Pursuing efficient text shape representations helps scene text detection models focus on compact foreground regions and optimize the contour reconstruction steps to simplify the whole detection pipeline. Current approaches either represent irregular shapes via box-to-polygon strategy or decomposing a contour into pieces for fitting gradually, the deficiency of coarse contours or complex pipelines always exists in these models. Considering the above issues, we introduce EdgeText to fit text contours compactly while alleviating excessive contour rebuilding processes. Concretely, it is observed that the two long edges of texts can be regarded as smooth curves. It allows us to build contours via continuous and smooth edges that cover text regions tightly instead of fitting piecewise, which helps avoid the two limitations in current models. Inspired by this observation, EdgeText formulates the text representation as the edge approximation problem via parameterized curve fitting functions. In the inference stage, our model starts with locating text centers, and then creating curve functions for approximating text edges relying on the points. Meanwhile, truncation points are determined based on the location features. In the end, extracting curve segments from curve functions by using the pixel coordinate information brought by truncation points to reconstruct text contours. Furthermore, considering the deep dependency of EdgeText on text edges, a bilateral enhanced perception (BEP) module is designed. It encourages our model to pay attention to the recognition of edge features. Additionally, to accelerate the learning of the curve function parameters, we introduce a proportional integral loss (PI-loss) to force the proposed model to focus on the curve distribution and avoid being disturbed by text scales.