Proposition. Translations and rotations form an exact plane-motor family [fcap-002S]
Proposition. Translations and rotations form an exact plane-motor family [fcap-002S]
Retain the real degenerate frame and projective encodings of
Convention [fcap-002Q]. Put \(E=e_{01}\). Since \(E^2=0\) and
\(\widetilde E=-E\), the one-axis motor
\[M_d=1-dE\]
satisfies
\[M_d^{-1}=\widetilde M_d=1+dE,\qquad
M_sM_t=M_{s+t}.\]
It belongs to the actual vector-generated Spin carrier because
\[M_d=e_1(e_1+de_0),\qquad
Q(e_1)=Q(e_1+de_0)=1\]
CliffordAlgebra.ι_mul_ι_mem_spinGroup_of_norm_mul_norm_eq_one.
Direct sandwich multiplication gives
\[\begin {aligned}
M_dP(w,x,y)M_d^{-1}&=P(w,x+2dw,y),\\
M_dL(d_0,a,b)M_d^{-1}&=L(d_0-2da,a,b).
\end {aligned}\]
Thus the radical shear of Example [fcap-002N] is, in the fixed projective
encoding, translation by \((2d,0)\). The packaged nondegenerate transvection
theorem is not used.
For a general translation vector \(t=(t_x,t_y)\), define \[T(t)=1-\frac {t_x}{2}e_{01}+\frac {t_y}{2}e_{20}.\] Each increment squares to zero, and their products in either order vanish, so \[T(t)T(v)=T(t+v),\qquad T(t)^{-1}=T(-t).\] More precisely, \[T(t)= \left [e_1\left (e_1+\frac {t_x}{2}e_0\right )\right ] \left [e_2\left (e_2+\frac {t_y}{2}e_0\right )\right ].\] Every displayed vector has quadratic value \(1\); hence \(T(t)\) lies in the same actual Spin carrier. This is a direct factorization in the full degenerate Clifford algebra, not membership in a radical quotient.
Let \(c^2+s^2=1\) and put \[U=c-se_{12}=e_1(ce_1-se_2),\qquad U^{-1}=\widetilde U=c+se_{12}.\] Again both vector factors have quadratic value \(1\). Its active spatial matrix is \[R_U= \begin {pmatrix} c^2-s^2&-2cs\\ 2cs&c^2-s^2 \end {pmatrix}.\] For the ordered factorization \(M(t,U)=T(t)U\), which applies the rotation before the translation, write \(p=(x,y)\) and \(n=(a,b)\). Then \[\begin {aligned} (w,p)&\longmapsto (w,R_Up+wt),\\ (d,n)&\longmapsto (d-t\mathbin {\cdot }R_Un,R_Un). \end {aligned}\] The second row is the line action. It preserves incidence because \[(d-t\mathbin {\cdot }R_Un)w +(R_Un)\mathbin {\cdot }(R_Up+wt)=dw+n\mathbin {\cdot }p.\] In particular, translations fix every ideal point \((0,p)\). The selected family is closed, with exact order and inverse \[\begin {aligned} M(t,U)M(v,V)&=M(t+R_Uv,UV),\\ M(t,U)^{-1}&=M(-R_U^{-1}t,U^{-1}). \end {aligned}\] These formulas prove the claims for this family; they do not assert that every motor is a single exponential.
Sokolov's negative-half-angle rotor and nilpotent translator fix the sign and half-factor in this convention [sokolov2013foundations, (51)--(52), p. 39]. Independently, [dereli2010degenerate, Lemma 2.2, p. 567] proves \(\exp (aF)=1+aF\) for a radical vector \(F\); taking \(F=e_0\) and \(a=(t_xe_1+t_ye_2)/2\) gives exactly the increment of \(T(t)\). That paper later defines a quotient carrier [dereli2010degenerate, Definition 2.4, p. 570]; no such quotient is imported here. The vector-pair factorizations above establish membership in the carrier actually used in these notes.