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Geometry in the 21 st Century Dynamic and Symbolic

Geometry in the 21 st Century Dynamic and Symbolic. Douglas B. Meade Department of Mathematics University of South Carolina E-mail: meade@math.sc.edu Homepage: www.math.sc.edu/~meade/. Outline. Euclidean Geometry Conic Sections Shrinking Circles. Euclidean Geometry.

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Geometry in the 21 st Century Dynamic and Symbolic

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  1. Geometry in the 21st CenturyDynamic and Symbolic Douglas B. Meade Department of Mathematics University of South Carolina E-mail: meade@math.sc.edu Homepage: www.math.sc.edu/~meade/

  2. Outline • Euclidean Geometry • Conic Sections • Shrinking Circles SC Junior Academy of Science Winter Conference (SVHS)

  3. Euclidean Geometry • Napoleon’s Theorem • Given any DABC • Construct an equilateral triangle on each side of DABC • Construct DGIK joining the incenters of the three equilateral triangles • What can be said about DGIK? SC Junior Academy of Science Winter Conference (SVHS)

  4. Conic Sections • Ellipse • Hyperbola • Parabola SC Junior Academy of Science Winter Conference (SVHS)

  5. Ellipse • DefinitionAn ellipse is a curve that is the locus of all points in the plane the sum of whose distances from two fixed points (the foci) is a given positive constant. SC Junior Academy of Science Winter Conference (SVHS)

  6. Hyperbola • DefinitionA hyperbola is a curve defined as the locus of all points in the plane the difference of whose distances from two fixed points (the foci) a given positive constant. SC Junior Academy of Science Winter Conference (SVHS)

  7. Parabola • DefinitionA parabola is the set of all points in the plane equidistant from a given line (the conic section directrix) and a given point not on the line (the focus). SC Junior Academy of Science Winter Conference (SVHS)

  8. Shrinking Circles • Construction • C* – unit circle centered at (1,0) • Cr – circle with radius r centered at origin • P – a (0, r) • Q – intersection of C* and Cr in first quadrant • R – intersection of the line PQ and the x-axis SC Junior Academy of Science Winter Conference (SVHS)

  9. Shrinking Circles • QuestionWhat happens to R as Cr shrinksi.e., in the limit as r decreases towards 0? SC Junior Academy of Science Winter Conference (SVHS)

  10. Generalization • Suppose C* is replaced with an ellipsethat passes through the origin: • What happens to R? as Cr shrinks? • Does it matter if an axis of the ellipse is on the x-axis. • What if C* is replaced with a general curve that passes through the origin? SC Junior Academy of Science Winter Conference (SVHS)

  11. Generalization • 3-D Construction • S* – unit sphere centered at (1,0,0) • Sr – sphere with radius r centered at origin • P – a (0, 0, r) • C – intersection (curve) of S* and Sr • R – intersection (curve) of cone with vertex P and cross-section C with the xy-plane SC Junior Academy of Science Winter Conference (SVHS)

  12. Conclusions • Geometry • is more than pencil and paper constructions • is more than two-column proofs • can be dynamic • is relevant to calculus, and beyond SC Junior Academy of Science Winter Conference (SVHS)

  13. Relevant Software • Geometry Expressions (beta)http://www.geometryexpressions.com/ • Geometer’s Sketchpadhttp://www.keypress.com/sketchpad/ • Cabri3Dhttp://www.cabri.com/ • Maplehttp://www.maplesoft.com/ SC Junior Academy of Science Winter Conference (SVHS)

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