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Intersection of Implicit and Explicit Surfaces

Intersection of Implicit and Explicit Surfaces
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2016-08-22 21:38
FunctionTrigonometryCoordinate GeometrySolid GeometryGraphing
In collections intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve
This example demonstrates the intersection points between the implicit function f(x,y,z) = cos(x) + cos(y) + cos(z) = 0 and the explicit function z = x^2 - y^2. Key mathematical concepts include implicit and explicit functions, and the intersection of surfaces in 3D space. GeoGebra commands used include If, Polygon, Textfield, Point, Zip, ImplicitCurve, IntersectPath, Sequence. Suitable for high school and above students or teachers for teaching 3D geometry problems involving surface intersections.

GeoGebra Commands & Parameters

  1. If(Length[RemoveUndefined[Join[Zip[Mirror[{Roots[f01(a, x), y_{min}, y_{max}]}, y = x] + (a, 0), a, Sequence[x_{min} + ((x_{max} - x_{min}) / N_P * i), i, 0, N_P]]]]] ≟ 0, false, true) → lb1
  2. If(lb1, RemoveUndefined[Join[Zip[Mirror[{Roots[f01(a, x), y_{min}, y_{max}]}, y = x] + (a, 0), a, Sequence[x_{min} + ((x_{max} - x_{min}) / N_P * i), i, 0, N_P]]]], Sequence[(10000, 10000), k, 1, 50]) → P1xxy
  3. If(lb1, RemoveUndefined[Join[Zip[Reverse[{Roots[f01(x, a), x_{min}, x_{max}]}] + (0, a), a, Sequence[y_{min} + ((y_{max} - y_{min}) / N_P * i), i, 0, N_P]]]], Sequence[(10000, 10000), k, 1, 50]) → P2yyx
  4. Polygon((x_{min}, y_{min}), (x_{max}, y_{min}), (x_{max}, y_{max}), (x_{min}, y_{max})) → poly1, b, c, d, e
  5. Textfield(x_{min}) → inputBox4
  6. Textfield(x_{max}) → inputBox5
  7. Textfield(y_{max}) → inputBox6
  8. Textfield(z_{max}) → inputBox7
  9. Textfield(y_{min}) → inputBox8
  10. Textfield(z_{min}) → inputBox9
  11. Textfield(N_P) → inputBox10
  12. Textfield(zoom_{3D}) → InputBox1
  13. Point(Pxy) → D
  14. Textfield(u) → InputBox4
  15. Textfield(f) → inputBox3
  16. Zip((x(a), y(a), u(x(a), y(a))), a, Join[{P1xxy, P2yyx}]) → list1
  17. ImplicitCurve(f01) → pp
  18. IntersectPath(Z_0, f(x, y, z0) + z0) → ic2z
  19. Textfield(t) → InputBox2
  20. Sequence(Sequence[(i * acos((-cos(x)) - cos(y))) + ((j * 2) * π), i, -1, 1, 2], j, -2, 2) → l_f