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Intersection of Sphere with Elliptic/Hyperbolic Cylinder

Intersection of Sphere with Elliptic/Hyperbolic Cylinder
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2021-07-02 01:19
Solid GeometryParametric Equation
This example demonstrates the quartic curve obtained when intersecting a sphere with an elliptic or hyperbolic cylinder. Key mathematical concepts include spheres, cylinders, ellipses, hyperbolic curves, and parametric equations. Construction involves GeoGebra commands such as Intersect, Sphere, CurveCartesian, and others. Suitable for high school students or college undergraduates learning solid geometry and parametric curve construction.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → o
  2. Point(xAxis) → e_{1}
  3. Circle(o, e_{1}) → c_E
  4. Intersect(c_E, yAxis, 2) → i
  5. Intersect(c_E, yAxis, 1) → i'
  6. Line(i, i') → g_y
  7. Intersect(c_E, xAxis, 2) → e'
  8. Line(e', e_{1}) → g_x
  9. Sphere(o, 1) → eku
  10. Point(zAxis) → N
  11. Point(g_x) → f
  12. Point(g_x) → f'
  13. Point(g_x) → s_x
  14. If(abs((1 / 2 * (f + f')) - s_x) > (1 / 2 * abs(f - f')), Ellipse[f, f', s_x], Hyperbola[f, f', s_x]) → ccon
  15. CurveCartesian(m + (a * cos(t)) + ((b * ί) * sin(t)), t, (-π), π) → eelli
  16. CurveCartesian(x(m + (a * cos(t)) + ((b * ί) * sin(t))), y(m + (a * cos(t)) + ((b * ί) * sin(t))), oProj_{+}(x(m + (a * cos(t)) + ((b * ί) * sin(t))), y(m + (a * cos(t)) + ((b * ί) * sin(t)))), t, (-π), π) → qq_{+}
  17. CurveCartesian(x(m + (a * cos(t)) + ((b * ί) * sin(t))), y(m + (a * cos(t)) + ((b * ί) * sin(t))), oProj_{-}(x(m + (a * cos(t)) + ((b * ί) * sin(t))), y(m + (a * cos(t)) + ((b * ί) * sin(t)))), t, (-π), π) → qq_{-}
  18. CurveCartesian(projSt_{+}(m + (a * cos(t)) + ((b * ί) * sin(t))), t, (-π), π) → stqq_{+}
  19. CurveCartesian(projSt_{-}(m + (a * cos(t)) + ((b * ί) * sin(t))), t, (-π), π) → stqq_{-}
  20. CurveCartesian(T(projSt_{+}(m + (a * cos(t)) + ((b * ί) * sin(t)))), t, (-π), π) → bizQu_{+}
  21. CurveCartesian(T(projSt_{-}(m + (a * cos(t)) + ((b * ί) * sin(t)))), t, (-π), π) → bizQu_{-}
  22. Line(P_{u+}, N) → g
  23. Line(N, P_{u-}) → h

AI Drawing Prompt Reference

Draw a unit sphere centered at the origin. In the horizontal plane, first construct a circle centered at the origin and find its intersections with the coordinate axes, then draw lines through these intersection points. Pick several points on these lines and, based on their relative positions, generate an ellipse or hyperbola in the horizontal plane as the directrix of a cylinder. Extend this directrix vertically to form a cylinder, and compute the intersection curve between the cylinder and the sphere. Finally, take a point above the sphere and draw projection lines from this point to different parts of the intersection curve, then project these lines onto the horizontal plane to show the resulting quartic curve.
The prompt above was compiled from the original drawing commands and is for reference only.