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The tangential angle φ for an arbitrary curve A in P. In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the x-axis. [1] (Some authors define the angle as the deviation from the direction of the curve at some fixed starting point.
The pedal curve is the first in a series of curves C 1, C 2, C 3, etc., where C 1 is the pedal of C, C 2 is the pedal of C 1, and so on. In this scheme, C 1 is known as the first positive pedal of C, C 2 is the second positive pedal of C, and so on. Going the other direction, C is the first negative pedal of C 1, the second negative pedal of C ...
Even with these restrictions, if the polar angle (inclination) is 0° or 180°—elevation is −90° or +90°—then the azimuth angle is arbitrary; and if r is zero, both azimuth and polar angles are arbitrary. To define the coordinates as unique, the user can assert the convention that (in these cases) the arbitrary coordinates are set to zero.
The point O is called the pedal point and the values r and p are sometimes called the pedal coordinates of a point relative to the curve and the pedal point. It is also useful to measure the distance of O to the normal p c (the contrapedal coordinate ) even though it is not an independent quantity and it relates to ( r , p ) as p c := r 2 − p ...
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
The distances shown are the ordinate (AP), tangent (TP), subtangent (TA), normal (PN), and subnormal (AN). The angle φ is the angle of inclination of the tangent line or the tangential angle. In geometry, the subtangent and related terms are certain line segments defined using the line tangent to a curve at a given point and the coordinate ...
The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question. The azimuthal angle is denoted by φ ∈ [ 0 , 2 π ] {\displaystyle \varphi \in [0,2\pi ]} : it is the angle between the x -axis and the projection of the radial vector onto the xy -plane.
The inverse of a sinusoidal spiral with respect to a circle with center at the origin is another sinusoidal spiral whose value of n is the negative of the original curve's value of n.
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