Ellipsoidals

An ellipsoid is a quadric surface that is the three-dimensional generalization of an ellipse. It can be viewed as an affine transformation of a sphere. The standard equation for an ellipsoid centered at the origin is given by x²/a² + y²/b² + z²/c² = 1, where a, b, and c are the lengths of the semi-axes along the x, y, and z directions, respectively. If a = b = c, the ellipsoid is a sphere.

Ellipsoidals have a variety of applications in mathematics, physics, and engineering. In geometry, they are fundamental shapes for studying curvature and surfaces. In celestial mechanics, the orbits of celestial bodies are often approximated as ellipsoids. In computer graphics, ellipsoids are used as basic building blocks for creating more complex shapes and for collision detection. The surface area and volume of an ellipsoid are more complex to calculate than those of a sphere, often involving elliptic integrals.

The cross-sections of an ellipsoid made by planes are ellipses. If the plane is parallel to one of the coordinate planes and passes through the interior of the ellipsoid, the resulting cross-section is an ellipse. If the plane is tangent to the ellipsoid, the intersection is a single point. If the plane does not intersect the ellipsoid, there is no intersection. The properties of an ellipsoid, such as its symmetry and curvature, are determined by the lengths of its semi-axes.