ellipsoidors

An ellipsoid is a quadric surface that is a generalization of a sphere. It is defined by the equation (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. Ellipsoids have three principal axes of symmetry.

In geometry, an ellipsoid is a three-dimensional analogue of an ellipse. It can be visualized as a stretched or compressed sphere. The shape is determined by the relative lengths of its semi-axes. For example, if one semi-axis is much longer than the other two, the ellipsoid will appear elongated, resembling a prolate spheroid. If two semi-axes are equal and shorter than the third, it will be oblate.

Ellipsoids appear in various scientific and engineering contexts. In astronomy, celestial bodies like planets and stars are often approximated as oblate spheroids due to their rotation. In computer graphics, ellipsoids are commonly used as simplified models for objects. The surface area and volume of an ellipsoid can be calculated using specific formulas, though they are more complex than those for a sphere. The calculation of volume is V = (4/3)πabc, and the surface area does not have a simple closed-form expression and often requires numerical integration.