Feature point detection based on convolutional neural network (CNN) has been studied widely. The effective approaches for improving detection accuracy are building a deeper network or using a multi‐network cascade structure. However, some potential capacity of CNN has not been excavated. In this study, the authors mainly analyse several factors influencing CNN performance from two aspects: (i) the position relationships between feature points and (ii) the normalisation methods of coordinates. Whether the network can learn the position relationships is also studied. For extracting the deep features of images, a network containing three convolution layers is constructed. The specific geometric relationship constraints are applied during calibration to maximise the capability of the CNN for learning the position relationship between feature points. Considering that different feature points only appear in various local regions of an image, local normalisation is proposed, which increases the mapping scope of the feature points and decreases the mapping error. The experimental results prove that the specific position relationship and local normalisation obviously improve the feature point detection based on CNN. At the detection error of 5%, the average detection accuracy of eyelid feature points is improved by 7.1% and single‐point detection receives a high accuracy of 97.96%.