scholarly journals Exploiting lattice structures in shape grammar implementations

2018 ◽  
Vol 32 (2) ◽  
pp. 147-161 ◽  
Author(s):  
Hau Hing Chau ◽  
Alison McKay ◽  
Christopher F. Earl ◽  
Amar Kumar Behera ◽  
Alan de Pennington
Keyword(s):  
Three Dimensional ◽  
Real Life ◽  
General Purpose ◽  
Three Dimensions ◽  
Shape Grammar ◽  
Shape Grammars ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Computation Process ◽  
Shape Computation

AbstractThe ability to work with ambiguity and compute new designs based on both defined and emergent shapes are unique advantages of shape grammars. Realizing these benefits in design practice requires the implementation of general purpose shape grammar interpreters that support: (a) the detection of arbitrary subshapes in arbitrary shapes and (b) the application of shape rules that use these subshapes to create new shapes. The complexity of currently available interpreters results from their combination of shape computation (for subshape detection and the application of rules) with computational geometry (for the geometric operations need to generate new shapes). This paper proposes a shape grammar implementation method for three-dimensional circular arcs represented as rational quadratic Bézier curves based on lattice theory that reduces this complexity by separating steps in a shape computation process from the geometrical operations associated with specific grammars and shapes. The method is demonstrated through application to two well-known shape grammars: Stiny's triangles grammar and Jowers and Earl's trefoil grammar. A prototype computer implementation of an interpreter kernel has been built and its application to both grammars is presented. The use of Bézier curves in three dimensions opens the possibility to extend shape grammar implementations to cover the wider range of applications that are needed before practical implementations for use in real life product design and development processes become feasible.

scholarly journals Quaternionic Bézier curves, surfaces and volume

2013 ◽  
Vol 54 ◽  
Author(s):  
Severinas Zube
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Subspace ◽  
Bezier Curves ◽  
Bézier Curves ◽  
The Real

We extended the rational Bézier construction for linear, bi-linear and threelinear map, by allowing quaternion weights. These objects are Möbius invariant and have halved degree with respect to the real parametrization. In general, these parametrizations are in four dimensional space. We analyse when a special the three-linear parametrized volume is in usual three dimensional subspace and gives three orthogonal family of Dupine cyclides.

Shape Optimisation of Curved Interconnecting Ducts

2015 ◽  
Vol 65 (4) ◽  
pp. 300 ◽  
Author(s):  
K. Srinivasan ◽  
V. Balamurugan ◽  
S. Jayanti
Keyword(s):  
Pressure Drop ◽  
Practical Interest ◽  
Three Dimensions ◽  
Shape Optimisation ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Process Plants ◽  
Angular Bisector ◽  
The One ◽  
Dynamics Simulations

<p class="AbstractText">Practical ducting layout in process plants needs to satisfy a number of on-site constraints. The search for an optimal flow path around the obstructions is a multi-parameter problem and is computationally prohibitively expensive. In this study, authors proposed a rapid and efficient methodology for the optimal linkage of arbitrarily oriented fluid flow ducts using a single-parameter quadratic/cubic Bézier curves in two/three dimensions to describe the centreline of the curved duct. A smooth interconnecting duct can then be generated by extruding the duct face along the curve. By varying the parameter either along the angular bisector or along the axes of the ducts, a family of Bézier curves is generated. Computational fluid dynamics simulations show that the relationship between pressure drop and the adjustable parameter is a unimodal curve and the optimal connecting duct is the one which has the least pressure drop while satisfying on-site constraints can be used for linking the ducts. The efficacy of the method is demonstrated by applying it to some cases of practical interest.</p><p class="AbstractText"><strong>Defence Science Journal, Vol. 65, No. 4, July 2015, pp. 300-306, DOI: http://dx.doi.org/10.14429/dsj.65.8353</strong></p><p class="AbstractText"> </p>

Designing Three-Dimensional Directional Well Trajectories Using Bézier Curves

2016 ◽  
Vol 139 (3) ◽  
Author(s):  
Jorge H. B. Sampaio
Keyword(s):  
Three Dimensional ◽  
Automated System ◽  
Hole Drilling ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Smooth Change ◽  
End Conditions ◽  
Planning Cycle ◽  
Three Space ◽  
End Set

A relatively simple, general, and very flexible method to design complex, three-dimensional hole trajectories can be obtained by using a 3D extension for Bézier curves. This approach offers superior results in terms of coding, use, and flexibility compared to other methods using double-arc, cubic functions, spline-in-tension functions, or constant curvature. The mathematics is surprisingly simple, and the method can be used to obtain trajectories for any of the four typical end conditions in terms of inclination and azimuth, namely: free-end, set-end, set-inclination/free azimuth, and free-inclination/set-azimuth. The resulting trajectories are smooth, with continuous and smooth change of curvature and toolface, better exploiting the expected delivery of modern rotary steerable deviation tools, particularly the point-the-bit and the push-the-bit systems. With the relevant parameters at any point of the trajectory (curvature and toolface angle) an automated system can steer the hole toward the defined targets in a smooth fashion. The beauty of the method is that the description of the trajectory is obtained with one single expression that handles the three space coordinates, instead of working with three separate coordinate functions. It uses a generalization of the well-known 2D Bézier curve. The concept is easy to understand, and implementation even using spreadsheets is straightforward. Besides, the conditions at both ends (coordinates and inclination/azimuth for set ends) the trajectory curve has up to two independent parameters. By playing suitably with these parameters, one can obtain a curve that favors the reduction of drag and torque during drilling, tripping, and casing running. In addition to the formulation for trajectory calculation, the paper presents the expressions to calculate the inclination, azimuth, curvature, and toolface at any point along the trajectory. Proper numerical examples illustrate the various end-conditions. The method can be used during the hole planning cycle as well as during the hole drilling for automatic and manual steerage.

Shape Grammar Approaches in Architecture

2020 ◽  
pp. 19-55
Keyword(s):  
Design Automation ◽  
Architectural Design ◽  
Three Dimensional ◽  
Automated Design ◽  
Shape Grammar ◽  
Shape Grammars ◽  
Recent Trends ◽  
Generative Algorithms ◽  
Shape Descriptions ◽  
The Relationship

This chapter reviews emerging Shape Grammar research, categorising it into three themes: design analysis and generation, automated design and generative algorithms, and algebraic Shape Grammars. The first theme consists of theoretical Shape Grammar approaches, two-dimensional architectural design, three-dimensional architectural design, urban design, and design in art and engineering. The second theme addresses four alternative perspectives to grammatical approaches based on design automation, procedural modelling, genetic algorithms, and other algorithmic generation and evaluation methods. The last theme examines research using algebraic shape descriptions and operations. The purpose of this chapter is to provide a critical summary of recent trends in Shape Grammar research and an overview of the relationship between grammatical and generative systems in architecture.

scholarly journals Quaternion rational Bézier curves

2011 ◽  
Vol 52 ◽  
Author(s):  
Severinas Zube
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Subspace ◽  
Space Curve ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Rational Bézier Curves

We extended the rational Bézier model for space curve, by allowing quaternion weights. These curves are Möbius invariant and have halved degree with respect to real Bézier curves. This simplify the analysis of curves. In general, these curves are in four dimensional space. We analyze when the quadratically parameterized quaternion curve is in usual three dimensional subspace.  

Shape grammars as design tools: an implementation of a multipurpose chair grammar

2018 ◽  
Vol 32 (2) ◽  
pp. 240-255
Author(s):  
Sara Garcia ◽  
António Menezes Leitão
Keyword(s):  
Three Dimensional ◽  
Real Life ◽  
Research Area ◽  
Design Tool ◽  
Shape Grammars ◽  
Combinatorial Explosion ◽  
Interface Elements ◽  
Practical Applications ◽  
Design Alternatives ◽  
Chair Design

AbstractThis paper presents a multipurpose chair grammar and its implementation in the design tool ChairDNA. This tool is oriented for the exploration of design alternatives in the early concept phase of the chair design process. This work addresses two shortcomings within the research area of shape grammars (SGs), namely, the lack of implementation of SGs applied to design domains, and the lack of practical applications in real-life design scenarios. To address these problems, a methodology is proposed for the implementation of a SG (more specifically, a set grammar) into a tool, comprising the translation of the grammar into user-interface elements oriented for design practitioners. By using the proposed tool, the user can add/delete chair components and edit shape parameters, while visualizing the effects on a three-dimensional digital model presented in a variety of CAD applications. Compared with other SG implementations, ChairDNA uses an approach that keeps under control the combinatorial explosion of rule applications, which simplifies the use of the tool by designers that do not have experience in SGs. The generative potential of the tool is demonstrated by generating chairs of different types, and its usability and utility in aiding the designer are evaluated by design students and design practitioners.

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