How to "diagonally" think about rotations.



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14 thoughts on “How to "diagonally" think about rotations.”

  1. You have to have both cos and sin to know the correct sense of the rotation.
    EDIT: This is a real-world issue e.g. in Doppler radar. You need to duplicate the analogue circuitry, known as In-phase and Quadrature channels. In this case you need the quadrature channel to distinguish approaching and receding targets.

    Actually for some purposes transcendental functions are overkill. You can parametrise
    cos(theta) = (1-t^2)/(1+t^2)
    sin(theta) = 2 t/ (1+t^2)
    where
    t = tan(theta/2)
    t real on (-inf, inf)
    There's a problem with theta = +/- 180 deg where t is singular. In practice just take cos(180 deg)= -1, sin(180 deg)=0.
    EDIT: Or take homogenous parameters t0, t1 such that t=t1/t0. Then
    cos(theta) = (t0^2-t1^2)/(t0^2+t1^2)
    sin(theta) = 2 t0 t1/(t0^2+t1^2)
    The 180 deg case is (t0, t1) = (0, 1)

  2. For proving that R_theta is a rotation matrix, isn’t it easier to write v = (r cos(phi), r sin(phi)) and apply trigonometric identities to show that R_theta v = (r cos(phi+theta), r sin(phi+theta))? That gets you around the issues with whether cos is properly injective too.

  3. Nice video Michael! Another nice picture for this is the Bloch sphere / Pauli algebra one from quantum mechanics, though that might be a stretch for Linear Algebra students. There the standard rotation matrix shows up as an exponential of the Pauli Y matrix (rotation around y axis of the Bloch sphere), and the two orthogonal eigenvectors reappear as the two antipodal points on the y axis of the sphere.

  4. Plane rotations can be viewed as a 1D projective space, i.e. line, known as an S1, with basis points 1 and k where k is a quaternion. The general rotation is
    u = cos(theta/2) + k sin(theta/2)
    To compound rotations u and v you take the product vu. v here can be understood as a projectivity on the S1.
    To rotate a plane point (x,y) by u you make the quaternion
    q = 1 + ix + jy
    and then
    q -> u q u*
    where * is quaternion conjugation.
    This generalises pretty well. In fact ALL lines in the projective 3-space, or S3, of the quaternions represent rotations about some axis in Euclidean 3-space (strictly an axis through the origin).
    Sorry to unload on Michael's channel, I just find it a fascinating topic.

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