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Orthoptic of Fermat4

Orthoptic of Fermat4
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2019-09-24 20:58
FunctionCoordinate GeometryParametric Equation
This example illustrates the orthoptic of Fermat4, a significant concept in geometry involving the properties of tangents to a curve. Using the GeoGebra command CurveCartesian, specific curves are constructed based on given parameters. This example is suitable for high school mathematics education, particularly in coordinate geometry and parametric equations.

GeoGebra Commands & Parameters

  1. CurveCartesian((u^(3) - (1 - u^(4))^(3 / 4)) / (u^(6) + (1 - u^(4))^(3 / 2)), (u^(3) + (1 - u^(4))^(3 / 4)) / (u^(6) + (1 - u^(4))^(3 / 2)), u, -1, 1) → a
  2. CurveCartesian(((-u^(3)) + (1 - u^(4))^(3 / 4)) / (u^(6) + (1 - u^(4))^(3 / 2)), ((-u^(3)) - (1 - u^(4))^(3 / 4)) / (u^(6) + (1 - u^(4))^(3 / 2)), u, -1, 1) → b
  3. CurveCartesian((-sin(t)^(0.5)), cos(t)^(0.5), t, 0, π / 2) → d
  4. CurveCartesian(cos(t)^(0.5), sin(t)^(0.5), t, 0, π / 2) → c
  5. CurveCartesian((-sin(t)^(0.5)), (-cos(t)^(0.5)), t, 0, π / 2) → e
  6. CurveCartesian(cos(t)^(0.5), (-sin(t)^(0.5)), t, 0, π / 2) → f

AI Drawing Prompt Reference

Draw a curve defined by parametric equations where the x-coordinate is (u^(3) - (1 - u^(4))^(3 / 4)) / (u^(6) + (1 - u^(4))^(3 / 2)) and the y-coordinate is (u^(3) + (1 - u^(4))^(3 / 4)) / (u^(6) + (1 - u^(4))^(3 / 2)) with parameter u ranging from -1 to 1, and another curve with x-coordinate (-u^(3)) + (1 - u^(4))^(3 / 4)) / (u^(6) + (1 - u^(4))^(3 / 2)) and y-coordinate ((-u^(3)) - (1 - u^(4))^(3 / 4)) / (u^(6) + (1 - u^(4))^(3 / 2)) also ranging from -1 to 1. Then, draw four additional curves defined by parametric equations with parameter t ranging from 0 to π/2, corresponding to the coordinates (-sin(t)^(0.5), cos(t)^(0.5)), (cos(t)^(0.5), sin(t)^(0.5)), (-sin(t)^(0.5), -cos(t)^(0.5)), and (cos(t)^(0.5), -sin(t)^(0.5)).
The prompt above was compiled from the original drawing commands and is for reference only.