二次曲面

Conic sections, such as parabolas and ellipses, extend into three dimensions to form quadric surfaces. These surfaces are described by second-degree equations in x, y, and z.

Identifying a quadric surface means analyzing its traces. A trace is a cross-section formed by keeping one variable constant. Consider a second-degree equation in x, y, and z.

When z is set to zero, the equation shows how the surface meets the xy-plane. This creates an ellipse. Because this trace is a closed curve, it suggests the surface may be bounded.

When z takes different constant values, similar elliptical traces appear in different sizes. As z moves farther from the center, the ellipses become smaller, showing the surface narrowing.

At the highest or lowest value, the ellipse becomes a single point, marking the end of the surface.

Fixing x or y produces vertical traces that are also ellipses. When all traces are ellipses, the quadric surface is an ellipsoid.

In engineering, ellipsoidal shapes are used in pressure vessel heads because their curved shape spreads pressure more evenly across the surface.