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Taxicab Conic Sections

Taxicab Conic Sections
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2020-11-23 08:08
TrigonometryCoordinate GeometryOptimizationGeometric Construction
This example demonstrates conic sections (circle, ellipse, hyperbola, parabola) defined under the Taxicab geometry (Manhattan distance). Key math concepts include Taxicab distance, conic section definitions, integral calculus, linear programming, angles, and unit circle. GeoGebra commands/objects used are points A, B, C, i, j, l, m, f, e, and sliders k, a, b, c, d for parameter control and dynamic visualization. Suitable for high school and above students to explore the properties of geometric shapes under different distance metrics and the construction of conic sections.

GeoGebra Commands & Parameters

  1. Point A(0.0, 3.0)
  2. Point B(5.0, 1.0)
  3. Numeric k(5.0)
  4. a: abs(x - x(A)) + abs(y - y(A)) + abs(x - x(B)) + abs(y - y(B)) = k
  5. b: abs(x - x(A)) + abs(y - y(A)) = k
  6. c: abs(x - x(A)) + abs(y - y(A)) - (abs(x - x(B)) + abs(y - y(B))) = k
  7. d: (-(abs(x - x(A)) + abs(y - y(A)))) + abs(x - x(B)) + abs(y - y(B)) = k
  8. Point C(2.0456540864086485, 6.444572022297527)
  9. Boolean i(false)
  10. Boolean j(true)
  11. Boolean l(false)
  12. Boolean m(false)
  13. y = y(C)
  14. e: abs(y - y(C)) = abs(y - y(A)) + abs(x - x(A))

AI Drawing Prompt Reference

Draw taxicab conic sections including circle, ellipse, hyperbola, and parabola under the Manhattan distance. Start by plotting points A at (0,3) and B at (5,1). Then, draw the following curves: one where the sum of the Manhattan distances from A and B equals k, one where the Manhattan distance from A to C equals k, one where the absolute difference of the Manhattan distances from A to B equals k, and one where the Manhattan distance from C to A equals the sum of the Manhattan distances from C to B.
The prompt above was compiled from the original drawing commands and is for reference only.