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Let u = (x 1, y 1) and v = (x 2, y 2). Consider the restrictions on x 1, x 2, y 1, y 2 required to make u and v form an orthonormal pair. From the orthogonality restriction, u • v = 0. From the unit length restriction on u, ||u|| = 1. From the unit length restriction on v, ||v|| = 1. Expanding these terms gives 3 equations:
For example, the three-dimensional Cartesian coordinates (x, y, z) is an orthogonal coordinate system, since its coordinate surfaces x = constant, y = constant, and z = constant are planes that meet at right angles to one another, i.e., are perpendicular. Orthogonal coordinates are a special but extremely common case of curvilinear coordinates.
Thus, the vector is parallel to , the vector is orthogonal to , and = +. The projection of a onto b can be decomposed into a direction and a scalar magnitude by writing it as a 1 = a 1 b ^ {\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} } where a 1 {\displaystyle a_{1}} is a scalar, called the scalar projection of a onto b , and b̂ is ...
If we condense the skew entries into a vector, (x,y,z), then we produce a 90° rotation around the x-axis for (1, 0, 0), around the y-axis for (0, 1, 0), and around the z-axis for (0, 0, 1). The 180° rotations are just out of reach; for, in the limit as x → ∞ , ( x , 0, 0) does approach a 180° rotation around the x axis, and similarly for ...
A square matrix is called a projection matrix if it is equal to its square, i.e. if =. [2]: p. 38 A square matrix is called an orthogonal projection matrix if = = for a real matrix, and respectively = = for a complex matrix, where denotes the transpose of and denotes the adjoint or Hermitian transpose of .
Consequently, the transformation matrix Q θ for rotations about the x-axis through an angle θ may be written in terms of Pauli matrices and the unit matrix as [6] Q θ = 1 cos θ 2 + i σ x sin θ 2 . {\displaystyle Q_{\theta }={\boldsymbol {1}}\,\cos {\frac {\theta }{2}}+i\,\sigma _{x}\sin {\frac {\theta }{2}}.}
The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre [3] as the coefficients in the expansion of the Newtonian potential | ′ | = + ′ ′ = = ′ + (), where r and r′ are the lengths of the vectors x and x′ respectively and γ is the angle between those two vectors.
Hence, coefficients can also be found by solving a linear system, for instance by matrix inversion. Fast algorithms to calculate the forward and inverse Zernike transform use symmetry properties of trigonometric functions, separability of radial and azimuthal parts of Zernike polynomials, and their rotational symmetries.