In the paper, the chaotic characteristic of the quadratic function in plane unit area is studied, and it is found that the standard quadratic mapping is Li-Yorke chaos, and also Devaney chaos, and that under certain conditions, there are a large number of quadratic functions that are chaotic. Some quadratic functions can transform into the standard quadratic functions by moving and zooming, without changing their chaotic characteristics. In addition, non-standard quadratic function is preliminary studied. The chaotic characteristic of the quadratic curve is analyzed by calculating Lyapunov exponents and drawing the bifurcation diagram of conic. The bifurcation diagram of the parameter variation and the area distributing diagram of parameter control points have certain research value. The study also shows that more conic curve cross iteration can generate a better chaotic sequence, and the chaotic sequence can be used to image encryption and other practical purposes.