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  2. Negative relationship - Wikipedia

    en.wikipedia.org/wiki/Negative_relationship

    Negative correlation can be seen geometrically when two normalized random vectors are viewed as points on a sphere, and the correlation between them is the cosine of the circular arc of separation of the points on a great circle of the sphere. [1] When this arc is more than a quarter-circle (θ > π/2), then the cosine is negative.

  3. Fourier shell correlation - Wikipedia

    en.wikipedia.org/wiki/Fourier_shell_correlation

    Typically, random halves are used, although some programs may use the even particle images for one half and the odd particles for the other half of the data set. Some publications quote the FSC 0.5 resolution cutoff, which refers to when the correlation coefficient of the Fourier shells is equal to 0.5.

  4. Spherical harmonics - Wikipedia

    en.wikipedia.org/wiki/Spherical_harmonics

    As a result, the sum of these spaces is also dense in the space L 2 (S n−1) of square-integrable functions on the sphere. Thus every square-integrable function on the sphere decomposes uniquely into a series of spherical harmonics, where the series converges in the L 2 sense.

  5. Bipolar coordinates - Wikipedia

    en.wikipedia.org/wiki/Bipolar_coordinates

    The circles of positive τ lie in the right-hand side of the plane (x > 0), whereas the circles of negative τ lie in the left-hand side of the plane (x < 0). The τ = 0 curve corresponds to the y-axis (x = 0). As the magnitude of τ increases, the radius of the circles decreases and their centers approach the foci.

  6. Spherical coordinate system - Wikipedia

    en.wikipedia.org/wiki/Spherical_coordinate_system

    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 ...

  7. Gaussian curvature - Wikipedia

    en.wikipedia.org/wiki/Gaussian_curvature

    The intersection of a normal plane and the surface will form a curve called a normal section and the curvature of this curve is the normal curvature. For most points on most “smooth” surfaces, different normal sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures , call these κ ...

  8. Genus (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Genus_(mathematics)

    Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing. The non-orientable genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps (i.e. a non-orientable surface of (non-orientable) genus n). (This number is also called the demigenus.)

  9. Osculating circle - Wikipedia

    en.wikipedia.org/wiki/Osculating_circle

    An osculating circle is a circle that best approximates the curvature of a curve at a specific point. It is tangent to the curve at that point and has the same curvature as the curve at that point. [2] The osculating circle provides a way to understand the local behavior of a curve and is commonly used in differential geometry and calculus.