Recently, uniformly valid asymptotic solutions of rod eutectic growth for liquid-solid interface slopes at triple phase junction being the normal order have been presented in reference (Li and Xu, 2017). In this work, analytical calculations of rod eutectic growth during directional solidification for liquid-solid interface of small order are performed. The purpose is to study the liquid-solid interfacial shape and average undercooling of rod eutectic growth. We give the mathematical formulation of rod eutectic growth for the general case. The steady solutions of rod eutectic growth are obtained by using asymptotic expansion method and matching method. The relationship of the overall average interfacial undercooling with the rod eutectic spacing is presented. The overall average interfacial undercooling always exists a minimum value at the minimum-undercooling spacing, and the minimum-undercooling spacing is slightly smaller than one given by Jackson and Hunt. We also analyze the effect of pulling speeds and thermal gradients on the interfacial shapes for the case of the minimum-undercooling spacing.