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Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3, 60°). In blue, the point (4, 210°). In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are
Thus, this inversion transforms the bipolar coordinate system defined by the Apollonian circles into a polar coordinate system. Obviously, the transformed pencils meet at right angles. Since inversion is a conformal transformation, it preserves the angles between the curves it transforms, so the original Apollonian circles also meet at right ...
where a is the radius of the circle, (,) are the polar coordinates of a generic point on the circle, and (,) are the polar coordinates of the centre of the circle (i.e., r 0 is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x axis to the line connecting the origin to the centre of ...
Any two polar circles of two triangles in an orthocentric system are orthogonal. [1]: p. 177 The polar circles of the triangles of a complete quadrilateral form a coaxal system. [1]: p. 179 The most important property of the polar circle is the triangle is self-polar; the polar of each side/point is the opposite side/point.
The north polar circle on a polar projection. The polar circle as lines on a modified cylindrical projection. The Arctic Circle in Finland, 1975. The Arctic Circle in Norway at Saltfjellet mountain plateau in July 2003. A polar circle is a geographic term for a conditional circular line (arc) referring either to the Arctic Circle or the ...
In the cylindrical coordinate system, a z-coordinate with the same meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple (r, θ, z). [8] Spherical coordinates take this a step further by converting the pair of cylindrical coordinates (r, z) to polar coordinates (ρ, φ) giving a triple (ρ, θ, φ). [9]
Relationship of Earth's axial tilt (ε) to the tropical and polar circles. The Arctic Circle is the southernmost latitude in the Northern Hemisphere at which the centre of the Sun can remain continuously above or below the horizon for twenty-four hours; as a result, at least once each year at any location within the Arctic Circle the centre of the Sun is visible at local midnight, and at least ...
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range ...