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Intersection of Implicit and Explicit Surfaces by Roman Chijner

Intersection of Implicit and Explicit Surfaces by Roman Chijner
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2016-08-22 21:58
FunctionCoordinate GeometrySolid GeometryCalculus
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 points of intersection between an implicit surface f(x,y,z)=0 and an explicit surface z=z(x,y). Key math concepts include implicit surface equations, explicit surface equations, intersection calculations, and 3D geometry. It is constructed using GeoGebra commands like If, Polygon, Textfield, Point, Zip, ImplicitCurve, and IntersectPath, making it suitable for university mathematics, geometry education, and 3D visualization purposes.

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

AI Drawing Prompt Reference

Create a 3D visualization showing the intersection points between an implicit surface defined by f(x,y,z)=0 and an explicit surface z=z(x,y). Start by constructing a rectangular frame with labeled corner points, then generate two sets of parallel lines inside the frame representing projections of the roots of the implicit surface in different directions. Next, create a grid of points at the intersections of these lines. Finally, combine the curve of the implicit surface equation with the grid of points to visualize the intersection points in 3D space, adjusting the view and zoom to highlight the features of the intersections.
The prompt above was compiled from the original drawing commands and is for reference only.