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In geometry, the compass equivalence theorem is an important statement in compass and straightedge constructions. The tool advocated by Plato in these constructions is a divider or collapsing compass , that is, a compass that "collapses" whenever it is lifted from a page, so that it may not be directly used to transfer distances.
Angle trisection is the construction, using only a straightedge and a compass, of an angle that is one-third of a given arbitrary angle. This is impossible in the general case. For example, the angle 2 π /5 radians (72° = 360°/5) can be trisected, but the angle of π /3 radians (60°) cannot be trisected. [8]
The triple-angle formula gives an expression relating the cosines of the original angle and its trisection: cos θ = 4 cos 3 θ / 3 − 3 cos θ / 3 . It follows that, given a segment that is defined to have unit length, the problem of angle trisection is equivalent to constructing a segment whose length is the root of a cubic ...
Direction determination refers to the ways in which a cardinal direction or compass point can be determined in navigation and wayfinding.The most direct method is using a compass (magnetic compass or gyrocompass), but indirect methods exist, based on the Sun path (unaided or by using a watch or sundial), the stars, and satellite navigation.
32-point compass rose. The points of the compass are a set of horizontal, radially arrayed compass directions (or azimuths) used in navigation and cartography.A compass rose is primarily composed of four cardinal directions—north, east, south, and west—each separated by 90 degrees, and secondarily divided by four ordinal (intercardinal) directions—northeast, southeast, southwest, and ...
The pons asinorum in Oliver Byrne's edition of the Elements [1]. In geometry, the theorem that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum (/ ˈ p ɒ n z ˌ æ s ɪ ˈ n ɔːr ə m / PONZ ass-ih-NOR-əm), Latin for "bridge of asses", or more descriptively as the isosceles triangle theorem.
Direction determination refers to the ways in which a cardinal direction or compass point can be determined in navigation and wayfinding. The most direct method is using a compass ( magnetic compass or gyrocompass ), but indirect methods exist, based on the Sun path (unaided or by using a watch or sundial ), the stars, and satellite navigation .
Here α, β, γ are the direction cosines and the Cartesian coordinates of the unit vector | |, and a, b, c are the direction angles of the vector v. The direction angles a, b, c are acute or obtuse angles, i.e., 0 ≤ a ≤ π, 0 ≤ b ≤ π and 0 ≤ c ≤ π, and they denote the angles formed between v and the unit basis vectors e x, e y, e z.