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In the half-plane model, antipodal points are reflections into the opposite half-plane (shaded gray). Hyperbolic triangles ABC (orange) share a base AB and area. The locus of apex C is a hypercycle (dashed green) passing through points antipodal to A and B.
A half-space can be either open or closed. An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space. A closed half-space is the union of an open half-space and the hyperplane that defines it. The open (closed) upper half-space is the half-space of all (x 1, x 2, ..., x n) such that x n > 0
In this case, the half planes can be described by a point P of their intersection, and three vectors b 0, b 1 and b 2 such that P + b 0, P + b 1 and P + b 2 belong respectively to the intersection line, the first half plane, and the second half plane. The dihedral angle of these two half planes is defined by
The metric of the model on the half-plane, { , >}, is: = + ()where s measures the length along a (possibly curved) line. The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose centers are on the x-axis) and straight vertical rays ...
The Poincaré half-plane model takes one-half of the Euclidean plane, bounded by a line B of the plane, to be a model of the hyperbolic plane. The line B is not included in the model. The Euclidean plane may be taken to be a plane with the Cartesian coordinate system and the x-axis is taken as line B and the half plane is the upper half ( y > 0 ...
These reference planes include: the observer's horizon, the galactic equator (defined by the rotation of the Milky Way), the celestial equator (defined by Earth's rotation), the plane of the ecliptic (defined by Earth's orbit around the Sun), and the plane of the earth terminator (normal to the instantaneous direction to the Sun).
It is the opposite of a Split S, which involves a half-roll followed by a half-loop, resulting in level flight in the opposite direction at a lower altitude. [ citation needed ] To successfully execute a roll-off-the-top turn, the pilot accelerates to sufficient airspeed to perform a loop in the aircraft.
The red oblate spheroid (flattened sphere) corresponds to μ = 1, whereas the blue half-hyperboloid corresponds to ν = 45°. The azimuth φ = −60° measures the dihedral angle between the green xz half-plane and the yellow half-plane that includes the point P. The Cartesian coordinates of P are roughly (1.09, −1.89, 1.66).