We investigate the complex geometry and Eulerian Beta--Gamma arc-length parameterization of the normalized Dirichlet eta terms. Whereas individual terms on the critical line undergo a nearly sixfold modulus oscillation driven by the Euler prefactor, the two-term partial sum $S_2(s) = t_1(s) + t_2(s)$ traces an exact M"obius circle for every fixed $sigma = operatorname{Re}(s) eq 1$; this circle is centered at the origin if and only if $operatorname{Re}(s) = 1/2$, where it reduces to a scaled Blaschke factor of constant radius. Factoring across odd--even dyadic pairs $P_m(s) = t_m(s) + t_{2m}(s)$ eliminates prefactor amplitude oscillations entirely, collapsing precessing annular rosettes into concentric origin-centered circles on the critical line with strictly decreasing radii. Removing all odd--even pairs leaves the complementary multiples-of-$4$ terms. We prove that on the critical line, the anchor term $t_4(s)$ traces an exact Lemniscate of Bernoulli with foci at $0$ and $1$ (uniquely for $operatorname{Re}(s) = 1/2$), pure powers of $2$ trace closed periodic multi-loop rational curves, and non-powers of $2$ trace precessing lemniscate rosettes. Meanwhile, the dyadic differences $D_{4m}(s) = t_{4m}(s) - t_{8m}(s)$ collapse both families into exact origin-centered circles, partitioning the series termwise and decomposing $zeta(s)$ via analytic continuation. Finally, we establish a pointwise velocity--modulus duality linking the central polar angle $alpha(b)$ of $t_4(1/2 + ib)$ to its fundamental perimeter $L(t_4)$, thereby parameterizing every individual term $t_n(s)$ and every concentric dyadic circle on the critical line in terms of the regularized inverse incomplete Beta function. As a geometric consequence, critical-line zeros are reframed as ordinates $b$ where the resultant of the inner concentric circles lands on the antipode of the initial circle of constant radius.