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English: This file illustrates the concept of coterminal angles. A blue angle AOI is represented, and its measure is seen as a blue arc, with an arrow of the same color pointing counterclockwise to indicate its direction, and the text 45°. Point I is on the x axis and on OI, point A is on OA.
Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles. The reference angle (sometimes called related angle) for any angle θ in standard position is the positive acute angle between the terminal side of θ and the x-axis (positive or negative).
The dual notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists exactly one morphism X → T. Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object.
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.
In formal language, gimbal lock occurs because the map from Euler angles to rotations (topologically, from the 3-torus T 3 to the real projective space RP 3, which is the same as the space of rotations for three-dimensional rigid bodies, formally named SO(3)) is not a local homeomorphism at every point, and thus at some points the rank (degrees ...
An early author of modern projective geometry G. B. Halsted introduced the terms copunctal and flat-pencil to define angle: "Straights with the same cross are copunctal." Also "The aggregate of all coplanar, copunctal straights is called a flat-pencil" and "A piece of a flat-pencil bounded by two of the straights as sides, is called an angle." [22]
Rotation of an object in two dimensions around a point O. Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point.
The sum of the measures of the angles of any triangle is less than 180° if the geometry is hyperbolic, equal to 180° if the geometry is Euclidean, and greater than 180° if the geometry is elliptic. The defect of a triangle is the numerical value (180° − sum of the measures of the angles of the triangle). This result may also be stated as ...