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Suppose a rectangular xyz-coordinate system is rotated around its z axis counterclockwise (looking down the positive z axis) through an angle , that is, the positive x axis is rotated immediately into the positive y axis. The z coordinate of each point is unchanged and the x and y coordinates transform as above.
In other words, either P and v are fixed while CS rotates (alias), or CS is fixed while P and v rotate (alibi). Any given rotation can be legitimately described both ways, as vectors and coordinate systems actually rotate with respect to each other, about the same axis but in opposite directions.
Pygame version 2 was planned as "Pygame Reloaded" in 2009, but development and maintenance of Pygame completely stopped until the end of 2016 with version 1.9.1. After the release of version 1.9.5 in March 2019, development of a new version 2 was active on the roadmap. [11] Pygame 2.0 released on 28 October, 2020, Pygame's 20th anniversary. [12]
The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the ...
The engineering and robotics communities typically use 3-1-3 Euler angles. Notice that after composing the independent rotations, they do not rotate about their axis anymore. The most external matrix rotates the other two, leaving the second rotation matrix over the line of nodes, and the third one in a frame comoving with the body.
For the convex polygon, a linear time algorithm for the minimum-area enclosing rectangle is known. It is based on the observation that a side of a minimum-area enclosing box must be collinear with a side of the convex polygon. [ 1 ]
Sequence of probes around the convex hull of a polygon to determine its diameter using Rotating Caliper method. In computational geometry, the method of rotating calipers is an algorithm design technique that can be used to solve optimization problems including finding the width or diameter of a set of points.
The method can be visualized by considering a thin vertical rectangle at x with height f(x) − g(x), and revolving it about the y-axis; it forms a cylindrical shell. The lateral surface area of a cylinder is 2πrh, where r is the radius (in this case x), and h is the height (in this case f(x) − g(x)). Summing up all of the surface areas ...