Cylinder Meets Hyperboloid: 3D Surface Intersection
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In collections
intersection: surface-surface, curve-curve
This GeoGebra example demonstrates the intersection of a cylinder and a hyperboloid in three-dimensional space, dynamically visualizing the spatial curve formed where the two quadric surfaces meet. Core mathematical concepts include solid geometry, three-dimensional analytic geometry, quadric surface equations, parametric curves, and vector algebra. The construction relies on GeoGebra's CurveCartesian command to generate parametric Cartesian curves and the Vector command to define directional vectors and geometric objects; it also employs algebraic objects K, A, d, a, e, f, t, b, c, r_+, r_-, g, h together with text labels text1 and text2 for dynamic annotation and interactive control. Suitable for senior high school and university STEM students, this resource supports spatial geometry instruction, parametric equation visualization, and classroom demonstrations of surface intersections.
GeoGebra Commands & Parameters
CurveCartesian(cos(u), sqrt(K + sin(u)^(2) - (A * cos(u)^(2))), sin(u), u, 0, t) → bCurveCartesian(cos(u), (-sqrt(K + sin(u)^(2) - (A * cos(u)^(2)))), sin(u), u, 0, t) → cVector(Point[{cos(t), sqrt(K + sin(t)^(2) - (A * cos(t)^(2))), sin(t)}]) → r_+Vector(Point[{cos(t), (-sqrt(K + sin(t)^(2) - (A * cos(t)^(2)))), sin(t)}]) → r_-
AI Drawing Prompt Reference
Draw a 3D intersection of a unit cylinder and a hyperboloid. Sweep a parameter angle from 0 to a variable upper limit to generate two symmetric space curves, one using the positive square-root branch and one using the negative branch, representing where the two quadric surfaces meet. Add position vectors from the origin to the endpoints of both curves to indicate their directions. Keep adjustable parameters and text labels for dynamic control and annotation of the curve range and intersection.
The prompt above was compiled from the original drawing commands and is for reference only.