Previous research has demonstrated particular inadequacies in conventional methods used to identify outlier cases in bivariate regression models. Only through a combination of methods can one detect all of the deviant points potentially overinfluencing a regression model's parameters. This paper investigates whether a range of data points might exist that could be influential to a regression estimate, yet remain undetected by current methods. A computer program written for prior research (DRAWREG) was expanded to provide graphical displays of rejection boundaries according to the standardized, studentized, leverage, and Cook's Distance methods. Overlapping these regions revealed four geometric areas where influential points might exist, yet remain outside the usual detection frames. The traditional homogeneity assumption underlying least-squares regression (a rectangular pattern) did not agree with the usual graphic representation of bivariate relationships (an ellipsoid). The DRAWREG program was enhanced to compute and display an ellipsoid around the regression line and perpendicular coordinates. While further work is needed to clarify recommended boundaries for this region, it has potential as a single, adequate measure for influential point detection. Ten figures illustrate the discussion. A 12-item list of references and an appendix listing formulas are included. (SLD)