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1. The Design and Manufacture of a Gear-Coupled Serial Chain to Trace the Butterfly Curve

   https://doi.org/10.1115/DETC2017-68388  

   Liu, Yang; Wang, Peter Lee-Shien; McCarthy, J. Michael (August 2017, International Design Engineering Technical Conferences and Computers and Information in Engineering Conference)

   This paper presents a design and manufacturing methodology for a mechanical system that draws trigonometric curves assembled from a series of links connected by gears. We demonstrate this technique using 11 gear-coupled links that trace a butterfly curve. The equation of a butterfly curve is converted to the relative rotations of the links of a coupled serial chain assembled so it operates with one input. We present a procedure to determine the adjustments to the gear ratios and link dimensions necessary for practical manufacture of the mechanism. The results are demonstrated by a working prototype. 

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2. An Invariant Representation of Coupler Curves Using a Variational AutoEncoder: Application to Path Synthesis of Four-Bar Mechanisms

   https://doi.org/10.1115/1.4063726  

   Nurizada, Anar; Purwar, Anurag (January 2024, Journal of Computing and Information Science in Engineering)

   Abstract This paper focuses on the representation and synthesis of coupler curves of planar mechanisms using a deep neural network. While the path synthesis of planar mechanisms is not a new problem, the effective representation of coupler curves in the context of neural networks has not been fully explored. This study compares four commonly used features or representations of four-bar coupler curves: Fourier descriptors, wavelets, point coordinates, and images. The results demonstrate that these diverse representations can be unified using a generative AI framework called variational autoencoder (VAE). This study shows that a VAE can provide a standalone representation of a coupler curve, regardless of the input representation, and that the compact latent dimensions of the VAE can be used to describe coupler curves of four-bar linkages. Additionally, a new approach that utilizes a VAE in conjunction with a fully connected neural network to generate dimensional parameters of four-bar linkage mechanisms is proposed. This research presents a novel opportunity for the automated conceptual design of mechanisms for robots and machines. 

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3. Finding straight line generators through the approximate synthesis of symmetric four-bar coupler curves

   https://doi.org/10.1016/j.mechmachtheory.2023.105310  

   Baskar, Aravind; Plecnik, Mark; Hauenstein, Jonathan D. (October 2023, Mechanism and Machine Theory)

   The approximate path synthesis of four-bar linkages with symmetric coupler curves is presented. This includes the formulation of a polynomial optimization problem, a characterization of the maximum number of critical points, a complete numerical solution by homotopy continuation, and application to the design of straight line generators. Our approach specifies a desired curve and finds several optimal four-bar linkages with a coupler trace that approximates it. The objective posed simultaneously enforces kinematic accuracy, loop closure, and leads to polynomial first order necessary conditions with a structure that remains the same for any desired trace leading to a generalized result. Ground pivot locations are set as chosen parameters, and it is shown that the objective has a maximum of 73 critical points. The theoretical work is applied to the design of straight line paths. Parameter homotopy runs are executed for 1440 different choices of ground pivots for a thorough exploration. These computations found the expected linkages, namely, Watt, Evans, Roberts, Chebyshev, and other previously unreported linkages which are organized into a 2D atlas using the UMAP algorithm. 

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4. On smooth plane models for modular curves of Shimura type

   https://doi.org/10.1007/s40993-022-00409-7  

   Anni, Samuele; Assaf, Eran; Lorenzo García, Elisa (June 2023, Research in Number Theory)

   Abstract In this paper we prove that there are finitely many modular curves that admit a smooth plane model. Moreover, if the degree of the model is greater than or equal to 19, no such curve exists. For modular curves of Shimura type we show that none can admit a smooth plane model of degree 5, 6 or 7. Further, if a modular curve of Shimura type admits a smooth plane model of degree 8 we show that it must be a twist of one of four curves. 

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5. Expressive curves

   https://doi.org/10.1090/cams/12  

   Fomin, Sergey; Shustin, Eugenii (August 2023, Communications of the American Mathematical Society)

   We initiate the study of a class of real plane algebraic curves which we callexpressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity. We prove that a plane curve  $C$ is expressive if (a) each irreducible component of  $C$ can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of $C$ in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of $C$ in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a)–(c), unless it exhibits some exotic behaviour at infinity. We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more. 

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