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In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly. [2] [3] A rotation of axes is a linear map [4] [5] and a rigid transformation.
where for every direction in the base space, S n, the fiber over it in the total space, SO(n + 1), is a copy of the fiber space, SO(n), namely the rotations that keep that direction fixed. Thus we can build an n × n rotation matrix by starting with a 2 × 2 matrix, aiming its fixed axis on S 2 (the ordinary sphere in three-dimensional space ...
A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops ...
In 2 dimensions the curl of a vector field is not a vector field but a function, as 2-dimensional rotations are given by an angle (a scalar – an orientation is required to choose whether one counts clockwise or counterclockwise rotations as positive); this is not the div, but is rather perpendicular to it.
This definition relies on the fact that every simple closed curve admits a well-defined interior, which follows from the Jordan curve theorem. The inner loop of a beltway road in a country where people drive on the right side of the road is an example of a negatively oriented ( clockwise ) curve.
The outer coin makes two rotations rolling once around the inner coin. The path of a single point on the edge of the moving coin is a cardioid.. The coin rotation paradox is the counter-intuitive math problem that, when one coin is rolled around the rim of another coin of equal size, the moving coin completes not one but two full rotations after going all the way around the stationary coin ...
When viewed at a position along the positive z-axis, the ¼ turn from the positive x-to the positive y-axis is counter-clockwise. For left-handed coordinates, the above description of the axes is the same, except using the left hand; and the ¼ turn is clockwise. Interchanging the labels of any two axes reverses the handedness.
In the standard (non-flipped) images, this is a counterclockwise measure relative to the axis into the direction of positive declination. In the case of observed visual binary stars , it is defined as the angular offset of the secondary star from the primary relative to the north celestial pole .