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Derivation of Kepler’s laws from Newton’s law
Published in G.A. Gurzadyan, Theory of Interplanetary Flights, 2020
The Hyperbola. For a hyperbola the eccentricity is larger than unity, e > 1, and the parameter p=a(e2−1) where a is a large half-axis – a=OII and p=SK in Fig. 15. The distance of the focus from the center O is also equal to c=SO=ae, however, bearing in mind that e > 1, the distance between focuses turns out to be larger than the half-axis, i.e. c > a. A hyperbola has perigee II; the apogee lies at infinity.
Compliances of Basic Flexible-Hinge Segments
Published in Nicolae Lobontiu, Compliant Mechanisms, 2020
The circle and the ellipse are not the only planar curves resulting in right corner-filleted flexure segments. Rational Bézier curves can also be used to generate right corner-filleted flexible hinges, and these curves are actually conic-section curves, as shown here. Conic-section curves include the parabola, hyperbola and ellipse (or circle, as a limit case of an ellipse with equal semi-axes). They result by intersecting a conic surface with a plane that is in one of the following positions: at an angle with respect to the cone axis and intersecting both the lateral surface and the circular base of the cone – this results in a parabola, parallel to the cone axis – producing a hyperbola, and at an angle with respect to the cone axis and intersecting the lateral cone surface but not the circular base of the cone–this results in an ellipse, perpendicular to the cone axis – producing a circle; the parabola, hyperbola and ellipse are illustrated in Figure 2.18.
Classical Mechanics and Field Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
Problem 10.33. Show that the unbound states of the Kepler problem, i.e., states with E > 0, result in the trajectory of the particle being hyperbolic and compute the deflection angle α, the distance of closest approach ρmin to the center, and the impact parameter d, see Fig. 10.36, in terms of the constants E and L. Hint: Like an ellipse, a hyperbola may be described by the relation ρ=ρ0/(1+ε cos (ϕ)) in polar coordinates, where ε is the eccentricity. Unlike an ellipse, the eccentricity for a hyperbola is ε > 1.
Optimizing Dimensions of Cassegrain Antenna System with Large Magnification Factor
Published in IETE Journal of Research, 2023
In designing subreflector, we use the governing equations on hyperbolic surfaces. A hyperbolic curve is shown in Figure 4. Hyperbola is created from points that their distance differences from 2 fixed points with spacing equal to 2C, called focal points, become a constant namely, 2A. So a hyperbola would be identified by its focal points and parameters C and A. If we assume that the origin of the polar coordinate system be located on focal point , then considering Figure 4, it can be shown simply that where e = C/A, is called the eccentricity of hyperbola and is related by Cassegrain antenna magnification “M” by (3). Subreflector diameter can be calculated by ,where is the subtended angle of the main reflector as indicated in Figure 3, therefore
Non-equivalent notions of the eccentricity of the conics: an exploratory study with high school teachers
Published in International Journal of Mathematical Education in Science and Technology, 2023
Antonio Rivera-Figueroa, Ernesto Bravo-Díaz
The BA is one where the circle is the locus in the plane of the points whose distance from a fixed point F is a constant. The point F is called the centre of the circle, and the distance from either point of the circle to the centre is called the circle's radius. The parabola is the locus of points in the plane whose distance from a fixed point F equals their distance from a fixed line l. Point F is called the parabola's focus, and line l is called the directrix of the parabola. The ellipse and the hyperbola are defined by two foci. The ellipse is the locus of points such that the sum of their distances from two fixed points is a constant, and the hyperbola is the locus of points such that the positive difference between their distances from two fixed points is a constant. In both cases, the fixed points, and , are called foci of the curves (Figure 1), and the respective constant is called the constant of the curve. The eccentricity of the ellipse is the quotient where c is the semi-distance between the foci, and a is the length of the semi-major axis (see, e.g. Larson & Edwards, 2010, p. 701; Lehmann, 1964, p. 176; Rider, 1947, p. 120; Simmons, 1996, p. 536). Some authors allow the equality , in which case c has the value zero and, consequently, e is also zero, and the locus is a circle. In this regard, Rider (1947) comments, ‘the circle can be considered an ellipse whose eccentricity is zero’ (p. 121), while Apostol (1966, p. 498), without alluding to eccentricity, points out that ‘if the foci coincide, the ellipse is reduced to a circle’. Other authors (e.g. Hahn, 1998) allow the equality , so we have e = 1, and ‘the ellipse consists of points on the segment ’ (p. 91). The eccentricity e of the hyperbola is the same quotient , where now c is the semi-distance between the foci and a is the length of the transverse semiaxis of the hyperbola (Larson & Edwards, 2010, p. 704; Lehmann, 1964, p. 176; Rider, 1947, p. 132; Simmons, 1996, p. 545). Since , we have . In the bifocal approach, the authors generally do not assign the value to any curve; in particular, it is not assigned to the parabola.