A constraint-based dynamic geometry system

Freixas, Marc; Arinyo, Robert Joan; Soto-Riera, Antoni

doi:10.1145/1364901.1364909

Cited by 6 publications

(6 citation statements)

References 18 publications

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“…Traditional constraint solution strategies like [TW06, FF09] often require computation times in the order of at least a few seconds even for moderately complex problems and are thus not applicable to realtime interactive editing systems. Also related to our work is the Dynamic Geometry system based on geometric constraints [FASR08]. However, this system is able to handle sets of constraints with exactly one degree of freedom only.…”

Section: Related Workmentioning

confidence: 99%

“…The central problem of interactive constrained editing is, given a user modification in the form of re‐positioning a set of vertices, to adjust the positions of the remaining vertices such that all constraints are satisfied. Various solutions to handle this problem have been proposed, ranging from simple (weighted) least squares solutions [XWY*09] over ad‐hoc propagate‐and‐fix approaches [ZFCO*11, GSMCO09] to elaborate strategies from the field of Computational Geometry [JTNM06, HL01, FASR08]. However, the main challenge of any incremental editing approach is not handled satisfyingly: In order to not destroy the results of earlier (manual or automatic) editing operations, it is of crucial importance to modify the positions of as few additional vertices as possible during the automatic constraint satisfaction phase.…”

Section: Introductionmentioning

confidence: 99%

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Linear Analysis of Nonlinear Constraints for Interactive Geometric Modeling

Habbecke

Kobbelt

2012

Computer Graphics Forum

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Thanks to its flexibility and power to handle even complex geometric relations, 3D geometric modeling with nonlinear constraints is an attractive extension of traditional shape editing approaches. However, existing approaches to analyze and solve constraint systems usually fail to meet the two main challenges of an interactive 3D modeling system: For each atomic editing operation, it is crucial to adjust as few auxiliary vertices as possible in order to not destroy the user's earlier editing effort. Furthermore, the whole constraint resolution pipeline is required to run in real-time to enable a fluent, interactive workflow. To address both issues, we propose a novel constraint analysis and solution scheme based on a key observation: While the computation of actual vertex positions requires nonlinear techniques, under few simplifying assumptions the determination of the minimal set of to-be-updated vertices can be performed on a linearization of the constraint functions. Posing the constraint analysis phase as the solution of an under-determined linear system with as few non-zero elements as possible enables us to exploit an efficient strategy for the Cardinality Minimization problem known from the field of Compressed Sensing, resulting in an algorithm capable of handling hundreds of vertices and constraints in real-time. We demonstrate at the example of an image-based modeling system for architectural models that this approach performs very well in practical applications.

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Linear Analysis of Nonlinear Constraints for Interactive Geometric Modeling

Habbecke

Kobbelt

2012

Computer Graphics Forum

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“…Dinaminė geometrija -priemonė arba priemonių visuma, kuri pakeičia braižymą pieštuku, liniuote ir skriestuvu popieriaus lape į galimybę tai daryti greičiau, tiksliau ir aprėpiant visą uždavinių aibę (keičiant parametrus) kompiuterio ekrane (Freixas, Joan-Arinyo, Soto-Riera, 2008). Dinaminės geometrijos vizualizavimo galimybės, palyginti su įprastu statiniu vizualizavimu, yra platesnės: geometriniai objektai gali būti interaktyviai tempiami ir jungiami ar dalijami, statiniai objektai paverčiami interaktyviais iš karto ekrane; be to, vienas brėžinys gali vaizduoti visą geometrinių objektų klasę (Scher, 2002).…”

Section: Dinaminės Geometrijos Paradigmaunclassified

Interaktyviojo geometrijos vizualizavimo modelis naudojant dinaminės geometrijos paradigmą

Jasutė¹,

Dagienė²

2011

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Straipsnyje analizuojama interaktyviojo geometrijos vizualizavimo samprata, atskleidžiamos dinaminės geometrijos konstravimo problemos. Pateikiami interaktyviojo vizualizavimo dinaminėje geometrijoje didaktiniai principai: bendrieji, skirti visai veiklos sričiai, ir specialieji, skirti interaktyviam vaizdui. Apibūdinami interaktyviojo vizualizavimo dinaminėje geometrijoje konstravimo principai: geometriniam brėžiniui ir interaktyviam vaizdui. Straipsnyje pateikiamas interaktyviojo vizualizavimo modelis naudojant dinaminės geometrijos paradigmą. Aprašomas modelio taikymo pavyzdys vizualizuojant pagrindinės mokyklos geometrijos temą. Pateikiamas modelio pagrindimas. Nusakomi tolimesnio tyrimo tikslai.

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“…The index in the construction plan in Figure 2.4 is I = {s 1 , s 2 , s 3 , s 4 }. For an in depth study of the index and the role it plays in geometric constraint solving see [24]. A similar definition can be found in [94].…”

Section: Constructive Geometric Constraint Problems Solvingmentioning

confidence: 99%

“…We present in this section an approach to solve the tracing problem, both the simple and the continuous one, in the framework of the dynamic geometry system based on constructive [24]. In this framework, after the analysis and computation of the domain, critical values and continuous transitions are identified, leading to a map of the domain which allows the user to plan which one of the possible paths the tracing is going to take.…”

Section: An Approach To the Solution Of The Tracing Problemmentioning

confidence: 99%

Geometric constraint solving in a dynamic geometry framework.

Hidalgo García

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Geometric constraint solving is a central topic in many fields such as parametric solid modeling, computer-aided design or chemical molecular docking. A geometric constraint problem consists of a set geometric objects on which a set of constraints is defined. Solving the geometric constraint problem means finding a placement for the geometric elements with respect to each other such that the set of constraints holds. Clearly, the primary goal of geometric constraint solving is to define rigid shapes. However an interesting problem arises when we ask whether allowing parameter constraint values to change with time makes sense. The answer is in the positive. Assuming a continuous change in the variant parameters, the result of the geometric constraint solving with variant parameters would result in the generation of families of different shapes built on top of the same geometric elements but governed by a fixed set of constraints. Considering the problem where several parameters change simultaneously would be a great accomplishment. However the potential combinatorial complexity make us to consider problems with just one variant parameter. Elaborating on work from other authors, we develop a new algorithm based on a new tool we have called h-graphs that properly solves the geometric constraint solving problem with one variant parameter. We offer a complete proof for the soundness of the approach which was missing in the original work. Dynamic geometry is a computer-based technology developed to teach geometry at secondary school, which provides the users with tools to define geometric constructions along with interaction tools such as drag-and-drop. The goal of the system is to show in the user's screen how the geometry changes in real time as the user interacts with the system. It is argued that this kind of interaction fosters students interest in experimenting and checking their ideas. The most important drawback of dynamic geometry is that it is the user who must know how the geometric problem is actually solved. Based on the fact that current user-computer interaction technology basically allows the user to drag just one geometric element at a time, we have developed a new dynamic geometry approach based on two ideas: 1) the underlying problem is just a geometric constraint problem with one variant parameter, which can be different for each drag-and-drop operation, and, 2) the burden of solving the geometric problem is left to the geometric constraint solver. Two classic and interesting problems in many computational models are the reachability and the tracing problems. Reachability consists in deciding whether a certain state of the system can be reached from a given initial state following a set of allowed transformations. This problem is paramount in many fields such as robotics, path finding, path planing, Petri Nets, etc. When translated to dynamic geometry two specific problems arise: 1) when intersecting geometric elements were at least one of them has degree two or higher, the solution is not unique and, 2) for given values assigned to constraint parameters, it may well be the case that the geometric problem is not realizable. For example computing the intersection of two parallel lines. Within our geometric constraint-based dynamic geometry system we have developed an specific approach that solves both the reachability and the tracing problems by properly applying tools from dynamic systems theory. Finally we consider Henneberg graphs, Laman graphs and tree-decomposable graphs which are fundamental tools in geometric constraint solving and its applications. We study which relationships can be established between them and show the conditions under which Henneberg constructions preserve graph tree-decomposability. Then we develop an algorithm to automatically generate tree-decomposable Laman graphs of a given order using Henneberg construction steps.

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A constraint-based dynamic geometry system

Cited by 6 publications

References 18 publications

Linear Analysis of Nonlinear Constraints for Interactive Geometric Modeling

Linear Analysis of Nonlinear Constraints for Interactive Geometric Modeling

Interaktyviojo geometrijos vizualizavimo modelis naudojant dinaminės geometrijos paradigmą

Geometric constraint solving in a dynamic geometry framework.

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