rotation in two dimensions
of a point (x,y)
of angle t radians
around the origin (0,0)

X = cost x - sint y
Y = sint x + cost y

1748 Euler
Z = z.e^it = (x+iy)(cost+isint) = X + iY


1775 Euler
rotation of a point r=(x,y,z) around an axis n, passing through origin
with directional cosines (a,b,c) such that a2+b2+c2=1

X =   (a2+(1-a2)cost)  x + (ab(1-cost)-csint) y + (ac(1-cost)+bsint) z
Y = (ba(1-cost)+csint) x +   (b2+(1-b2)cost)  y + (bc(1-cost)-asint) z
Z = (ca(1-cost)-bsint) x + (cb(1-cost)+asint) y +   (c2+(1-c2)cost)  z

it is basically along three directions n, r and nxr=( bz-cy, cx-az, ay-bx )

X=(1-cost)a(ax+by+cz)+(cost)x+(sint)(bz-cy)
Y=(1-cost)b(ax+by+cz)+(cost)y+(sint)(cx-az)
Z=(1-cost)c(ax+by+cz)+(cost)z+(sint)(ay-bx)

in vector notation
R=(1-cost)(n.r)n+(cost)r+(sint)(nxr)

using BAC-CAB identity (Lagrange 1773) we can rewrite it
along 3 directions r, nxr, nx(nxr)
R=r+(sint)(nxr)+(1-cost)(nxnxr)
not orthogonal yet

n nxr nx(nxr) orthogonal



1840 Rodrigues: more symmetry through half angle t/2

l=asin(t/2)
m=bsin(t/2)
n=csin(t/2)
r=cos(t/2)
l2+m2+n2+r2=1

X = (r2+l2-m2-n2) x +   2(lm-rn)    y +   2(ln+rm)    z
Y =   2(lm+rn)    x + (r2-l2+m2-n2) y +   2(mn-rl)    z
Z =   2(ln-rm)    x +   2(mn+rl)    y + (r2-l2-m2+n2) z 

this is quaternions 3 years before Hamilton!


1843 Hamilton: quaternions

quaternion sandwich
R= q.r.q' the same as Rodrigues



1858 Cayley: Matrix

      bz-cy    0 -c  b     x
nxr = cx-az =  c  0 -a  *  y = Kr
      ay-bx   -b  a  0     z

     -(b2+c2)    ab       ac        a2-1  ab   ac
K2 =    ab    -(a2+c2)    bc     =   ab  b2-1  bc
        ac       bc    -(a2+b2)      ac   bc  c2-1

K3 = -K

R = (I + sint K + (1-cost) K2) r


1827 Mobius
Barycentric coordinates

1844 Grassmann: wedge product, bivectors
he defined a directed line segment as a subtraction of points
then he defined wedge product and bivectors

1878 Cliford: rotation in 4 dimensions, rotors, e^-B/2 r e^B/2
He died next year! 

1880 Gibbs