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Curvature of a 3D Surface

Curvature of a 3D Surface
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2020-11-18 23:41
Solid GeometryCalculusDerivativeParametric EquationGraphing
① This GeoGebra example demonstrates the curvature of a 3D surface at a selected point, visually indicating how the surface bends in different directions. ② It covers core concepts from differential geometry: normal vectors, principal curvatures, Gaussian curvature, and mean curvature. ③ The construction uses Textfield for parameter input, Circle to create auxiliary circles, Point to define surface points, OrthogonalLine to generate normal lines, Segment for connecting elements, and CurveCartesian to plot the parametric surface. Objects include the surface function f, points A, B, C, D, and lines g, h, i, k, with dynamic updates via sliders or input fields. ④ This is ideal for university-level differential geometry courses, advanced calculus visualization, and learners interested in 3D surface analysis.

GeoGebra Commands & Parameters

  1. Textfield(f) → Eingabefeld1
  2. Circle(A, r, xOyPlane) → c
  3. Point(c) → B
  4. OrthogonalLine(B, xOyPlane) → g
  5. Segment(B, C) → h
  6. Segment(D, A) → i
  7. CurveCartesian(x(A) + (r * cos(t)), y(A) + (r * sin(t)), f(x(A) + (r * cos(t)), y(A) + (r * sin(t))), t, 0, (2 * π)) → k
  8. Segment(A, B) → j
  9. CurveCartesian(x(A) + (t * (x(B) - x(A))), y(A) + (t * (y(B) - y(A))), f(x(A) + (t * (x(B) - x(A))), y(A) + (t * (y(B) - y(A)))), t, 0, 1) → l

AI Drawing Prompt Reference

Draw a 3D surface defined by a user-input function f, and visualize its curvature at a chosen point. Create an auxiliary circle in the horizontal plane centered at the selected point using radius r, pick a point on this circle as the reference on the surface, and construct the normal line through that point perpendicular to the horizontal plane. Connect the reference point with the other two surface reference points using segments. Plot the surface near the selected point using parametric coordinates that sweep in the circular direction via angle t. Also draw a curve segment on the surface by interpolating between two reference points with a parameter from 0 to 1.
The prompt above was compiled from the original drawing commands and is for reference only.