e= –1 for the hyperbolic roulette (hyperbola with semi-axesaandb). Cartesian equation:. Cartesian parametrization in theellipticcase: where, e=c/a. Curvilinear abscissa:. Cartesian parametrization in thehyperboliccase:. The notion of Sturmrouletterefers to the locus of the centre of a centred ...
A circle is a special form of an ellipse in which the semimajor and semiminor axes are equal in length. The equation of a circle is (7.7)x2+y2=r2, where r is the radius of the circle, and the circle has its center at the origin of the x, y coordinate system. If we are deali...
We know that the equation of a circle centered at (0,0) with a radius r is (see Fig. 26.2) x2+y2=r2. In order to calculate the curvature at any point (x,y) along the circle, we have to calculate y′ and y″. Using implicit differentiation with respect to x, we have 2x+2y...
X, Y, and Z are functions of T (parametric equations). For example: Let x be defined by: sin(t) Let y be defined by: cos(t) For t1 = 0 and t2 = pi, the result is a semi-circle. (Closed geometry is not allowed.) Z is for 3D sketches only. Under Parameters, specify the ...
The eccentricity of ellipse, e = c/a Where c is the focal length and a is length of the semi-major axis. Since c ≤ a the eccentricity is always less than 1 in the case of an ellipse. Also, c2= a2– b2 Therefore, eccentricity becomes: ...
Finding the Equation of a Circle When we need to find the equation of a circle, the radius of the circle and the coordinates of its center are required. Then, we express the equation of the circle as (x−a)2+(y−b)2=r2 where the radius is r and ...
On the rational difference equation y n +1 = A + y n / y n-k We find conditions for the global asymptotic stability, periodicity and semi-cycle analysis of the unique positive equilibrium y = 1 + A of the equation y... M Saleh,M Aloqeili - 《Applied Mathematics & Computatio...
We give an equivalent condition for the existence of a semi-conjugacy to an irrational rotation for conservative homeomorphisms of the two-torus. This lead... T Jäger - 《Inventiones Mathematicae》 被引量: 70发表: 2009年 Numerical Invariants for Semiconjugacy of Homeomorphisms of the Circle ...
Existence of resolvents (or equivalently, dense image for ); Existence of a contractive heat propagator semigroup (in the positive case); Existence of a unitary Schrödinger propagator group ; Existence of a unitary wave propagator group (in the positive case); Existence of a “reasonable” fun...
The real and imaginary parts of the amplification factor are (11.83d)AR=1−σ+σcosϕ,AI=−σsinϕ, which geometrically describe a circle of radius σ with its center at (1−σ,0) in the complex plane, that is, (11.83e)[AR−(1−σ)σ]2+[AIσ]2=1. Fig. 11.7 shows...