Cayleyovalsoffsetsenvelopessingularpointscrunodescuspscomputeralgebradynamicgeometry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper explores the offsets of Cayley ovals, which are smooth degree-8 plane curves defined by a bifocal harmonic-mean condition. It claims that even though the progenitor oval is nonsingular, its offsets develop cusps and self-intersection points (crunodes), and that changing the offset distance leads to substantial changes in shape and topology. The authors use a dynamic geometry system for exploration and a computer algebra system for symbolic computation, producing conjectures and classifications of the singular points that appear on the offsets.

What carries the argument

The central machinery is the parametric representation of the Cayley oval given by $$x = \frac{$b^{2}$}{a}\cdot\frac{\cos 2t}{\$sin^{4}$ 2t}, \quad y = \pm \sqrt{\frac{$b^{2}$}{4\$cos^{4}$ t}-(x-a)^2},$$ which determines the external component of the oval. Offsets are constructed by adding a multiple $d$ of a unit normal vector to this parametrization, and singular points are found by solving either the derivative conditions $dx/dt = dy/dt = 0$ or the curvature condition $k = -1/d$ for cusps, and by solving the system of equations that identifies two different parameter values giving the same offset point for crunodes.

What would settle it

One could derive a parametrization for the internal loops of a Cayley oval, compute the offsets of those internal loops, and check whether new cusps or crunodes appear that are not captured by the external-loop parametrization used in the paper.

Watch

Extended reading notes

Core claim

For Cayley ovals, the paper's central claim is that offsets at various distances exhibit cusps and crunodes whose numbers and positions depend on the offset distance, with topological transitions as the distance grows. The authors classify these singularities according to the classical taxonomy of crunodes (self-intersections), acnodes (isolated points), and cusps. They find that the curve's own smoothness does not prevent its offsets from becoming significantly more complicated, and that a one-to-one correspondence between singularities of the progenitor and of the offset does not exist.

Load-bearing premise

The singular-point analysis relies on a parametrization that determines only the external component of the Cayley oval, so the catalog of cusps and crunodes may be incomplete if the internal loops were included.

Editorial extensions

If this is right

  • If the paper's classification is correct, engineers and designers who use offsets of smooth curves should expect cusps and self-intersections to appear even when the progenitor is smooth.
  • The topology of an offset of a Cayley oval is not determined by the topology of the progenitor; changes in offset distance can switch between oval-shaped, looped, astroid-like, and sand-clock-like components.
  • The lack of a one-to-one correspondence between progenitor singularities and offset singularities implies that smoothness of a curve is not a sufficient condition for smoothness of its offsets.
  • For Cayley ovals with eccentricity $e$ below, equal to, or above 1, the number and arrangement of crunodes and cusps depends on both the shape parameter $e$ and the offset distance $d$.
  • The polynomial equations obtained for the Cayley ovals and their envelopes are irreducible, which explains why automated factoring tools cannot separate the irrelevant internal loops from the true geometric locus.
  • The paper's computational methods—using both a dynamic geometry system and a computer algebra system—provide a workflow for exploring other families of curves whose equations are too complex for hand calculation.
  • Because the parametrization used in the Maple sessions determines only the external loop of the Cayley oval, the catalogue of singularities reported might change if the internal loops were included; this is an open limitation.
  • A natural testable extension would be to derive a parametrization for the internal loops and repeat the singular-point analysis, or to seek a closed-form condition on $d$ that predicts exactly when each topological transition occurs.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper studies offsets (parallel curves) of Cayley ovals, the degree-8 plane curves defined by the bifocal harmonic-mean condition. Using GeoGebra for dynamic exploration and Maple for symbolic and numeric computation, the authors plot envelopes of centered circles and offsets for four eccentricity regimes and investigate singularities of the offsets. Their central claim is that, although Cayley ovals are nonsingular (except for the lemniscatic case), their offsets develop cusps and self-intersections, and that the number and shape of these singularities change as the offset distance d varies. The statements are formulated as conjectures supported by computational experiments rather than as proved theorems.

Significance. If the computational conclusions are correct, the paper offers interesting evidence that offsets of a smooth degree-8 curve can have a much richer singularity structure than the progenitor, complementing existing studies of offsets of classical and singular curves. The manuscript has concrete strengths: it provides the full Maple code, uses the dynamic geometry system to cross-check the bifocal definition against superfluous algebraic components, and is unusually explicit about its own limitations (gaps in plots, parameter-interval sensitivity, restriction to the external loop). However, its significance as a mathematical contribution is currently limited by the absence of exact proofs and, more seriously, by defects in the two cusp computations, which are load-bearing for the abstract's main assertion.

major comments (4)
  1. [§4.2] The cusp computation solves only `solve(dxi = 0, t)` and `solve(dxo = 0, t)` for the derivative of the x-coordinate of the offset parametrization, and never imposes dy/dt = 0. Consequently, the red points in Figure 13 may be ordinary points with vertical tangents rather than cusps. A cusp requires simultaneous vanishing (or undefinedness) of both derivatives, or a reparametrization-invariant criterion. Please solve the system {dx/dt = 0, dy/dt = 0}, or use the curvature blow-up criterion on the correct parametrization, and revise the figure and the associated claims accordingly.
  2. [§4.3] The curvature-based cusp computation is performed on the curve `x := cos(t); y := sin(t)^3`, which is not the Cayley oval parametrization (3) nor any Cayley oval. Figure 14 therefore displays cusps of offsets of a different curve, and the claim that Cayley oval offsets have cusps is not supported by this computation. Please rerun the curvature computation with the parametrization (3) over an appropriate domain, or remove this figure and the cusp part of the conclusions.
  3. [§4.1] The authors explicitly state that the parametrization (3) 'determines the external component of the Cayley oval' and 'does not determine the internal components', so the singular-point analysis in Section 4 treats only offsets of the external loop. The abstract, however, makes assertions about offsets of Cayley ovals without this qualification. Please either extend the parametrization to cover the internal loops or qualify the paper's claims throughout as being about offsets of the external loop only.
  4. [§4.4] The crunode search depends on user-supplied solve intervals, and the paper concedes that 'irrelevant points may be obtained for certain intervals' and that 'some points may be missed for a wrong choice of the interval' (Figure 16). Without a completeness argument, the catalogue of crunodes and hence the claimed topological transitions as d varies are not established. Please use an interval-independent method, such as symbolic elimination or certified root isolation, or present the crunode findings explicitly as provisional.
minor comments (5)
  1. [§4.3] The sentence about 'substituting −y instead of −y' is self-contradictory; it should presumably read 'instead of y'.
  2. [§4.4] Equation (6) is typeset incorrectly: the dot symbols and missing square-root signs make the system ambiguous; please rewrite it in the same notation as the Maple code.
  3. [§2] There are several typographical errors, e.g., 'peformed' for 'performed' and 'ahem been' near the end of Section 5.
  4. [References] References [25] and [28] appear to be the same work; please merge or distinguish them.
  5. [Figures 12 and 13] The plots show gaps and irrelevant horizontal segments; the authors acknowledge this, but a note explaining which arcs are spurious would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the offset and singularity computations derive from explicit definitions and standard formulas, with no fitted parameter renamed as a prediction.

full rationale

The paper's central claims concern the shapes and singularities of offsets of Cayley ovals. The offsets are defined directly in Definition 1.2 as geometric loci, and the singular-point analysis in Section 4 is based on the explicit parametric offset curves, with derivative conditions and the standard curvature formula (5). No parameter is fitted to data and then reused as a 'prediction'; the offset distance d is an independent input that is varied to observe topological changes. The parametrization (3) is taken from an external source, mathcurve.com, and its restriction to the external component is explicitly acknowledged in Section 4.1; this is a completeness limitation, not a circular dependence. The self-citations, e.g., [25] for the curvature method, are not load-bearing as unverified authorities: the curvature formula is printed in the paper and is a standard result, so the derivation does not reduce to the self-citation. The 'wrong curve' substitution in the Maple code of Section 4.3 (x=cos(t), y=sin(t)^3) is a correctness concern about the evidence for cusps, not a circularity, because the conclusion is not made equivalent to an input by construction. Overall, no step in the derivation chain is forced by definition, by fitted parameters, or by an unverified self-citation chain.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

Central claim rests on standard definitions of envelopes and offsets, a parametrization of the Cayley oval from an external web resource, a standard curvature condition for offset cusps, and a visual classification of shapes into four regimes. One manual choice is treated as a free parameter: the fsolve intervals used to locate crunodes, because the paper shows that different intervals give different point sets. No invented entities are introduced.

free parameters (1)
  • Solve ranges for crunode equations = varies; e.g. [4π,10π] for d=1, a=b=1
    Manual interval choices in fsolve determine which self-intersections are found; the paper shows wrong ranges produce missing or spurious points.
assumptions (5)
  • standard math Envelope of a 1-parameter family is the set solving F=0 and ∂F/∂t=0 (Definition 1.1).
    Standard analytical definition adopted from Bruce-Giblin and Berger; used to compute envelope polynomials in Section 1.2.
  • standard math Offset of a curve at distance d is the locus of points at signed distance d along the unit normal (Definition 1.2).
    Standard definition; the basis for all offset parametrizations and locus computations.
  • domain assumption Parametrization (3) describes the relevant component of a Cayley oval.
    Taken from mathcurve.com without proof; the authors acknowledge it covers only the external loop, so computations in Section 4 are incomplete for internal loops.
  • standard math Cusps of an offset occur where the progenitor curvature k satisfies k = -1/d, with k from formula (5).
    Standard offset singularity condition, cited from the authors' earlier Cassini oval paper; used in Section 4.3.
  • domain assumption Cayley ovals split into four topological regimes according to e=b/a.
    Based on dynamic experiments and plots in Section 2; no algebraic proof is given, but the paper's whole exploration is organized around this classification.

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Pith. "Pith review of Exploration of offsets of Cayley ovals and their singularities." pith.science (2026). https://pith.science/paper/WITKAF7L

@misc{pith2026250521028,
  author       = {Pith},
  title        = {Pith review of: Exploration of offsets of Cayley ovals and their singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WITKAF7L}},
  note         = {Machine review of arXiv:2505.21028}
}
read the original abstract

We explore offsets of Cayley ovals, by networking with different kinds of software. Using their specific abilities, algebraic, geometric, dynamic, we conjecture interesting properties of the offsets. For a given progenitor (the given plane curve whose offsets are studied), changes in the offset distance induce great changes in the shape and the topology of the offset. Such a study has been performed in the past for classical curves, and recently for non classical ones.Here we relate to Cayley ovals; despite them being non singular, their offsets have intriguing properties, cusps, and self-intersections. We begin with a short study of envelopes of families of circles with constant radius centered on the oval (these constructs are often studied together with offsets, but they are different objects). Then we study the offsets, which are defined as geometric loci. Both approaches are supported by the automated methods provided by the software.

Figures

Figures reproduced from arXiv: 2505.21028 by the authors.

Figure 1
Figure 1. Envelopes of two families of circles centred in the same ellipse [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Offsets at distance 1 of two different ellipses [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Comparison of envelope and offset of a singular curve [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Cayley ovals, obtained with the symbolic non polynomial equation. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Cayley ovals, obtained with GD’s LocusEquation [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Cayley ovals with superfluous components. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: The offset drawing technique. WLOG, we experiment with Cayley ovals, whose foci are on the x−axis, symmetric about the y-axis, and at a distance of 4 from each other. We started with small offset distance and increased it afterwards. In the following figures, we show s…
Figure 8
Figure 8. Figure 8: shows a partial plot of the offset. Each loop provides two parallel loops [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: Offsets of a Cayley oval with e = 1 at various distances In each case, note the crunodes (i.e. the points of self-intersection). We study them in Section 4. 3.0.3 1 < e < √ 3. For small offset distances we get curves looking ”really parallel” (Figure 10a), but for some…
Figure 10
Figure 10. Figure 10: Offsets of a Cayley oval with 1 < e < √ 3 at various distances 6An ongoing work with GeoGebra-Discovery developer Z. Kov´acs (Linz, Austria) is aimed at improving the plots and eliminating the gaps in the plots 10 [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: Offsets of a Cayley oval with e > √ 3 at various distances 4 Singular points The singular points are dispatched into three categories (according to the taxon￾omy fixed by[46]). Crunodes are points of self-intersection, Acnodes are isolated singular points and Cusps ar…
Figure 12
Figure 12. Figure 12: Examples of offsets enhancing singular points [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]
Figure 13
Figure 13. Figure 13: Examples of offsets at various distances enhancing singular points [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]
Figure 14
Figure 14. Figure 14: Cusps which have been found according to the curvature [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]
Figure 15
Figure 15. Figure 15: Examples of offsets at various distances enhancing crunodes [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]
Figure 16
Figure 16. Figure 16: Missing or unwanted points for wrong ranges when d=5 [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]

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Works this paper leans on

54 extracted references · 54 canonical work pages

  1. [1]

    Botana, A

    Ab´ anades, M.A. Botana, A. Montes and T. Recio (2014). An algebraic tax- onomy for locus computation in dynamic geometry , Computational Aided Design 56, 22–33

    work page 2014

  2. [2]

    Alcazar, J.R

    J.G. Alcazar, J.R. Sendra (2017). Local shape of offsets to algebraic curves , Journal of Symbolic Computation 42, 338-351

    work page 2017

  3. [3]

    Bagno, T

    E. Bagno, T. Dana-Picard and S. Reches (2024): ChatGPT in Linear Al- gebra: Strides Forward, Steps to Go , Open Educational Studies 6 (1), pp. 20240031. DOI:https://doi.org/10.1515/edu-2024-0031

    work page doi:10.1515/edu-2024-0031 2024

  4. [4]

    Berger (1994)

    M. Berger (1994). Geometry, Springer Verlag

    work page 1994

  5. [5]

    Blaˇ zek, P

    J. Blaˇ zek, P. Pech (2019).Locus Computation in Dynamic Geometry Envi- ronment, Mathematics in Computer Science 13 (1-2), 31-40

    work page 2019

  6. [6]

    Botana, M.A

    F. Botana, M.A. Ab´ anades (2014). Automatic deduction in (dynamic) ge- ometry: Loci computation , Computational Geometry 47 (1), 75-89

    work page 2014

  7. [7]

    Botana, T

    F. Botana, T. Recio (2013). A prop´ osito de la envolvente de una familia de elipses, Boletin de la sociedad Puig Adam 95, 15–30

    work page 2013

  8. [8]

    Botana, T

    F. Botana, T. Recio , M/P/ V´ elez (2024).On Using GeoGebra and ChatGPT for Geometric Discovery . Computers 13, 187. DOI: https://doi.org/10. 3390/computers13080187

    work page 2024

Show all 54 references
  1. [9]

    Botana, J.L

    F. Botana, J.L. Valcarce (2004). Automatic determination of envelopes and other derived curves within a graphic environment , Mathematics and Com- puters in Simulation 67, 3–13. 21

    2004

  2. [10]

    Botana, T

    F. Botana, T. Recio (2017). Computing envelopes in dynamic geometry environments, AMAI (Annals of Mathematics and Artificial Intelligence) 80 (1), 3–20

    2017

  3. [11]

    Botana, T

    F. Botana, T. Recio (2019). A proposal for the automatic computation of envelopes of families of plane curves , Journal of Systems Science and Com- plexity 32 (1), 150–157

    2019

  4. [12]

    Bruce, P.J

    J.W. Bruce, P.J. Giblin (2012). Curves and Singularities , Cam- bridge University Press (1992). Online https://doi.org/10.1017/ CBO9781139172615

    2012

  5. [13]

    D. Cox, J. Little, D. O’Shea (1992). I deals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics, Springer

    1992

  6. [14]

    Dana-Picard (2020)

    T. Dana-Picard (2020). Safety zone in an entertainment park: Envelopes, offsets and a new construction of a Maltese Cross , In: Wei-Chi Yang (Ed.), The Proceedings of Asian Technology Conference in Mathematics ATCM 2020. Online https://atcm.mathandtech.org/EP2020/invited/ 21794.pdf

    2020

  7. [15]

    Dana-Picard (2020)

    T. Dana-Picard (2020). Automated study of isoptic curves of an astroid , Journal of Symbolic Computation 97, 56–68

    2020

  8. [16]

    Dana-Picard (2021)

    T. Dana-Picard (2021). Envelopes and offsets of two algebraic plane curves: exploration of their similarities and differences , Mathematics in Computer Science 15, 757–774, Springer. DOI: 10.1007/s11786-021-00504-5

    2021 doi

  9. [17]

    Dana-Picard (2025)

    T. Dana-Picard (2025). Dynamic constructions of hyperbolisms of plane curves, Electronic Journal of Technology in Mathematics 19(1). Online: https://ejmt.mathandtech.org/Contents/eJMT_v19n1p4.pdf

    2025

  10. [18]

    Dana-Picard (2025)

    T. Dana-Picard (2025). Pedal curves of conics: an automated exploration of some cubics, sextics, octics and more . arXiv: http://arxiv.org/abs/ 2503.15135

    2025 arXiv

  11. [19]

    Dana-Picard and Z

    T. Dana-Picard and Z. Kov´ acs (2024). Estrella Solitaria in offset curves , ACA 29 (Applications of Computer Algebra), Havana, Cuba. DOI:https:// doi.org/10.13140/RG.2.2.36667.84006. Video: https://www.youtube. com/watch?v=3WmjsrBfy-0

    2024

  12. [20]

    Dana-Picard, G

    T. Dana-Picard, G. Mann, N. Zehavi (2011). From conic intersections to toric intersections: the case of the isoptic curves of an ellipse , The Montana Mathematical Enthusiast 9 (1), 59–76

    2011

  13. [21]

    Dana-Picard, D

    T. Dana-Picard, D. Tsirkin (2025). Envelopes of Circles Centered on a Kiss Curve, to appear in Maple Transaction (2025). 22

    2025

  14. [22]

    Dana-Picard, N

    T. Dana-Picard, N. Zehavi and G. Mann (2014). Bisoptic curves of hy- perbolas, International Journal of Mathematical Education in Science and Technology 45 (5), 762–781

    2014

  15. [23]

    Dana-Picard, Z

    T. Dana-Picard, Z. Kov´ acs (2018). Automated determination of isoptics with dynamic geometry , in Intelligent Computer Mathematics (F. Rabe, W. Farmer, G. Passmore, A. Youssef , Eds.), Lecture Notes in Artificial Intelli- gence (a subseries of Lecture Notes in Computer Scienc...

    2018

  16. [24]

    Dana-Picard, Z

    T. Dana-Picard, Z. Kov´ acs (2021). Experimental study of isoptics of a plane curve using dynamical coloring , in (P. Richard, M.P. Velez and S. van Vaerenbergh, edts) Mathematics Education in the Age of Artificial Intelli- gence, Mathematics Education in the Digital Era Serie...

    2021

  17. [25]

    Dana-Picard, Z

    T. Dana-Picard, Z. Kov´ acs (2022).Offsets of Cassini ovals , The Electronic Journal of Mathematics and Technology (eJMT) 16(1), 25-39

    2022

  18. [26]

    Dana-Picard, W

    T. Dana-Picard, W. Mozgawa (2020). Automated exploration of inner isop- tics of an ellipse, Journal of Geometry 111, 33. https://doi.org/10.1007/ s00022-020-00546-3

    2020

  19. [27]

    Dana-Picard, Z

    T. Dana-Picard, Z. Kov´ acs (2021). Networking of technologies: a dialog between CAS and DGS , The electronic Journal of Mathematics and Tech- nology 15 (1), 43–59

    2021

  20. [28]

    Dana-Picard, Z

    T. Dana-Picard, Z. Kov´ acs (2022).Offsets of Cassini ovals , The Electronic Journal of Mathematics and Technology (eJMT) 16 (1), 25-39. Avail- able https://php.radford.edu/~ejmt/deliveryBoy.php?paper=eJMT_ v16n1p

    2022

  21. [29]

    Dana-Picard, T

    T. Dana-Picard, T. Recio (2023). Dynamic construction of a family of octic curves as geometric loci , AIMS Mathematics 8 (8), 19461-19476

    2023

  22. [30]

    Dana-Picard, T

    T. Dana-Picard, T. Recio (2023). Second Thales Theorem: a basic but involved example on the need for DGS and CAS cooperation for geometric locus computation, Bolet ´ ın de la Sociedad Puig Adam 116, 44-59

    2023

  23. [31]

    Dana-Picard, N

    T. Dana-Picard, N. Zehavi (2016). Revival of a classical topic in Differen- tial Geometry: the exploration of envelopes in a computerized environment , International Journal of Mathematical Education in Science and Technology 47 (6), 938–959

    2016

  24. [32]

    Emprin, P.R

    F. Emprin, P.R. Richard (2024). Intelligence Artificielle et Didac- tique des Math´ ematiques : Etat des Lieux et Questionnements , An- nales de Didactique et de Sciences Cognitives 28, 131–181, IREM de Strasbourg. https://mathinfo.unistra.fr/websites/math-info/irem/ Publicati...

    2024

  25. [33]

    Ferr´ eol (2017)

    R. Ferr´ eol (2017). Cassinian ovals. Online https://mathcurve.com/ courbes2d.gb/cassini/cassini.shtml

    2017

  26. [34]

    Jareˇ s and P

    J. Jareˇ s and P. Pech (2025). Exploring loci of points by DGS and CAS in teaching geometry, The Electronic Journal of Mathematics and Technology 7 (2). Online: https://ejmt.mathandtech.org/Contents/eJMT_v7n2a4. pdf

    2025

  27. [35]

    Kock (2007)

    A. Kock (2007). Envelopes – notion and definiteness , Beitr¨ age zur Algebra und Geometrie (Contributions to Algebra and Geometry) 48, 345–350

    2007

  28. [36]

    Kov´ acs (2019)

    Z. Kov´ acs (2019). Achievements and Challenges in Automatic Locus and Envelope Animations in Dynamic Geometry, Mathematics in Computer Sci- ence 13, 131–141

    2019

  29. [37]

    Kov´ acs, B

    Z. Kov´ acs, B. Parisse (2015).Giac and GeoGebra-improved Gr¨ obner basis computations, In: Computer Algebra and Polynomials, Lecture Notes in Computer Science 8942, 126–138

    2015

  30. [38]

    Lawrence (2014)

    J.D. Lawrence (2014). A Catalogue of Special Plane Curves, 2/e, Dover

    2014

  31. [39]

    Leibnitz (1692)

    G.W. Leibnitz (1692). Generalia de Natura Linearum, Anguloque Contac- tus et Osculi Provocationibis Aliisque Cognatis et Eorum Usibus Nonnullis , Acta Eruditorum

  32. [40]

    Mazotti (2017)

    A.A. Mazotti (2017). All Sides to an Oval: Properties, Parameters, and Borromini’s Mysterious Construction , Cham: Springer Nature

    2017

  33. [41]

    Montes (2018)

    A. Montes (2018). The Gr¨ obner Cover, Algorithms and Computations in Mathematics 27, Springer

    2018

  34. [42]

    Pottmann, S

    H. Pottmann, S. Leopoldseder (2002). Geometries for CAGD , Handbook of Computer Aided Geometric Design

    2002

  35. [43]

    Recio, M.P

    T. Recio, M.P. V´ elez (2021). Towards an Ecosystem for Computer- Supported Geometric Reasoning , International Journal of Technology in Mathematics Education

    2021

  36. [44]

    Roanes-Lozano, E

    E. Roanes-Lozano, E. Roanes-Mac ´ ıas, M. Vilar-Mena, M. (2003).A Bridge Between Dynamic Geometry and Computer Algebra , Mathematical and Computer Modelling 37, 1005–1028

    2003

  37. [45]

    Roanes-Lozano (2002)

    E. Roanes-Lozano (2002). Boosting the Geometrical Possibilities of Dy- namic Geometry Systems and Computer Algebra Systems through Cooper- ation, In: M. Borovcnik, H. Kautschitsch (Eds.), Technology in Mathematics Teaching. Proceedings of ICTMT-5, Schriftenreihe Didaktik der M...

    2002

  38. [46]

    ¨ obv & hpt, Vienna, 335–348. 24

  39. [47]

    Salmon (1879)

    G. Salmon (1879). A treatise on the higher plane curves: intended as a sequel to A treatise on conic sections . Dublin: Hodges, Foster & Figgis,

  40. [48]

    San Segundo, J.R

    F. San Segundo, J.R. Sendra (2009). Partial degree formulae for plane offset curves, Journal of Symbolic Computation 44, 635–654

    2009

  41. [49]

    San Segundo, J.R

    F. San Segundo, J.R. Sendra (2005). Degree formulae for offset curves , Journal of Pure and Applied Algebra 195, 301–335

    2005

  42. [50]

    Thrainer (2020)

    S. Thrainer (2020). Drawing algebraic curves with LEGO-linkages. Diploma thesis, Johannes Kepler University, Linz (2020)

    2020

  43. [51]

    Sendra, F

    J.R. Sendra, F. Winkler, S. P´ erez-D ´ ıaz (2008).Rational Algebraic Curves: A Computer Algebra Approach , Springer

    2008

  44. [52]

    R. Yates. (1974). Curves and Their Properties , NCTM. 25

    1974

  45. [53]

    Zeitoun, T

    D.G. Zeitoun, T. Dana-Picard (2022). Delineation of the Zone of In- fluence of Pumping Wells using CAS and DGS , Electronic Proceedings of the Asian Conference on Technology in Mathematics ACTM 2021, Mathematics and Technology. https://atcm.mathandtech.org/EP2021/ regular/21918.pdf

    2022

  46. [1873]

    org/0/items/117724690/117724690.pdf

    Retrieved February 4th, 2025 from https://dn790005.ca.archive. org/0/items/117724690/117724690.pdf

    2025

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.