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The direction of vector rotation is counterclockwise if θ is positive (e.g. 90°), and clockwise if θ is negative (e.g. −90°) for ().Thus the clockwise rotation matrix is found as
Railroad reverse ("S") curve. In civil engineering, a reverse curve (or "S" curve) is a section of the horizontal alignment of a highway or rail route in which a curve to the left or right is followed immediately by a curve in the opposite direction.
This is Rodrigues' formula for the axis of a composite rotation defined in terms of the axes of the two component rotations. He derived this formula in 1840 (see page 408). [3] The three rotation axes A, B, and C form a spherical triangle and the dihedral angles between the planes formed by the sides of this triangle are defined by the rotation ...
A transition curve can connect a track segment of constant non-zero curvature to another segment with constant curvature that is zero or non-zero of either sign. Successive curves in the same direction are sometimes called progressive curves and successive curves in opposite directions are called reverse curves.
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]
The "improper rotation" term refers to isometries that reverse (flip) the orientation. In the language of group theory the distinction is expressed as direct vs indirect isometries in the Euclidean group, where the former comprise the identity component. Any direct Euclidean motion can be represented as a composition of a rotation about the ...
The horizontal alignment (or alinement in the United States) is done by using a predefined length of string line (such as 62-foot in the US and 20 meters in Australia [5]) to measure along the gauge side of the reference rail. It is the distance (in inches or millimeters) from the midpoint of the string line to the gauge of the reference rail.
Ackermann geometry. The Ackermann steering geometry (also called Ackermann's steering trapezium) [1] is a geometric arrangement of linkages in the steering of a car or other vehicle designed to solve the problem of wheels on the inside and outside of a turn needing to trace out circles of different radii.