In the paper, the chaotic characteristics of two functions are studied by a quadratic surface mapping in spatial unit area. When a surface is the standard surface in spatial unit area and another surface is generated randomly, the probability that the two functions are in the chaos can be greater than one-tenth, so this is a better method of generating chaos. The chaotic characteristics are analyzed by calculating the Lyapunov exponent and drawing the bifurcation diagram. According to the bifurcation diagram of the changing parameter and the characteristics of the regional distribution of the chaotic surface control points, the chaotic mapping function can be found and a lot of two-dimensional chaotic attractor graphics can be obtained. Besides, gray scale image is regarded as a discrete two-dimensional function for the first time. The study of image as an iteration expression shows some chaotic characteristics. The study shows that the same or similar image converges to the cycle point easily, which can be used in some research areas such as image recognition.