Mathematical Modeling of Eyebrow Curvature

The aim of the study is to describe a mathematical model for analyzing eyebrow curvature that can be applied broadly to curvilinear facial features. A total of 100 digital images (50 men, 50 women) were obtained from standardized headshots of medical professionals. Images were analyzed in ImageJ by plotting either 8 or 15 points along the inferior-most row of contiguous brow cilia. A best-fit curve was automatically fit to these points in Microsoft Excel. The second derivative of the second-degree polynomial and a fourth-degree polynomial were used to evaluate brow curvature. Both techniques were subsequently compared with each other. A second-degree polynomial and fourth-degree polynomial were fit to all eyebrows. Plotting 15 points yielded greater goodness-of-fit than plotting 8 points along the inferior brow and allowed for more sensitive measurement of curvature across all images. A fourth-degree polynomial function provided a closer fit to the eyebrow than a second-degree polynomial function. This method provides a simple and reliable tool for quantitative analysis of eyebrow curvature from images. Fifteen-point plots and a fourth-degree polynomial curve provide a greater goodness-of-fit. The authors believe the described technique can be applied to other curvilinear facial features and will facilitate the analysis of standardized images.