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It is not doubtful on the success of shape grammar formalism, especially the calculation of straight lines, which is implemented by multiple robust interpreters such as SGI (Krishnamurti in SGI: a shape grammar interpreter. Centre for configurational studies, design discipline, 1982), GRAPE (Grasl and Economou in Environ Planning B: Planning Des 40:905–922, 2013), SortAl GI (Dy and Stouffs in Combining geometries and descriptions a shape grammar plug-in for grasshopper, 2018), Shape Machine (Hong and Economou in Autom Constr (AiC) 146, 2023) and sort forth. Calculation of straight lines is adopted in multiple fields such architecture, industrial design, mechanical engineering and sort forth (McKay et al. in Artif Intell Eng Des Anal Manuf 26:143–159, 2012), however, calculation of curves, for example, arcs, circles, parabola, hyperbola, ellipse, elliptical arcs, piece-wise curves, NURB curves, is still a task that remains unresolved. Countable applications that tackled this task can be found in the literature, for instance, Curve-based SGI (McCormack and Cagan in Environ Planning B: Planning Des 33:523–540, 2006), QI (Jowers and Earl in Environ Planning B: Planning Des 38:616–635, 2011), and GRAPE (Grasl and Economou in Environ Planning B: Planning Des 40:905–922, 2013; Grasl and Economou in Artif Intell Eng Des Anal Manuf 32(2):208–224, 2018) But still, it is not ready to adopt curve calculation in the practical projects since curves, especially arbitrary curves, are hardly represented with point boundaries since the boundary representation of curves is not unique (Jowers and Earl in Environ Planning B: Planning Des 38:616–635, 2011). This study proposed an algorithm which is currently implemented in the Shape Machine, for maximization of circular arcs and circles in computer-aided design systems, with a calibration that can mitigate the precision errors in geometry modeler (Hong and Economou in Autom Constr (AiC) 146, 2023), as well as an algorithm for shape embedding of circular arcs and circles. By calculating circular arcs, and circles in shape grammar interpreter, great number of applications are possible. In the second section of this paper, one example is introduced to demonstrate the applications of curve calculation. In the end, a discussion of visual equivalency is given to provide an alternative perspective about the standard for shape calculation.

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From Lines to Curves: Implementation of Shape Embedding for Circular Arcs and Circles in 2D CAD Systems

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  • 发表日期：

    2025年

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  • Tzu-Chieh Kurt Hong

摘要

It is not doubtful on the success of shape grammar formalism, especially the calculation of straight lines, which is implemented by multiple robust interpreters such as SGI (Krishnamurti in SGI: a shape grammar interpreter. Centre for configurational studies, design discipline, 1982), GRAPE (Grasl and Economou in Environ Planning B: Planning Des 40:905–922, 2013), SortAl GI (Dy and Stouffs in Combining geometries and descriptions a shape grammar plug-in for grasshopper, 2018), Shape Machine (Hong and Economou in Autom Constr (AiC) 146, 2023) and sort forth. Calculation of straight lines is adopted in multiple fields such architecture, industrial design, mechanical engineering and sort forth (McKay et al. in Artif Intell Eng Des Anal Manuf 26:143–159, 2012), however, calculation of curves, for example, arcs, circles, parabola, hyperbola, ellipse, elliptical arcs, piece-wise curves, NURB curves, is still a task that remains unresolved. Countable applications that tackled this task can be found in the literature, for instance, Curve-based SGI (McCormack and Cagan in Environ Planning B: Planning Des 33:523–540, 2006), QI (Jowers and Earl in Environ Planning B: Planning Des 38:616–635, 2011), and GRAPE (Grasl and Economou in Environ Planning B: Planning Des 40:905–922, 2013; Grasl and Economou in Artif Intell Eng Des Anal Manuf 32(2):208–224, 2018) But still, it is not ready to adopt curve calculation in the practical projects since curves, especially arbitrary curves, are hardly represented with point boundaries since the boundary representation of curves is not unique (Jowers and Earl in Environ Planning B: Planning Des 38:616–635, 2011). This study proposed an algorithm which is currently implemented in the Shape Machine, for maximization of circular arcs and circles in computer-aided design systems, with a calibration that can mitigate the precision errors in geometry modeler (Hong and Economou in Autom Constr (AiC) 146, 2023), as well as an algorithm for shape embedding of circular arcs and circles. By calculating circular arcs, and circles in shape grammar interpreter, great number of applications are possible. In the second section of this paper, one example is introduced to demonstrate the applications of curve calculation. In the end, a discussion of visual equivalency is given to provide an alternative perspective about the standard for shape calculation.