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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.
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.
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 .
Translation is done by shearing parallel to the xy plane, and rotation is performed around the z axis. To represent affine transformations with matrices, we can use homogeneous coordinates . This means representing a 2-vector ( x , y ) as a 3-vector ( x , y , 1), and similarly for higher dimensions.
Changing orientation of a rigid body is the same as rotating the axes of a reference frame attached to it.. In geometry, the orientation, attitude, bearing, direction, or angular position of an object – such as a line, plane or rigid body – is part of the description of how it is placed in the space it occupies. [1]
The axis of rotation is a line of its fixed points. They exist only in n = 3. The plane of rotation is a plane that is invariant under the rotation. Unlike the axis, its points are not fixed themselves. The axis (where present) and the plane of a rotation are orthogonal.
Translation is done by shearing along over the z axis, and rotation is performed around the z axis. Using an augmented matrix and an augmented vector, it is possible to represent both the translation and the linear map using a single matrix multiplication. The technique requires that all vectors be augmented with a "1" at the end, and all ...
The axes of the original frame are denoted as x, y, z and the axes of the rotated frame as X, Y, Z.The geometrical definition (sometimes referred to as static) begins by defining the line of nodes (N) as the intersection of the planes xy and XY (it can also be defined as the common perpendicular to the axes z and Z and then written as the vector product N = z × Z).