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The Locus Story of a Rocking Camel in a Medical Center in the City of Freistadt

Anna Käferböck, Zoltán Kovács

TL;DR

The paper addresses the problem of characterizing the motion-induced locus of a point on a real 4-bar linkage realized by a rocking camel and demonstrates how automated geometry reasoning can derive and verify the locus of a point using GeoGebra Discovery. It combines precise measurements, GeoGebra modeling, and computer algebra to produce a symbolic locus that is a degree $6$ polynomial and demonstrates a computer-assisted proof via elimination, facilitated by the Dilate command and parameter sliders. Key contributions include linking a public artifact to rigorous algebraic geometry in education, showcasing symbolic-locus computation and Eliminate-based proofs within GeoGebra Discovery, and proposing scalable classroom activities that blend real-world problems with automated reasoning. The work highlights the practical impact of integrating artifacts-driven modeling with automated proof tools to enrich STEM/STEAM learning and illustrate advanced geometry concepts in an educational setting.

Abstract

We give an example of automated geometry reasoning for an imaginary classroom project by using the free software package GeoGebra Discovery. The project is motivated by a publicly available toy, a rocking camel, installed at a medical center in Upper Austria. We explain how the process of a false conjecture, experimenting, modeling, a precise mathematical setup, and then a proof by automated reasoning could help extend mathematical knowledge at secondary school level and above.

The Locus Story of a Rocking Camel in a Medical Center in the City of Freistadt

TL;DR

The paper addresses the problem of characterizing the motion-induced locus of a point on a real 4-bar linkage realized by a rocking camel and demonstrates how automated geometry reasoning can derive and verify the locus of a point using GeoGebra Discovery. It combines precise measurements, GeoGebra modeling, and computer algebra to produce a symbolic locus that is a degree polynomial and demonstrates a computer-assisted proof via elimination, facilitated by the Dilate command and parameter sliders. Key contributions include linking a public artifact to rigorous algebraic geometry in education, showcasing symbolic-locus computation and Eliminate-based proofs within GeoGebra Discovery, and proposing scalable classroom activities that blend real-world problems with automated reasoning. The work highlights the practical impact of integrating artifacts-driven modeling with automated proof tools to enrich STEM/STEAM learning and illustrate advanced geometry concepts in an educational setting.

Abstract

We give an example of automated geometry reasoning for an imaginary classroom project by using the free software package GeoGebra Discovery. The project is motivated by a publicly available toy, a rocking camel, installed at a medical center in Upper Austria. We explain how the process of a false conjecture, experimenting, modeling, a precise mathematical setup, and then a proof by automated reasoning could help extend mathematical knowledge at secondary school level and above.
Paper Structure (14 sections, 1 equation, 9 figures)

This paper contains 14 sections, 1 equation, 9 figures.

Table of Contents

  1. Introduction
  2. GeoGebra Discovery and its Automated Reasoning Tools
  3. A Rocking Camel
  4. The History of the Rocking Camel
  5. A STEM Activity
  6. Exact Measurements
  7. Modeling in GeoGebra
  8. A First Conjecture
  9. A Numerical Locus
  10. A Second Conjecture
  11. A Symbolic Locus
  12. A Proof
  13. Generalization
  14. Final Thoughts and Conclusion

Figures (9)

  • Figure 1: The exhibited toy in the medical center of the city of Freistadt, Upper Austria.
  • Figure 2: A former version of a GeoGebra applet that suggests that the motion of the hump of the camel is a part of an ellipse.
  • Figure 3: A small lamp mounted on a battery with cables. It can be attached to a moving object by a glue tape.
  • Figure 4: Motion of the lamp, by using two steps of preprocessing.
  • Figure 5: An attempt to identify the motion as an ellipse.
  • ...and 4 more figures