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Cissoid of Diocles traced by points M with ¯ = ¯ Animation visualizing the Cissoid of Diocles. In geometry, the cissoid of Diocles (from Ancient Greek κισσοειδής (kissoeidēs) 'ivy-shaped'; named for Diocles) is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio.
This is a rotation around the vector (x, y, z) by an angle 2θ, where cos θ = w and |sin θ| = ‖ (x, y, z) ‖. The proper sign for sin θ is implied, once the signs of the axis components are fixed. The 2:1-nature is apparent since both q and −q map to the same Q.
Rotation formalisms are focused on proper (orientation-preserving) motions of the Euclidean space with one fixed point, that a rotation refers to.Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame, or motions of a joint), this approach creates a knowledge about all motions.
The resulting orientation of Body 3-2-1 sequence (around the capitalized axis in the illustration of Tait–Bryan angles) is equivalent to that of lab 1-2-3 sequence (around the lower-cased axis), where the airplane is rolled first (lab-x axis), and then nosed up around the horizontal lab-y axis, and finally rotated around the vertical lab-z ...
Then multiplying the numerator and denominator inside the square root by (1 + cos θ) and using Pythagorean identities leads to: = + . Also, if the numerator and denominator are both multiplied by (1 - cos θ), the result is:
Sets of rotation axes associated with both proper Euler angles and Tait-Bryan angles are commonly named using this notation (see above for details). Sometimes, the same sequence is simply called z-x-z, Z-X-Z, or 3-1-3, but this notation may be ambiguous as it may be identical to that used for extrinsic rotations. In this case, it becomes ...
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle ) is called the reference plane (sometimes fundamental plane ).
Cnoidal wave solution to the Korteweg–De Vries equation, in terms of the square of the Jacobi elliptic function cn (and with value of the parameter m = 0.9). Numerical solution of the KdV equation u t + uu x + δ 2 u xxx = 0 (δ = 0.022) with an initial condition u(x, 0) = cos(πx).