Hotspot detection is an important exploratory technique to identify areas with high concentrations of crime and help deploy crime-reduction resources. Although a variety of methods have been developed to detect crime hotspots, few studies have systematically evaluated the performance of various methods, especially in terms of the ability to detect complex-shaped crime hotspots. Therefore, in this study, a comparative study of hotspot detection approaches while simultaneously considering the concentration and shape characteristics was conducted. Firstly, we established a framework for quantitatively evaluating the performance of hotspot detection for cases with or without the "ground truth". Secondly, accounting for the concentration and shape characteristics of the hotspot, we additionally defined two evaluation indicators, which can be used as a supplement to existing evaluation indicators. Finally, four classical hotspot-detection methods were quantitatively compared on the synthetic and real crime data. Results show that the proposed evaluation framework and indicators can describe the size, concentration and shape characteristics of the detected hotspots, thus supporting the quantitative comparison of different methods. From the selected methods, the AMOEBA (A Multidirectional Optimal Ecotope-Based Algorithm) method was more accurate in describing the concentration and shape characteristics and was powerful in discovering complex hotspots.