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In mathematics, a translation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x'y'-Cartesian coordinate system in which the x' axis is parallel to the x axis and k units away, and the y' axis is parallel to the y axis and h units away.
Cleavers, found in many Acheulean assemblages such as Africa, were similar in size and manner of hand axes. The differences between a hand axe and a cleaver is that a hand axe has a more pointed tip, while a cleaver will have a more transverse "bit" that consists of an untrimmed portion of the edge oriented perpendicular to the long axis of the tool.
While geometric translation is often viewed as an active transformation that changes the position of a geometric object, a similar result can be achieved by a passive transformation that moves the coordinate system itself but leaves the object fixed. The passive version of an active geometric translation is known as a translation of axes.
3D visualization of a sphere and a rotation about an Euler axis (^) by an angle of In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called the Euler axis) that runs through the fixed point. [6]
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In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and y axes counterclockwise through an angle .
If the third axis is shorter, the ellipsoid is a sphere that has been flattened (called an oblate spheroid). If the third axis is longer, it is a sphere that has been lengthened (called a prolate spheroid). If the three axes have the same length, the ellipsoid is a sphere.
In the active transformation (left), a point P is transformed to point P ′ by rotating clockwise by angle θ about the origin of a fixed coordinate system. In the passive transformation (right), point P stays fixed, while the coordinate system rotates counterclockwise by an angle θ about its origin.