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Pringles Modeling: Hyperbolic Paraboloid

Pringles Modeling: Hyperbolic Paraboloid
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2020-10-25 16:28
Coordinate GeometrySolid GeometryParametric EquationGeometric Construction
This example demonstrates how to model Pringles chips using a hyperbolic paraboloid, viewed as a ruled surface. It involves core math concepts like hyperbolic paraboloids, parametric equations, spatial surfaces, and analytical geometry. Construction highlights include commands like Intersect, IntersectConic, Point, Rotate, OrthogonalLine, Locus, Circle, Cylinder, CurveCartesian, Polygon, Sequence, and Segment in GeoGebra. Suitable for high school students and geometry enthusiasts learning surface modeling.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. IntersectConic(g, eq1) → c
  3. IntersectConic(eq1, g) → d
  4. Point(xAxis) → B
  5. Rotate(B, (e * 360°)) → B'
  6. OrthogonalLine(B', xOyPlane) → i
  7. Intersect(i, g) → C,
  8. Locus(C, e) → lieu1
  9. Locus(C, e) → lieu2
  10. Circle(A, B) → k
  11. Point(zAxis) → D
  12. Point(zAxis) → E
  13. Cylinder(D, E, 2) → j, p, q, l
  14. IntersectConic(g, l) → r
  15. IntersectConic(z = x^(2) / a^(2) - y^(2) / a^(2), x^(2) + y^(2) + (0 * z) = 4) → s
  16. CurveCartesian(((2 * cos(t)), (2 * sin(t)), pr((2 * cos(t)), (2 * sin(t)))), t, 0, (2 * 3.141592653589793)) → m
  17. Point(xAxis) → F
  18. Polygon(F, G, 4) → poly1, n, t, f_1, g_1, H, I
  19. Polygon(J, K, L, M) → q1, j_1, k_1, l_1, m_1
  20. Sequence(Segment[L + (k * (K - L)), M + (k * (J - M))], k, 0, 1, 0.1) → l1
  21. Sequence(Segment[K + (k * (J - K)), L + (k * (M - L))], k, 0, 1, 0.1) → l2
  22. Segment(L + (e * (K - L)), M + (e * (J - M))) → h_1
  23. Segment(Intersect[Segment[L + (e * (K - L)), M + (e * (J - M))], l, 1], Intersect[Segment[L + (e * (K - L)), M + (e * (J - M))], l, 2]) → i_1
  24. Sequence(Segment[Intersect[Segment[L + (ee * (K - L)), M + (ee * (J - M))], l, 1], Intersect[Segment[L + (ee * (K - L)), M + (ee * (J - M))], l, 2]], ee, 0, e, 0.01) → l3
  25. Rotate(Mirror[l3, xOyPlane], 90°) → l4
  26. Segment(Intersect[Segment[M + (e * (L - M)), J + (e * (K - J))], l, 1], Intersect[Segment[M + (e * (L - M)), J + (e * (K - J))], l, 2]) → n_1
  27. Segment(Intersect[Segment[M + ((1 - e) * (L - M)), J + ((1 - e) * (K - J))], l, 1], Intersect[Segment[M + ((1 - e) * (L - M)), J + ((1 - e) * (K - J))], l, 2]) → p_1

AI Drawing Prompt Reference

Create a 3D ruled hyperbolic paraboloid and use it to model a Pringles-like chip. Start from the intersection of the x- and y-axes as reference, then generate key spatial points via rotation around the vertical axis and orthogonal projection onto the xOy plane. Construct circle and conic intersections that define intersection curves tied to the paraboloid. Draw the hyperbolic paraboloid surface using a parameterized Cartesian curve in 3D, with z given by a difference of squared x and y over a common scale. Sample the resulting boundary directions with short segments to form the curved rim, then mirror or rotate the rim to obtain the opposite symmetric side, yielding the full Pringles-like surface with matching boundary curves.
The prompt above was compiled from the original drawing commands and is for reference only.