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In algebraic geometry, the parabola is generalized by the rational normal curves, which have coordinates (x, x 2, x 3, ..., x n); the standard parabola is the case n = 2, and the case n = 3 is known as the twisted cubic. A further generalization is given by the Veronese variety, when there is more than one input variable.
The trajectory then generalizes (without air resistance) from a parabola to a Kepler-ellipse with one focus at the center of the Earth (shown in fig. 3). The projectile motion then follows Kepler's laws of planetary motion. The trajectory's parameters have to be adapted from the values of a uniform gravity field stated above.
For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the "unrolled" tube (which is flat). [ 1 ] [ page needed ] On the other hand, since a sphere of radius R has constant positive curvature R −2 and a flat plane has constant curvature 0, these two surfaces are not isometric, not even locally.
The green path in this image is an example of a parabolic trajectory. A parabolic trajectory is depicted in the bottom-left quadrant of this diagram, where the gravitational potential well of the central mass shows potential energy, and the kinetic energy of the parabolic trajectory is shown in red. The height of the kinetic energy decreases ...
Extra-vehicular activity (EVA), working outside the space vehicle, was one of the goals of the Gemini Program during the 1960s. The astronauts were trained in the “zero gravity” condition by flying a parabolic trajectory in an aircraft that caused reduced gravity for thirty second intervals.
In fact, it can be enjoyable to have zero gravity in the cockpit. To produce 0g, the aircraft has to follow a ballistic flight path, which is essentially an upside down parabola. This is the only method to simulate zero gravity for humans on earth. In helicopters. In contrast, low-g conditions can be disastrous for helicopters.
Jacobi constant, a Zero Velocity Surface and Curve (also Hill's curve) [1] A zero-velocity surface is a concept that relates to the N-body problem of gravity. It represents a surface a body of given energy cannot cross, since it would have zero velocity on the surface. It was first introduced by George William Hill. [2]
In differential geometry and the study of Lie groups, a parabolic geometry is a homogeneous space G/P which is the quotient of a semisimple Lie group G by a parabolic subgroup P. More generally, the curved analogs of a parabolic geometry in this sense is also called a parabolic geometry: any geometry that is modeled on such a space by means of ...
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