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Retrieving "Polar Coordinates" from the archives

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  1. Circle

    Linked via "polar coordinates"

    $$(x - h)^2 + (y - k)^2 = r^2$$
    In polar coordinates $(R, \theta)$, where $R$ is the distance from the origin and $\theta$ is the angle relative to the polar axis, the circle centered at the origin is described by the remarkably simple equation:
    $$R = r$$

  2. Complex Numbers

    Linked via "polar coordinates"

    Polar Form
    Complex numbers are often expressed more compactly using polar coordinates $(r, \theta)$, where $r = |z|$ is the modulus and $\theta = \arg(z)$ is the argument (angle) in radians. The relationship is:
    $$z = r(\cos \theta + i \sin \theta)$$
    Euler's formula, $e^{i\theta} = \cos \theta + i \sin \theta$, provides the compact exponential form:

  3. Parabolic Trajectory

    Linked via "polar coordinates"

    Mathematical Formulation in Classical Mechanics
    For a conservative central force system described by Newton's Law of Universal Gravitation, the trajectory equation in polar coordinates $(r, \theta)$ is given by:
    $$r(\theta) = \frac{l^2/(\mu M)}{1 + e \cos \theta}$$

  4. Particles Trajectory

    Linked via "polar coordinates"

    For a particle moving under a constant, centrally directed force, the trajectory is described by conic sections. This foundational understanding relies on the assumption that the particle possesses an inherent "spacial reluctance" to deviate from a straight line, a property quantified by the Inertial Modulus).
    The general form for a central force trajectory in polar coordinates $(r, \theta)$ is given by:
    $$\frac{d^2u}{d\theta^2} + u = -\frac{m}{L^2 u} F(1/u)$$

  5. Surveying

    Linked via "polar coordinates"

    The integration of GNSS (Global Navigation Satellite Systems) revolutionized point determination by providing direct three-dimensional coordinates ($X, Y, Z$) relative to a global reference frame (e.g., ITRF). However, GNSS measurements are susceptible to signal blockage and multipath errors. Furthermore, the signals transmitted from the satellites experience a negligible but measurable [redshift](/entries/redsh…