elipsoidní

An elipsoidní (Czech for "ellipsoidal") shape refers to a three-dimensional geometric surface that is an ellipsoid, a smooth, closed surface resembling a squashed or stretched sphere. Ellipsoids are a generalization of spheres, where the three principal axes may have different lengths, resulting in varying degrees of flattening or elongation. This shape is fundamental in mathematics, physics, and engineering due to its ability to approximate complex surfaces with simpler models.

In mathematics, an ellipsoid is defined as the set of points in three-dimensional space where the sum of the weighted squared distances from three fixed axes is constant. The general equation of an ellipsoid centered at the origin is given by:

(x²/a²) + (y²/b²) + (z²/c²) = 1,

where *a*, *b*, and *c* are the semi-axes lengths along the x, y, and z axes, respectively. If all three axes are equal (*a* = *b* = *c*), the ellipsoid reduces to a sphere.

Ellipsoids are widely used in geodesy and cartography to model the Earth's shape, which is not a perfect sphere but an oblate spheroid—a type of ellipsoid flattened at the poles. This approximation is essential for accurate mapping, navigation, and surveying. In physics, ellipsoids describe shapes like planetary bodies, atomic orbitals, and stress distributions in materials. Engineering applications include designing optical lenses, aerodynamic shapes, and mechanical components where precise surface curvature is required.

Ellipsoidal coordinates are a coordinate system adapted to the ellipsoid’s geometry, useful in solving partial differential equations in physics and engineering. Additionally, ellipsoids appear in computer graphics for rendering complex surfaces and in statistics, where they model multivariate normal distributions in high-dimensional spaces. Their versatility makes them a key concept in both theoretical and applied sciences.