Coniflex cutting, a popular mass production method for straight bevel gears, employs two giant interlocked circular cutters to generate tooth surfaces. By controlling the tool pressure angle, Coniflex cutting enables profile and lengthwise crownings that result in advantageously low assembly sensibility. Coniflex gears are therefore widely used in the manufacture of a variety of gearboxes. This method is generally used on a Gleason dedicated traditional machine. Although the new CNC machine can implement this method through Gleason software, the technical details of their application are never revealed. Gleason has now stopped producing traditional machines. But the durability, low cost, and ease of setting up these machines have ensured their continued widespread use in the industry. As a result, their users, having no mathematical model of tooth surfaces, have traditionally reduced manufacturing errors by observing the position of the tooth contact bearing and applying a trial-and-error technique. This method is not only time consuming but also results in unpredictable contact performance. To address these issues, this paper proposes a mathematical model of a Coniflex bevel gear produced on a dedicated machine, whose coordinate systems between cutters and work gear are empirically well-defined. This mathematical model can also be used to derive the cutting position of the CNC machine. Once the tooth surface is derived from coordinate transformation and gear enveloping theory, ease off and tooth contact analysis can be conducted numerically; after which flank correction is achieved using sensitivity analysis and optimization methods. Both the proposed model and the flank correction are validated using cutting experiments on the Gleason Number 104 Straight Bevel Coniflex^® Generator.