Intersection Lines in 3D: Surfaces and Curves

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2017-10-14 21:19

① This example demonstrates the intersection lines formed when a surface meets multiple parametric curves in three-dimensional space, visualizing their spatial relationships. ② Core mathematical concepts include spatial analytic geometry, surface equations, parametric curves, and solving for intersection curves. ③ Construction highlights: using GeoGebra's Surface command to create the surface, CurveCartesian to define parametric curves, and objects such as k, M, a, b, dy, dz to control positions; the intersections are marked by inters1, inters2, inters3. ④ Suitable for senior high school or early university students studying solid geometry and analytic geometry, and for teachers to dynamically illustrate how surfaces and curves intersect, enhancing understanding of parametric equations and spatial intersections.

GeoGebra Commands & Parameters

Surface(u, (k / 10 * sqrt(100 - u^(2) - v^(2))) - k, v, u, (-M), M, v, (-M), M) → a
Surface(u, v, (((-k)) / 10 * sqrt(100 - u^(2) - v^(2))) + k, u, (-M), M, v, (-M), M) → b
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
Surface((k / 10 * sqrt(100 - u^(2) - v^(2))) - k, u, v, u, (-M), M, v, (-M), M) → c
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
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