Hyperboloid of one sheet
Hyperboloid of One Sheet: Geometry, Cross-Sections and Planes
This interactive GeoGebra activity explores the hyperboloid of one sheet, a quadric surface given by the equation x²/a² + y²/b² − z²/c² = 1.
Use the sliders for a, b and c to change the semi-axis lengths and see how the shape of the surface responds, and adjust the opacity slider to view the surface and its asymptotic cone together.
Activate the cross-sections panel to investigate how planes parallel to the coordinate planes (x = h, y = k, z = g) intersect the hyperboloid, producing ellipses or hyperbolas depending on the choice of plane.
An arbitrary plane, defined by a draggable point and a draggable normal vector in the 3D view, can also be explored: reposition and reorient it to see how the resulting cross-section changes shape, with its equation displayed dynamically.