OpenGeo: An Open Geometric Knowledge Base
Dongming Wang, Xiaoyu Chen, Wenya An, Lei Jiang, and DanSong
Beihang University, China
August 6, 2014
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20141 / 30
Motivation
Outline
1 Motivation
2 Geometric knowledge base: design methodology
3 OpenGeo: an enhanced version of GeoData
4 Conclusion and future work
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20142 / 30
Motivation
Geometric knowledge
Geometric knowledge is
rich in content: definitions, axioms, theorems, proofs,problems,solutions, and algorithms;
sophisticated in structure: from basic concepts to derivedconcepts,from simple diagrams to complicated configurations.
Problem
How to digitalize geometric knowledge and make it easilyaccessible,presentable, interoperable, and processable on advancedcomputingmachines and communication devices?
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20143 / 30
Motivation
A geometric knowledge base is a special database for storingandmanaging geometric knowledge data.
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20144 / 30
Motivation
GeoData: a geometric knowledge base
Resourcesµ
H. S. M. Coxeter and S. L. Greitzer. Geometry Revisited. TheMathematicalAssociation of America, Washington D.C., 1967
S. Chou. Mechanical Geometry Theorem Proving. Reidel, Dordrecht,1988
J. Hadamard. Lessons in Geometry: I. Plane Geometry. AmericanMathematicalSociety, Providence, 2008
GeoData currently includes
- 849 Euclidean plane geometric theorems
- 104 definitions of geometric concepts
- introductions to the historical background of somewell-knowntheorems (e.g., Simson’s theorem)
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20145 / 30
Motivation
http://geo.cc4cm.org/geodata/
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20146 / 30
http://localhost/geodata
Geometric knowledge base: design methodology
Outline
1 Motivation
2 Geometric knowledge base: design methodology
3 OpenGeo: an enhanced version of GeoData
4 Conclusion and future work
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20147 / 30
Geometric knowledge base: design methodology
Geometric knowledge base
The following aspects are needed to be studied for constructingageometric knowledge base.
Geometric knowledge representation
Meta-knowledge representation(the knowledge about geometricknowledge)
See AlsoATHLETIC PLACEMENT PROCESS FOR INTERSCHOOL ATHLETIC … · ATHLETIC PLACEMENT PROCESS . FOR . INTERSCHOOL ATHLETIC PROGRAMS . ... D. Coach’s Sport Skill Evaluation Form ... volleyball, - [PDF Document]Miami-Dade County Public Schools hiring FT-PARA I-THERAPEU(P)_HIALEAH MIDDLE(1623100) in Miami, Florida, United States | LinkedInGeometry: Chapter 7 Review: ANSWER KEY 1) A. …nyhsmath.weebly.com/uploads/4/2/2/9/42294853/geoch7review_answe… · Geometry: Chapter 7 Review: ANSWER KEY ... (think back to the - [PDF Document]Miami-Dade County Public Schools Office of Human Capital Management - [PPT Powerpoint]X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20148 / 30
Geometric knowledge base: design methodology
Represent geometric knowledge: multiple forms
Natural languageµa circle with center O andradius r
Algebraic expressionµ
(x, y)|x2 + y2 = r2 or
x = r · 1− t
2
1 + t2
y = r · 2t1 + t2
Drawing instructionµCircle[O, r]
Degeneracy conditionµr = 0
Image:
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20149 / 30
Geometric knowledge base: design methodology
Represent geometric knowledge: multiple forms (cont.)
Formalization:- Definition(intersection(l::Line,m::Line),[A::Point where and(incident(A, l),
incident(A,m))], not(parallel(l,m)))
- Theorem([A:=point(), B:=point(), C:=point(), D:=point(),incident(D,
circumcircle(triangle(A,B,C)))], [collinear(foot(D,line(A, B)),foot(D,line(A,
C)), foot(D, line(B, C)))])
Dynamic diagram:
Multimedia: video, audio
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201410 / 30
Geometric knowledge base: design methodology
Represent the meta-knowledge: encapsulationandclassification
A knowledge object is individual knowledge unit that can berecognized,differentiated, understood, and manipulated in theprocess of management.
Knowledge objects are used to encapsulate interrelatedgeometricknowledge data.
Knowledge classes are used to define the internal structureofknowledge objects.
- Definition, Axiom, Lemma, Theorem, Corollary, Conjecture,Problem,Example, Exercise, Proof, Solution, Algorithm,Introduction, Remark.
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201411 / 30
Geometric knowledge base: design methodology
Definition class
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201412 / 30
Geometric knowledge base: design methodology
Other classes
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201413 / 30
Geometric knowledge base: design methodology
Organize knowledge objects
Catalog is used to describe how knowledge objects areclustered.
Chapter: Points and Lines Connected with a Triangle
Section: Points of interest
Definition of orthocenter
Knowledge graph is used to describe how knowledge objectsarerelated.
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201414 / 30
Geometric knowledge base: design methodology
Knowledge graph: Section 1.5 from ”Geometry Revisited”
C: Points and Lines Connected with aTriangle
T1: Steiner-Lehmus theorem
P1: Steiner-Lehmus theorem’s proof
L1, L2: Lemma used in P1
E1, E2: Exercise for T1
S1, S2: Solution to the exercises
I1, R1: Introduction and remark on T1
D1: Definition of bisector
T2: Theorem: the three innerbisectors of a triangle areconcurrent
D2: Definition of incenter of a triangle
D3: Another definition of incenter ofa triangle
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201415 / 30
Geometric knowledge base: design methodology
Knowledge graph: inheritance relations between concepts
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201416 / 30
Geometric knowledge base: design methodology
Knowledge graph: inheritance relations between concepts
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201417 / 30
Geometric knowledge base: design methodology
Types of relations
Inclusion
A→include B
Inheritance
A→inherit B
Dependance
A→contextOf BA→deriveFrom BA→imply BA→hasProperty BA→decideBA→introduce BA→remarkOn BA→complicate BA→solve BA→exerciseOf B
Association
A→justify BA→applyOn BA→exampleOf BA↔associate BA↔equal B
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201418 / 30
OpenGeo: an enhanced version of GeoData
Outline
1 Motivation
2 Geometric knowledge base: design methodology
3 OpenGeo: an enhanced version of GeoData
4 Conclusion and future work
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201419 / 30
OpenGeo: an enhanced version of GeoData
OpenGeo is an enhanced version of GeoData, which is equippedwith
web-based interfaces,
new management facilities, and
made open online.
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201420 / 30
OpenGeo: an enhanced version of GeoData
Open to users
knowledge objects can be edited or deleted;
meta-information (e.g., language, format, and keyword) canbeannotated for organizing and classifying knowledge objects;
revisions of knowledge objects can be recorded;
knowledge objects can be retrieved in meta-information-basedways;
knowledge objects can be rated and commented forscreeninghigh-quality versions;
new knowledge objects can be created and added to OpenGeo.
*Creative Commons Attribution-ShareAlike license is adopted asits main content license.
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201421 / 30
OpenGeo: an enhanced version of GeoData
Implementation techniques: meta-knowledgerepresentation
We adopt ontology (OWL) to formally specify geometricknowledgeobjects and relations among them.
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201422 / 30
OpenGeo: an enhanced version of GeoData
Implementation techniques: meta-knowledgerepresentation
knowledge object7→ ontology instanceknowledge class7→ ontologyclass
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201423 / 30
OpenGeo: an enhanced version of GeoData
Implementation techniques: meta-knowledgerepresentation
knowledge class structure7→ ontology attributeknowledge graph7→ontology relation
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201424 / 30
OpenGeo: an enhanced version of GeoData
Implementation techniques: database schema
Database schema (relational data tables) can be automaticallygeneratedfrom the ontologies.
−→
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201425 / 30
OpenGeo: an enhanced version of GeoData
Implementation techniques: user interface
The LAMP (Linux Apache MySQL PHP/Perl/Python) framework
MathEdit: editing formatted formulas in a WISIWIG style
Sketchometry: drawing and exporting dynamic diagrams
GeoGebra: constructing and rendering dynamic diagrams
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201426 / 30
http://localhost:8000
Conclusion and future work
Outline
1 Motivation
2 Geometric knowledge base: design methodology
3 OpenGeo: an enhanced version of GeoData
4 Conclusion and future work
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201427 / 30
Conclusion and future work
Conclusion
OpenGeo is created for the purpose of research and education,and mayserve as
a public resource for users to test, for instance, geometrictheoremprovers and problem solvers; and
an infrastructure for developing new educational applications(e.g.,generation of textbooks and courses) in online learningenvironments.
We are
formalizing geometric theorems in the OpenGeo collection and
developing semantic querying tools based on images ofdiagrams.
We expect to complete these tasks and release a preliminaryversion ofOpenGeo in early 2015.
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201428 / 30
Conclusion and future work
Automated knowledge acquisition
Input Output
If the points A,B, and C are arbitrary, the point D is onthecircumcircle of the triangle ABC, F is theperpendicular foot of theline AC to the line DF , G isthe perpendicular foot of the line BCto the line DG,and E is the perpendicular foot of the line BA totheline DE, then the point F is on the line EG.
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201429 / 30
Conclusion and future work
Thanks
X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201430 / 30
MotivationGeometric knowledge base: design methodologyOpenGeo:an enhanced version of GeoDataConclusion and future work
OpenGeo: An Open Geometric Knowledge Basesites.nlsde.buaa.edu.cn/~chenxiaoyu/slides/icms2014.pdfOpenGeo: An Open Geometric Knowledge Base Dongming Wang, Xiaoyu Chen, Wenya An, Lei - [PDF Document] (2024)
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