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Bringing More Intelligence to Dynamic Geometry by Using Symbolic Computation

Explore the use of Symbolic Computation to enhance Dynamic Geometry, solving problems and discovering geometric properties.

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Bringing More Intelligence to Dynamic Geometry by Using Symbolic Computation

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  1. Bringing More Intelligence to Dynamic Geometry by Using Symbolic Computation Francisco BotanaUniv. Vigo (Spain) http://webs.uvigo.es/fbotana

  2. Outline • DG problems: continuity, loci, proof • GDI (Intelligent Dynamic Geometry) • (partial) Solutions to DG problems • webDiscovery: breaking up the Algebra and Geometry parts • Intercommunication

  3. Discontinuity ...

  4. ... or non determinism

  5. Geometric loci(sampling approach pitfalls)

  6. Geometric loci(a posteriori conditions)

  7. Proof

  8. GDIGeometría Dinámica Inteligente

  9. GDI(textfile)

  10. GDI(geometric properties)

  11. GDI(CoCoA)

  12. GDI(discovering and/or proving)

  13. GDI(discovering) The triangle ABX: 1.- Is impossible 2.- Is isosceles 3.- Is equilateral 4.- Is a right triangle 5.- None of the above

  14. GDI(discovering)

  15. GDI(discovering)

  16. webDiscovery

  17. webDiscovery(An Euler’s formula) Midpoint(X1,A,B) Midpoint(X2,A,C) Aligned(X3,A,B) Aligned(X4,A,C) Aligned(X5,B,C) Perp(A,B,X1,Ci) Perp(A,C,X2,Ci) Perp(A,B,I,X3) Perp(A,C,I,X4) Perp(B,C,I,X5) d(I,X3)=d(I,X4) d(I,X3)=d(I,X5)

  18. webDiscovery(An Euler’s formula)

  19. webDiscovery(An Euler’s formula) Use R::=Q[hu[5..10]drc]; Elim(h..u[10],Ideal(h u[6]-1, d^2-((u[9]-u[7])^2+(u[10]-u[8])^2), r-u[8], c^2-(u[9]^2+u[10]^2), 2u[9] - 1, ... )); Ideal(1/2d^4 - d^2c^2 - 2r^2c^2 + 1/2c^4) -------------------------------

  20. Intercommunication

  21. Intercommunication {1} Point(-31,227)[hidden]; {2} Point(590,227)[hidden]; {3} Point(206,107); {4} Point(296,107); {5} Line(2,1)[black]; {6} Segment(4,3)[black]; {7} Point on object(5,0.32528180)[label('A')]; {8} Point on object(5,0.58454108)[label('B')]; {9} Circle by radius(7,6)[black]; {10} Circle by radius(8,6)[black]; {11} Intersect2(10,9)[traced,label('P')];

  22. Intercommunication nash.sip.ucm.es/CabriOM

  23. References http://webs.uvigo.es/fbotana Thank you.

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