A non-rigid cluster rewriting approach to solve systems of 3D geometric constraints

Meiden, Hilderick A. van der; Bronsvoort, Willem F.

doi:10.1016/j.cad.2009.03.003

Cited by 12 publications

(6 citation statements)

References 24 publications

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“…These techniques are a subset of graph-based synthetic methods [12], [34], [37] in which the constraint graph is decomposed and recombined to extract construction steps that are then solved by algebraic methods [3], [13], [39]. Notable improvements include Sitharam's frontier algorithm [21], [22], which generalizes the graph decomposition, treatment for over-and under-constrained problems [23], [29], [32], [40], [42], and parameter ranges for dimensional constraints [26], [30], [31], [38], [41]. Despite the success of these methods for CAD applications, these constructive approaches generally do not translate to complex geometric configurations in theorem proving.…”

Section: B Geometric Constraint Problemmentioning

confidence: 99%

Automatically Building Diagrams for Olympiad Geometry Problems

Krueger¹,

Han²,

Selsam³

2020

Preprint

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We present a method for automatically building diagrams for olympiad-level geometry problems and implement our approach in a new open-source software tool, the Geometry Model Builder (GMB). Central to our method is a new domain-specific language, the Geometry Model-Building Language (GMBL), for specifying geometry problems along with additional metadata useful for building diagrams. A GMBL program specifies (1) how to parameterize geometric objects (or sets of geometric objects) and initialize these parameterized quantities, (2) which quantities to compute directly from other quantities, and (3) additional constraints to accumulate into a (differentiable) loss function. A GMBL program induces a (usually) tractable numerical optimization problem whose solutions correspond to diagrams of the original problem statement, and that we can solve reliably using gradient descent. Of the 39 geometry problems since 2000 appearing in the International Mathematical Olympiad, 36 can be expressed in our logic and our system can produce diagrams for 94% of them on average. To the best of our knowledge, our method is the first in automated geometry diagram construction to generate models for such complex problems.

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Section: B Geometric Constraint Problemmentioning

confidence: 99%

Automatically Building Diagrams for Olympiad Geometry Problems

Krueger¹,

Han²,

Selsam³

2020

Preprint

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“…3.1. We randomly project the node positions into 2D, and then use the geometric constraint solver proposed by van der Meiden and Bronsvoort [51] (Fig. 4.3) to resolve any geometric constraints.…”

Section: Geometric Reasoning and Constraint Satisfactionmentioning

confidence: 99%

“…(2) Traverse the graph, adding candidate angle constraints between a node and two of its neighbors (for all permutations of neighbors). (3) Evaluate candidates using either a solver [51] or the loop mobility criterion (Sec. 4.4) [2] to determine if the added constraint will determine the system.…”

Section: Geometric Reasoning and Constraint Satisfactionmentioning

confidence: 99%

“…If the input graph is overconstrained, or if the user specifies infeasible geometry, the algorithm raises an error to the user, who can modify the original input file. In cases where the model is correctly constrained, but the constraint solver [51] cannot find a solution, we restart with randomized positions to explore different local optima and then terminate if we are still unable to resolve the constraints.…”

Section: Geometric Reasoning and Constraint Satisfactionmentioning

confidence: 99%

“…Fig. 4 How we progressively resolve ambiguities in the user input (1): (2) Create an undirected mechanism graph, (3) solve for the ideal 2D positions using a geometric constraint solver [51], (4) substitute in user-selected components from the library, if specified, or appropriate defaults given the AM test (Sec. 3.3), (5) create an interface graph from the selected components and optimize build volume using MILP [53], and (6) locally modify the geometry at the selected interfaces to ensure movement when printed Fig.…”

Section: Detecting Manufacturing Constraintsmentioning

confidence: 99%

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The MechProcessor: Helping Novices Design Printable Mechanisms Across Different Printers

Fuge

Carmean

Cornelius

et al. 2015

Journal of Mechanical Design

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Additive manufacturing (AM), or 3D-printing, sits at the heart of the Maker Movement—the growing desire for wider-ranges of people to design physical objects. However, most users that wish to design functional moving devices face a prohibitive barrier-to-entry: they need fluency in a computer-aided design (CAD) package. This limits most people to being merely consumers, rather than designers or makers. To solve this problem, we combine advances in mechanism synthesis, computer languages, and design for AM to create a computational framework, the MechProcessor, which allows novices to produce 3D-printable, moving mechanisms of varying complexity using simple and extendable interfaces. The paper describes how we use hierarchical cascading configuration languages, breadth-first search, and mixed-integer linear programming (MILP) for mechanism synthesis, along with a nested, printable test-case to detect and resolve the AM constraints needed to ensure the devices can be 3D printed. We provide physical case studies and an open-source library of code and mechanisms that enable others to easily extend the MechProcessor framework. This encourages new research, commercial, and educational directions, including new types of customized printable robotics, business models for customer-driven design, and STEM education initiatives that involve nontechnical audiences in mechanical design. By promoting novice interaction in complex design and fabrication of movable components, we can move society closer to the true promise of the Maker Movement: turning consumers into designers.

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