Elicycloids

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

• Separate non-intercepting disciplines
• Each has a set of subdisciplines

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Mathematics and Informatics

• Complementary disciplines(co-disciplines)
• Two different views of the same discipline

Mathematics and Informatics
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Different Views
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Problem from the Real World
Wheel reflector
What is the curve when the bicycle moves
horizontally?
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Solution and a New Problem

• Mathematicians answer Its a trochoid.
• New problem How to model it?
• Mathematicians answer Use its parametric
  equation

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Implementation

• Implementation of the mathematicians approach

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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 trochoids properties
• The parametric equation must be known in advance

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Constructionists Approach

• Broadly used in few applications (maybe only in
  Geomland)?
• Descriptive construction
• Uses the natural relations between objects

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Transformational Approach

• Uses canonical elements, like a point at (0,0,0)
• Uses canonical transformations, like rotation
  around coordinate systems axes

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Implementation

• Mathematicians response So, what?

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The Little Prince

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

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Answer Epitrochoid

• Mathematician Its 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

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TransformationalEpitrochoids and Hypotrochoids

• Minimal changes in the source code
• No formulae

• Could be explained using common words

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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)

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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)

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More trochoids?

• (Epi-hypo)2-epi-trochoid
• (Hypo-epi)2-hypo-trochoid
• Hypo2epi3-trochoid

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More Problems - 1

• What is the curve of the pedalsrelatively to the
  ground?

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More Problems - 2

• What is the curve of the red nose of the lying
  person?

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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?

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The End
Whatis this curve?