Lead curve detection in design drawings is a critical problem in a wide range of applications ranging from checking similar drawings in patent granting to constructing hyperlinks between image and text description in digitalization. The difficulty of the problem are two folds: unknown end point of lead curve and complex crossings. However, most previous curve detection algorithms are usually applied in simple or no crossing situations. We make four contributions in solving the problem: (1) we transform the problem into a new problem that finds an optimal path with the best score in the cross-point graph. We introduce the "cross-point graph" representation which captures the topology of cross-point connectedness. Based on the original drawing and the corresponding cross-point graph, we introduce the coupling concept "curve-path", which correlates the curve in the original drawing with the corresponding path in the cross-point graph. (2) we design a set of joint feature representations for curve-path which describes different characteristics of a curve and its corresponding path. (3) we define a task specific loss function for our customized structured SVM. We propose a mixed negative instance sampling strategy to learn the weights of different joint feature representations. We prune the search space effectively for fast lead curve detection. (4) we build a software to efficiently facilitate manual lead curve labeling. We release the patent drawing dataset with groundtruth to public for lead curve detection research. The extensive experimental results prove the effectiveness of our methods.