Heart rate variability (HRV) is concerned with the analysis of the intervals between heartbeats. An emerging analysis technique is the Poincaré plot, which takes a sequence of intervals and plots each interval against the following interval. The geometry of this plot has been shown to distinguish between healthy and unhealthy subjects in clinical settings. The Poincaré plot is a valuable HRV analysis technique due to its ability to display nonlinear aspects of the interval sequence. The problem is, how do we quantitatively characterize the plot to capture useful summary descriptors that are independent of existing HRV measures? Researchers have investigated a number of techniques: converting the two-dimensional plot into various one-dimensional views; the fitting of an ellipse to the plot shape; and measuring the correlation coefficient of the plot. We investigate each of these methods in detail and show that they are all measuring linear aspects of the intervals which existing HRV indexes already specify. The fact that these methods appear insensitive to the nonlinear characteristics of the intervals is an important finding because the Poincaré plot is primarily a nonlinear technique. Therefore, further work is needed to determine if better methods of characterizing Poincaré plot geometry can be found.