Pyörähdysellipsoidi

A pyörähdysellipsoidi, which translates to spheroid or ellipsoid of revolution in English, is a three-dimensional shape formed by rotating an ellipse around one of its principal axes. If the ellipse is rotated around its major axis, the resulting shape is a prolate spheroid, which is elongated along the axis of rotation. If the ellipse is rotated around its minor axis, the resulting shape is an oblate spheroid, which is flattened at the poles.

The mathematical equation for a spheroid centered at the origin can be expressed in Cartesian coordinates. For a prolate spheroid rotated around the z-axis, the equation is (x^2/a^2) + (y^2/a^2) + (z^2/b^2) = 1, where 'a' is the radius of the equatorial circle and 'b' is the semi-major axis along the axis of rotation, with b > a. For an oblate spheroid rotated around the z-axis, the equation is (x^2/a^2) + (y^2/a^2) + (z^2/b^2) = 1, where 'a' is the semi-major axis in the equatorial plane and 'b' is the semi-minor axis along the axis of rotation, with a > b.

Spheroids are common in nature and science. For instance, many planets and stars are approximated as oblate spheroids due to their rotation. Prolate spheroids can describe the shape of American footballs or some types of molecules. The study of spheroids is important in fields such as astronomy, physics, and geometry, particularly in understanding gravitational fields and calculating moments of inertia for non-spherical celestial bodies.