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Intersections of Surfaces

Intersections of Surfaces
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2017-10-14 16:58
Coordinate GeometrySolid GeometryGeometric Construction
This example demonstrates the intersections of surfaces. Key mathematical concepts include intersections in coordinate geometry. Using the Surface and CurveCartesian commands, surfaces are constructed and their intersections are found. Suitable for high school and above students or educational purposes.

GeoGebra Commands & Parameters

  1. Surface(u, (k / 10 * sqrt(100 - u^(2) - v^(2))) - k, v, u, (-M), M, v, (-M), M) → a
  2. Surface(u, v, (((-k)) / 10 * sqrt(100 - u^(2) - v^(2))) + k, u, (-M), M, v, (-M), M) → b
  3. CurveCartesian(u, ((-100 / k)^(2) + sqrt((100 / k)^(2) - ((100 / k^(2) + 1) * u^(2)))) / (100 / k^(2) + 1) - dy, ((-100 / k)^(2) - sqrt((100 / k)^(2) - ((100 / k^(2) + 1) * u^(2)))) / (100 / k^(2) + 1) - dz, u, (-M), M) → inters1
  4. Surface((k / 10 * sqrt(100 - u^(2) - v^(2))) - k, u, v, u, (-M), M, v, (-M), M) → c
  5. CurveCartesian(((-100 / k)^(2) + sqrt((100 / k)^(2) - ((100 / k^(2) + 1) * u^(2)))) / (100 / k^(2) + 1) - dy, u, ((-100 / k)^(2) - sqrt((100 / k)^(2) - ((100 / k^(2) + 1) * u^(2)))) / (100 / k^(2) + 1) - dz, u, (-M), M) → inters2
  6. CurveCartesian(((-100 / k)^(2) + sqrt((100 / k)^(2) - ((100 / k^(2) + 1) * u^(2)))) / (100 / k^(2) + 1) - dy, ((-((-100 / k)^(2) - sqrt((100 / k)^(2) - ((100 / k^(2) + 1) * u^(2)))))) / (100 / k^(2) + 1) + dz, u, u, (-M), M) → inters3

AI Drawing Prompt Reference

Draw two 3D surfaces that intersect in space, forming curves. First, plot the surfaces varying with u and v, then find and display their intersection curves in different colors, and finally combine all the graphics to clearly show the intersection relationships in 3D space.
The prompt above was compiled from the original drawing commands and is for reference only.