The physical mechanism leading to the formation of the blue loop in the Hertzsprung-Russell (HR) diagram is not satisfactorily explained by the evolutionary track of single stars. Rapid rotation and low metallicity drastically modify the internal structures and surface compositions of stars. Therefore, they provide a very significant pattern to investigate the evolutionary properties of the blue loop. In this paper, we mainly explore how rapid rotation and low metallicity have an important impact on the occurrence and extension of the blue loop. To this end, we implemented the rotating stellar evolution model, including the angular momentum transportation and chemical element mixing. We incorporated several initial rotational velocities and two characteristic metallicities in various models to explore the blue loop extension. The blue loop can occur when the hydrogen burning shell merges with the hydrogen--helium abundance discontinuity. We find that the blue loop extension strongly depends on the amplitude and gradient of the hydrogen--helium discontinuity. The hydrogen--helium discontinuity is created by the intermediate convective region or the convective dredge-up. A steeper hydrogen gradient in association with a greater amplitude of the hydrogen abundance discontinuity may favour a hotter star. Both the low metallicity and rapid rotation tend to restrain the development of the outer convective envelope and thus disfavour the occurrence and extension of the blue loop. There are three main reasons for this occurrence. Firstly, the helium core and its core potential can be enlarged by rotational mixing or low metallicity. Secondly, rapid rotation reduces the convective dredge-up depth in the star with $ Z=0.014$ and the mass extension of the intermediate convective region in the star with $ Z=0.0008$. Both of these phenomena lead to a reduction of the amplitude of the hydrogen abundance gradient. Thirdly, strong rotational mixing in the model (i.e. $ ini =350$ Km/s) with $ Z=0.0008$ reduces the energy generation rate from the hydrogen burning shell. Without bending towards higher effective temperature in the HR diagram, the additional helium brought near the H-burning shell associated with the larger He core can cause the star to expand towards becoming a red giant star directly after the core hydrogen burning. Rapid rotation and low metallicity tend to produce surface enrichment of the ratio of nitrogen to carbon and reduce the $ C$ left in the core; this has an important influence on the stellar compactness of the supernovae progenitor.

Euclid preparation XXXVI. Modelling the weak lensing angular power spectrum