![]() See the following Matlab code to perform some rotations clear allīut remember with 2D, the rotation matrix does fairly a good job however, with 3D the importance of complex numbers represented via quaternion is significant. As you can see no trigonometric functions are being used. multiply the X part of the vector by -1, and then swap X and Y values. no need for an angle), you basically multiple it by a complex number $z=i$, the rotated vector is then (0,1). Normally, rotating vectors requires matrix math, but there is a really simple trick for rotating a 2D vector by 90 clockwise: multiply the X part of the vector by -1, then swap the X and Y values. For example, if you need to rotate a vector (1,0) and make it points upward (i.e. after represents it as a complex number) by a complex number that represents the rotation. This technique is so simple that you can even use it to rotate vectors by hand on paper The imaginary number i is the square root of -1. With complex numbers you could avoid that by multiplying the vector (i.e. In this post I share a technique that lets you use imaginary numbers (complex numbers more specifically) to be able to rotate vectors in 2d. sine and cosine with an independent variable known as the angle) which internally implemented by Taylor series (i.e. ![]() In the rotation matrix, you actually use trigonometric functions (i.e. The rotation options form can be activated from the tool icon on the Drawing Tab. Another way to see it is that complex numbers have only two degrees of freedom, while 2×2 matrices have four degrees of freedom. Selected items in the 2D View can be rotated to a new orientation using this tool. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO (3), the group of all rotation matrices. Matrices can represent those, but also nonuniform scaling and shearing. In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. For the sake of exmaple, lets say we keep y fixed. Now I want to apply that same rotation to a 3D vector, keeping 1 chosen dimension fixed. Complex numbers can only represent rotation and uniform scaling. From linear algebra, I know that the matrix will rotate a 2D vector v by 90 degrees clockwise when I take the matrix-vector product Mv. This corresponds to rotation by the phase of $z$ combined with scaling by the magnitude of $z$. Both methods end up doing the same calculations when you break it down.
0 Comments
Leave a Reply. |