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四元法

四元法


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  • linear algebra - How can one intuitively think about quaternions . . .
    The unit quaternions also act via left and right multiplication as rotations of the 4d space of all quaternions This gives a homomorphism from SU(2) × SU(2) onto the 4d rotation group SO(4) The kernel of this homomorphism is {±(1, 1)}, so we see SU(2) × SU(2) is a double cover of SO(4)
  • Understanding quaternions - Mathematics Stack Exchange
    Quaternions have real and imaginary parts, or one may call them a scalar and vector part That is, we can interpret $\mathbb{H}$ (named after Hamilton) as $\mathbb{R}\oplus\mathbb{R}^3$ We already know how to multiply a scalar by a scalar, and a vector by a scalar, so it remains to describe how to multiply two 3D vectors
  • Quaternions +Geometric (Clifford) Algebra: What Is the Proper . . .
    IV Historical Fun Facts About Quaternions and the Truth About Maxwell Theory Oliver Heavside and his side-kick Gibbs back in the day called Quaternions, “pure evil”, and “the work of the devil”… no joke! True story! I reference Grant Sanderson and Ben Eater’s YouTube video The reason why vector calculus won the day back in the
  • 3d - Averaging quaternions - Mathematics Stack Exchange
    If quaternions represent similar rotations, and the quaternions are normalized, and a correction has been applied for the "double-cover problem", then the quaternions can be directly averaged and then the result normalized again, treating them as 4-dimensional vectors, to produce a quaternion representing a roughly-average rotation
  • matrices - Matrix representation of the Quaternions? - Mathematics . . .
    $\begingroup$ Maybe helpful: Think of the Quaternions as a four dimensional vector space over $\mathbb{R}$ with basis ${1,i,j,k}$ Then think of the natural action of the Quatenions on this vector space
  • complex numbers - What exactly does a quaternion represent . . .
    Unit quaternions can be identified with rotations of three-dimensional space, which is often the best way to think about them Specifically, take a point in the three-dimensional sphere If it's either the origin of three p-space or the extra point, it represents the trivial rotation
  • Concise description of why rotation quaternions use half the angle
    The quaternions can do such a generalization identifying a $3D$-vector with a pure imaginary quaternion
  • Quaternions multiplication order (to rotate unrotate)
    Multiplication by quaternions in the conventional way transforms the world coordinates of whatever object you apply it to In order to achieve the effect of first performing the parent rotation, then performing the child rotation relative to the rotated coordinate system that resulted from the "parent" rotation, you should apply the "child
  • Quaternions: why does ijk = -1 and ij=k and -ji=k
    Stack Exchange Network Stack Exchange network consists of 183 Q A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers





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