On-machine measurement by a line laser scanner has a great potential in machining large complex components in aerospace and automation applications. This paper proposes a novel approach for path planning of measuring curved surfaces in a 5-axis machine tool, which takes scanning overlap, visual angle, and working distance into consideration. A conformal mapping algorithm is employed to transform a three-dimensional (3D) mesh surface to a 2D parametric plane. The shortest equidistant paths calculated in the 2D plane are transformed into smooth collision-free scanning paths in the 3D surface. The experiments verify that this method can improve the efficiency by 21.4% and the accuracy by 24.9%.