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Cosine & cosh are PG duals

Cosine & cosh are PG duals
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2018-07-25 20:54
FunctionTrigonometryCoordinate Geometry
This example demonstrates the duality between cosine and hyperbolic cosine functions in the context of Projective Geometry (PG). It involves core mathematical concepts such as cosine, hyperbolic cosine, projective geometry, similarity transformation, and coordinate geometry. The construction utilizes GeoGebra commands/objects like Plane, Sequence, and Intersect, along with objects like f, c, b, Pts1, Pts2, Pts3, g, l, A, Pts4, L_6, L_7, L_8, L_9, text1, text2, and Curves. It is suitable for high school and above students to explore the properties of functions in projective geometry and the application of similarity transformations.

GeoGebra Commands & Parameters

  1. Plane((0, c, 0), (1, c, 1), (-1, c, 3)) → b
  2. Sequence(((n * π), c, acosh(c)), n, -6, 6, 2) → Pts1
  3. Sequence(((n * π), c, (-acosh(c))), n, -6, 6, 2) → Pts2
  4. Sequence(((n * π), c, acosh(abs(c))), n, -5, 7, 2) → Pts3
  5. Intersect(f, l, (-6 * π), (6 * π)) → A, , , , , , , , , , ,
  6. Sequence(((n * π), c, (-acosh(abs(c)))), n, -5, 7, 2) → Pts4
  7. Sequence(((n * π), acosh(c)), n, -6, 6, 2) → L_6
  8. Sequence(((n * π), (-acosh(c))), n, -6, 6, 2) → L_7
  9. Sequence(((n * π), acosh(abs(c))), n, -5, 7, 2) → L_8
  10. Sequence(((n * π), (-acosh(abs(c)))), n, -5, 7, 2) → L_9
  11. Sequence(CurveCartesian[(n * π), ((-1)^(n) * cosh(t)), t, t, -25, 25], n, -5, 5) → Curves

AI Drawing Prompt Reference

Draw the graphs of the cosine and hyperbolic cosine functions to demonstrate their duality in projective geometry. First, draw the graph of the hyperbolic cosine function, then draw the graph of the cosine function. Next, use similarity transformations to show the duality between these two functions in projective geometry. Finally, use curves to represent the graphs of these functions.
The prompt above was compiled from the original drawing commands and is for reference only.