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For fixed θ they describe circles on the 2-sphere which are perpendicular to the z-axis and these circles may be viewed as trajectories of a point on the sphere. A point {θ 0, φ 0} on the sphere, under a rotation about the z-axis, will follow a trajectory {θ 0, φ 0 + φ} as the angle φ varies.
The basic triangle on a unit sphere. Both vertices and angles at the vertices of a triangle are denoted by the same upper case letters A, B, and C. Sides are denoted by lower-case letters: a, b, and c. The sphere has a radius of 1, and so the side lengths and lower case angles are equivalent (see arc length).
The three possible line-sphere intersections: 1. No intersection. 2. Point intersection. 3. Two point intersection. In analytic geometry, a line and a sphere can intersect in three ways: No intersection at all; Intersection in exactly one point; Intersection in two points.
The sphere, with various loxodromes shown in distinct colors. The loxodromes of the sphere map to curves on the plane of the form = /, where the parameter a measures the "tightness" of the loxodrome. Thus loxodromes correspond to logarithmic spirals. These spirals intersect radial lines in the plane at equal angles, just as the loxodromes ...
A sphere is the surface of a solid ball, here having radius r. In mathematics, a surface is a mathematical model of the common concept of a surface.It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line.
IB2d: Immersed Boundary Method for MATLAB and Python in 2D with 60+ examples, N.A. Battista, TCNJ; ESPResSo: Immersed Boundary Method for soft elastic objects; CFD IBM code based on OpenFoam; sdfibm: Another CFD IBM code based on OpenFoam; SimScale: Immersed Boundary Method for fluid mechanics and conjugate heat transfer simulation in the cloud
an inversion is a reflection in a sphere – various operations that can be achieved using such inversions are discussed at inversive geometry. In particular, the combination of inversion together with the Euclidean transformations translation and rotation is sufficient to express any conformal mapping – i.e. any mapping that universally ...
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system—shown here in the mathematics convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored ...