The accuracy of eye gaze estimation using image information is affected by several factors which include image resolution, anatomical structure of the eye, and posture changes. The irregular movements of the head and eye create issues that are currently being researched to enable better use of this key technology. In this paper, we describe an effective way of estimating eye gaze from the elliptical features of one iris under the conditions of not using an auxiliary light source, a head fixing equipment, or multiple cameras. First, we provide preliminary estimation of the gaze direction, and then we obtain the vectors which describe the translation and rotation of the eyeball, by applying a central projection method on the plane which passes through the line-of-sight. This helps us avoid the complex computations involved in previous methods. We also disambiguate the solution based on experimental findings. Second, error correction is conducted on a back propagation neural network trained by a sample collection of translation and rotation vectors. Extensive experimental studies are conducted to assess the efficiency, and robustness of our method. Results reveal that our method has a better performance compared to a typical previous method.