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Elicycloids Informatics – Bridge to Mathematics

Elicycloids Informatics – Bridge to Mathematics. Assoc. prof. Pavel Boytchev , KIT, FMI, Sofia University. Mathematics and Informatics. Informatics. Mathematics. Separate non-intercepting disciplines Each has a set of subdisciplines. Mathematics and Informatics.

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Elicycloids Informatics – Bridge to Mathematics

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  1. Elicycloids Informatics –Bridge to Mathematics Assoc. prof. Pavel Boytchev, KIT, FMI, Sofia University

  2. Mathematics and Informatics Informatics Mathematics • Separate non-intercepting disciplines • Each has a set of subdisciplines

  3. Mathematics and Informatics • Complementary disciplines(co-disciplines) • Two different views of the same discipline Mathematics and Informatics

  4. Mathematician: This is ! Mathematician: Huh? Informatician: Huh? Informatician: This is 0! Different Views

  5. Problem from the Real World Wheel reflector What is the curve when the bicycle moves horizontally?

  6. Solution and a New Problem • Mathematician’s answer: It’s a trochoid. • New problem: How to model it? • Mathematician’s answer: Use its parametric equation:

  7. Implementation Implementation of the mathematician’s approach

  8. Comments • Pros: • Quick and easy modeling • Representation is close to the mathematical one • Almost all graphical applications support this approach • Cons: • Not suitable for non-mathematicians • Hard to explain trochoid’s properties • The parametric equation must be known in advance

  9. T s a O R K Constructionist’s Approach • Broadly used in few applications (maybe only in Geomland)? • Descriptive construction • Uses the natural relations between objects

  10. Transformational Approach • Uses canonical elements, like a point at (0,0,0) • Uses canonical transformations, like rotation around coordinate system’s axes

  11. Implementation • Mathematician’s response: So, what?

  12. The Little Prince • A new problem: The Little Prince rides his bicycle on his small planet. What will be the curve of the reflector?

  13. Answer: Epitrochoid • Mathematician: It’s an epitrochoid with equation • The educational value of directly using the equation is rather disputable • Finding the equation might be a hard task for students

  14. TransformationalEpitrochoids and Hypotrochoids • Minimal changes in the source code • No formulae • Could be explained using common words

  15. Epi-epi-trochoid • Transformational approach can easily generate an epi-epi-trochoid (i.e. three circles, the second rolls over the first, and the third rolls over the second)

  16. Epi-epi-epi-epi-epi-trochoid • A simple change in code can produce any level of trochoidal epism • Epi5-trochoid – these are 6 circles (5 of them are rolling)

  17. More trochoids? • (Epi-hypo)2-epi-trochoid • (Hypo-epi)2-hypo-trochoid • Hypo2epi3-trochoid

  18. The curve of a reflector attached to the smallest wheel will be …? More Problems - 1 What is the curve of the pedalsrelatively to the ground?

  19. Adam is on the right gear, Eve – on the left. While rotating them what would Eve think? Is Adam epi- or hypo-trochoiding around her? More Problems - 2 What is the curve of the red nose of the lying person?

  20. What curves can a spirograph do? More Problems - 3 A double ferries wheel. What is the curve in respect to someone on the ground? Or someone in the other half of the wheel?

  21. The End Whatis this curve?

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