Three-dimensional (3D) imaging technology obtains massive points (MPs) of a surface in a very short time. In this study, we recognize points on 3D edge contour lines (ECLs) from MPs of a surface. First, the position features of the points on the ECLs, which are called edge contour points (ECPs), are analyzed, and it is proposed to detect ECPs from the vertical section lines (VSLs) of the ECLs. Then, the position of an ECP needs to be calculated based on the middle point of the chamfered arc (MPCA) on a VSL. Considering the distribution character of points on the CA (PsCA), three methods are proposed here to recognize the MPCA: parabola fitting for its vertex method, calculating the accumulated normal angle (AcNA) of each PsCA for the point whose AcNA is closest to the half central angle of the CA, and calculating the approximate arc length (AAL) of each PsCA for the point whose AAL is closest to the half arc length of the CA. Finally, the effectiveness and accuracy of the three methods are verified on several ECLs.