Abstract
This study focuses on the dynamic pattern of heart rate variability in the frequency range of respiration, the socalled respiratory sinus arrhythmia. Forty experimental time series of heart rate data from four healthy adult volunteers undergoing a paced respiration protocol were used as an empirical basis. For pacingcycle lengths >8 s, the heartbeat intervals are shown to obey a rule that can be expressed by a onedimensional circle map (nextangle map). Circle maps are introduced as a new type of model for time series analyses to characterize the nonlinear dynamic pattern underlying the respiratory sinus arrhythmia during voluntary paced respiration. Although these maps are not chaotic, the dynamic pattern shows typical imprints of nonlinearity. By starting from a piecewise linear model, which describes the different circle maps obtained from the empirical time series for various pacing frequencies, time invariant measures can be introduced that characterize the dynamic pattern of heart rate variability during voluntary slowpaced respiration.
 sinus arrhythmia
 cardiovascular model
 periodicity
 system physiology
 nonlinear dynamics
the variation of heart rate in the frequency range of respiration, known as respiratory sinus arrhythmia (RSA), was already described by Ludwig in 1847 (18). Since then, many studies have been carried out to explore the underlying physiological mechanisms (17,19). The most important ones are the modulation of cardiac filling pressure by respiratory movements (1), the direct respiratory modulation of parasympathetic and sympathetic neural activity in the brain stem (10), and the respiratory modulation of the baroreceptor feedback control (9).
There are different models summarizing the comprehensive results of the RSA analysis. Whereas mechanistic models (6, 22, 23, 26,28) support the qualitative understanding of the interplay of the RSAgenerating mechanisms, time series analyses (2, 23, 24, 28) result in quantitative estimates that are phenomenologically related to the strength of autonomic cardiac chronotropic innervation. Estimates based on linear time series analyses focus on amplitudes in either the time or the frequency domain (3, 15, 21). They can be used to quantify the strength of coupling between the respiratory and the cardiovascular system and, in particular, to estimate the sympathetic/parasympathetic balance (8).
As has already been pointed out by various authors (5, 16, 20), the dynamics of cardiorespiratory synchronization features essential nonlinearities that cannot be described by using these estimates. We hypothesize that the characteristic pattern of successive heartbeat intervals within one respiratory cycle (pattern of the RSA) can be used to quantify additional information about the nonlinear dynamics of the cardiac chronotropic control.
The theory of nonlinear dynamic systems provides many methods for the identification and classification of the dynamic pattern in an observed time series. Because biological and physiological systems are characterized by stable dynamic structures far from equilibrium, such methods can be used to analyze these systems (7, 11).
Complexity measures have been introduced to differentiate distinct geometric structures that are representations of the dynamics in an abstract (embedding) space (12, 16, 30). This differentiation was possible even in those cases in which the dynamic structure (the attractors) cannot be identified. One intrinsic drawback of this approach is the difficulty of giving a physiological interpretation of the complexity measures. However, these measures are already of clinical relevance. With regard to cardiology, these measures are discussed in the context of risk stratification of sudden cardiac death. With a significance level of ∼90%, Skinner et al. (27) showed that, for the heartbeat intervals, a reduction of the correlation dimension (a special complexity measure related to the dimension of the attractor) precedes ventricular fibrillation in highrisk patients by hours. However, most applications of nonlinear time series analyses in cardiology do not pay special attention to the cardiorespiratory coupling, which manifests itself in the RSA.
The complexity of spontaneous respiratory movements obviously precludes the identification of finite dimensional attractors in heart rate fluctuations within the frequency range of breathing (14). One way to overcome this problem is to introduce paced breathing. This standardized condition has already been used in physiological and clinical studies to analyze and quantify the RSA (4, 15, 20). The main result of the present paper is that heart rate fluctuations during voluntarily induced slowpaced breathing obey a onedimensional, nonlinear law of motion. Neither the structure of the onedimensional deterministic process nor the deviation from it depends on the mean RR interval length or the RSA amplitude. However, they do depend on the respiratory cycle length and vary significantly between individuals. For all subjects, the onedimensional law of motion breaks down for cycle lengths close to that of spontaneous breathing.
The onedimensional structure becomes apparent when the dynamics of the amplitudes of the RSA is separated from its form. This can be achieved by introducing circle maps. The circle map results from the introduction of polar coordinates into the twodimensional scatter plot (embedding) of successive heartbeat intervals. Circle maps have been used previously (13) to study the oscillatory behavior of externally stimulated heart tissue.
The different circle maps can be approximated by a piecewise linear model specified by a few parameters. From these parameters, new measures characterizing respirationrelated heart rate variability can be derived and interpreted from a physiological point of view.
METHODS
Subjects.
Four men, with ages between 29 and 35 yr and without any history of cardiopulmonary diseases, participated in this study. All subjects were nonsmokers for >1 yr. The measurements were taken between 9:00 AM and 2:00 PM, 2–3 h after the subject’s last meal or caffeinic beverage. Care was taken to ensure that subjects were in a relaxed state. All subjects were informed about the purpose of the study and gave their consent before the examination.
Experimental design.
The subjects were asked to breathe voluntarily in phase with a growing (inspiration) and vanishing (expiration) light circle at 10 different cycle lengths (Tt = 5, 5.5, 6, 6.5, 7, 8, 9, 10, 11, and 12 s) for 70 cycles at each frequency. To check for transient phenomena, the pacing frequencies were arranged to also include large frequency jumps. The ratio of expiration to inspiration length was kept constant (4:3). The tidal volume was not controlled because the subjects were allowed to adjust their alveolar minute ventilation to maintain the acidbase homeostasis. Note that tidal volume is considered to have a minor influence on the RSA (15). The paced respiration experiment was split into three sessions of nearly equal duration (Table 1
Data.
The cardiac cycle length (RR interval) was determined algorithmically with an accuracy of ∼1 ms as the time interval between two successive maxima of R waves of the QRS complexes of the original ECG. The data set contains only one extrasystolic heartbeat. In a previous study, the same data set was analyzed with the use of various methods including symbolic dynamics and the analysis of phase shifts (24).
To focus on the relevant timescale of the RSA [the respiratory cycle length (Tt)], a highpass filter was applied to the heartbeat lengths. The filter was implemented as the difference to a moving average with an averaging period equal to the respiratory period. This procedure removes lowfrequency nonstationary baseline trends. A typical piece of the highpass filtered RR interval time series, recorded at Tt = 9 s, is shown in Fig.1 A.
Respiratory flow data were used to check the breathing performance of the subjects. As can be seen from Table 2, the relative standard deviation of the Tt intervals is generally small. It is caused mainly by short periods of reduced attention to the optical metronome.
RESULTS
From time series to circle maps.
Typical features of the RSA dynamics of a single healthy subject are the pronounced variability of the amplitude of the RSA (defined as the difference between the maximal and minimal RR interval within one respiratory cycle) and the considerable stability of the relative partitioning of the respiratory cycle by heartbeat intervals (2, 23), which we call the characteristic pattern of the RSA dynamics. Comparing the RR interval time series of two subjects (Fig. 1,A andC), this pattern becomes visible: whereas subject 1 has a clustering of heartbeats at longer RR intervals, subject 2 has clusters of heartbeats at shorter intervals. Another feature is the asymmetry between the number of heartbeats with increasing and decreasing RR intervals, respectively (2, 22). In this paper, we introduce a more general method for describing and quantifying the RSA dynamics.
A first attempt at identifying the underlying process generating this stable and characteristic pattern of the RSA is to plot the same data as a twodimensional scatter plot of successive RR intervals (embedding of RR intervals) (Fig. 2).
However, no sharply defined pattern is revealed in this plot; the points scatter around an elliptical structure. All RR interval time series contain this feature, but they do not show a simple nonrandom structure. The elliptical structure mirrors the basic periodicity of the RR intervals corresponding to the periodicity of respiration.
As a next step in the analysis, we suppress the fluctuations of the RSA amplitude and focus instead on the angular motion of the points in the plane. This is done by introducing polar coordinates and by neglecting the fluctuations of the radii. The transformation requires the definition of a center (radius = 0) (cf. Fig.3
B). After application of the abovementioned highpass filter on the RR interval time series, the mean RR interval length (
For every pair of successive RR intervals, an angle Φ is calculated. Each rotation angle Φ corresponds to a unique phase of the RSA cycle. However, because this phase is not identical to the respiratory phases (inspiration and expiration), we reserve the term “phase” for the physiological phase. The respirationrelated angles are defined to vary between 0 and 2, corresponding to 0 − 2π
The transformation is illustrated in Fig. 3 for one respiratory cycle. The function shown in Fig. 3 C is called a circle map or nextangle map.
Identifying a onedimensional deterministic process.
If we transform the RR interval data from subject 1, as shown in Fig. 2, to polar angles (Eq. 1 ) and plot the data as a nextangle map, this twodimensional embedding of the angles leads to a onedimensional curve, and the points cluster close to the line (Fig.4 A). This curve describes the deterministic dynamics of the rotation angles around the origin in theL _{n} _{+1}versus L_{n} plane. The law of motion, which is represented by the graph in Fig. 4, allows us to read off the next angle when the preceding one is known. From comparison of the scattering of points in the twodimensional embedding of the angles (Fig. 4 A) with the scatter plot of the RR intervals (Fig. 2), it is obvious that the dynamics of the angles are much less affected by noise than are the dynamics of the RR intervals. Thus the restriction on the dynamics of the angles leads to the disclosure of a onedimensional deterministic law of motion. In Test for nonlinear deterministic process using surrogate data, we show formally that this visual impression is legitimate. Furthermore, this circle map will be shown to have essential nonlinear features.
As Fig. 5 shows, a onedimensional law of motion is typical for cardiorespiratory dynamics during slowpaced breathing. For Tt > ∼7 s, a staircaselike structure of the circle map is observed for all subjects under examination.
However, a comparison of the circle maps for different subjects (Fig.5, from left toright) and for different pacing cycle lengths (Fig. 5, from A toC), also establishes distinct interindividual differences. Nonetheless, the example pairs with the same Tt, shown in Fig. 5, are similar in mean RR interval length (deviation <5%) and mean RSA amplitude (deviation <15%). Different degrees and locations of clustering are observed: whereas subject 3 exhibits a clustering of successive rotation angles preferentially near the maximum (Φ_{n} = 0.3),subject 2 shows more clustering near the minimum (Φ_{n} = 1.3). This could already be seen in Fig. 1. A second difference is the varying degree of scattering (e.g., Fig. 5, Band C), indicating that the transition to the onedimensional dynamics occurs at different respiratory cycle lengths and to a different extent. Finally,subject 3 shows a characteristic kink between Φ_{n} = 0.0 and Φ_{n} = 0.4 for all cycle lengths that is absent in all other subjects. It is remarkable that there are cases in which the circle maps of different subjects differ qualitatively, whereas the commonly used descriptive variables of RSA analyses, the mean RR interval, and the mean RSA amplitude do not differ substantially. Thus we conclude that the analysis of the nextangle map provides additional information.
Test for nonlinear deterministic process using surrogate data.
To demonstrate whether the analysis of successive rotation angles (Fig.4 A) in fact reveals a deterministic structure and does not simply result from the embedding or the highpass filtering, we used surrogate data from Monte Carlo realizations of stochastic processes, which preserve certain linear properties of the original time series (25, 29).
To estimate the effect of the embedding, we first tested the hypothesis that the time series is generated by a white noise stochastic process. On the basis of this hypothesis, surrogate 1 data are obtained by random shuffling of the empirical RR intervals. This leads to a process that preserves the histogram.
In the recurrence plot of rotation angles forsurrogate 1 data, the originally observed structure is clearly lost (Fig.4 B). This means that this first hypothesis must be rejected, i.e., the onedimensionality seen in Fig.4 A is not an artifact of the transformation used. However, the embedding does have an effect, because Φ_{n} and Φ_{n} _{+1} are not independent variables (Eq. 1 ). Only part of the plane is filled, and the apparently deterministic points at (0/−0.5) and (1/0.5) are introduced by the embedding.
To show that the rejection of surrogate 1 data is not only due to the complete destruction of the autocorrelation, we generated a second surrogate process that does preserve both the spectrum (i.e., the linear autocorrelation) and the histogram. This second (null) hypothesis is equivalent to the assumption of an underlying linear autoregressive process of unlimited order that is observed by a nonlinear measurement function. The nonlinearity is defined by the nonGaussian property of the original process (25). Because of the highpass filtering of the original data, the effective dimension of the linear autoregressive model is of the order of the number of heartbeats per respiratory cycle. This kind of autocorrelationadapted surrogate process generates a spectrum that cannot be distinguished from the original one.
Compared with the scatter plot of the original data (Fig.4 A), the onedimensional structure in the recurrence plot of rotation angles forsurrogate 2 data (Fig.4 C) appears less obvious. Furthermore, the asymmetry between the minimum and the maximum is lost. Figure 1 shows the comparison between the original time series (A) and an example of this surrogate process (B).
To get a quantitative estimate of the distinctness of the two plots, we performed a χ^{2} test using a sum of squared residuals as a discriminator. The residuals were chosen as the deviations of the data from an optimally adapted piecewise linear model consisting of six line segment (see Piecewise linear model). This special choice of six lines is suggested by the data presented in Figs. 4 and 5. Two of the segments are chosen a priori as fixed, and the eight parameters of the other four segments are estimated independently for each surrogate process. Because of the large number of data points, the sampling distribution of the square root of the χ^{2} was assumed to be normal. The mean and average of this distribution were estimated from a sample of 10 surrogate data sets. The corresponding χ^{2} test results in the rejection of the second null hypothesis with a probability of <0.001. The square root of the χ^{2} of the original process is 7.75 standard deviations from the average of the corresponding surrogate cases. This means that the second null hypothesis must also be rejected. Furthermore, the fluctuations of the empirical heart rate data are influenced by nonlinear deterministic dynamics, which can be described significantly better by a piecewise linear model than by a linear autoregressive moving average model with roughly the same number of parameters.
In summary, we identified a onedimensional, nonlinear deterministic process underlying the dynamics of the RR intervals during slowpaced respiration.
Piecewise linear model.
As mentioned in Test for nonlinear deterministic process using surrogate data, a piecewise linear model can be used to quantify the structure of the individual circle maps and their specific changes due to the protocol parameter (respiratory cycle length). As Fig. 6shows, the model consists of six straight lines (12 parameters), two of which are chosen a priori to have a fixed distance to the bisector (Φ = 0.5) and fixed slopes parallel to it. This reflects the deterministic transitions due to the embedding (line segments 3 and6; see Fig. 4 B). To fix the remaining four lines of the model, we must specify eight parameters. One possible choice comprises the six positions of the corner points on the abscissa and two vertical distances of those corner points to the bisector: x _{1},x _{2},x _{3},x _{4},x _{5},x _{6},h _{2}, andh _{5}. The four fixed parameters areh _{1} =h _{4} = 0.5 andh _{3} =h _{6} = 0.5 (Fig. 6; for details see Ref. 28).
The free model parameters were estimated by iterative linear regression for all respiratory cycle lengths and all subjects. For simplicity, only the error of the ordinate was taken into account. For example, forsubject 1 at Tt = 9 s, the following parameter values were determined:x _{1} = 0.066,x _{2} = 0.248,x _{3} = 0.814,x _{4} = 1.030, x _{5} = 1.388, x _{6} = 1.948, h _{2} = 0.051, and h _{5} = 0.089.
For all cycle lengths and subjects, the estimated parameters correspond to invertible circle maps. Because maps generating chaotic dynamics are noninvertible (11), it is clear that we do not encounter deterministic chaos in our analysis.
There is a second embedding effect that can be used as a consistency check for the estimation. This second effect can best be understood from Fig. 2. There are two ways to obtain the number of heartbeats below
As a second test for the validity of the model, the root mean square error of the fit was calculated for all 40 time series (Fig.7). This error is a measure of how well the different line segments approximate the empirical data. It is particularly small for high pacing length. In the range between 8 and 12 s, the angle dynamics (and therefore the pattern of the RSA) are described well by the onedimensional model. For respiratory cycle lengths <8 s, the mean square error increases strongly; the onedimensional description starts breaking down (cf. Fig. 5,B andC). This result is in agreement with the identification of two different cardiorespiratory synchronization regimes (5–7 s and 8–12 s) within the same data set using symbolic dynamics (24).
In addition, the piecewise linear model defined herein can be used to reconstruct a time series on the basis of the estimated model parameters (28).
Characteristics of heart rate pattern.
To be able to quantify the pattern of the heartbeat intervals within one respiratory cycle, it is preferable to construct measures that are sensitive to all possible characteristics of the pattern. Furthermore, it should be possible to obtain these measures directly from the original time series as well as from the model introduced inPiecewise linear model. The invariance of the law of motion of the angles (Eq.1 ) implies that the pattern of the RSA dynamics has invariant characteristics that can be used as descriptors. Because the model is defined by seven independent parameters, the maximal number of descriptors should not be greater than that. It is convenient to introduce the following characteristic measures.
Asymmetry index 1(A
_{1}) describes the relative asymmetry between the number of heartbeats with interval lengths longer than
Two other measures,C
_{1} andC
_{2}, are used to quantify the degree of clustering at the accumulation points. Accumulation points are points near the bisector, because the fixed points of a onedimensional map are on the bisector. Empirically, the degrees of clustering can be obtained as the logarithmic ratio of the maximal step length of all angles compared with the average minimal step length for heartbeat intervals either longer than
Two other invariants,x _{2} andx _{5}, specify the degree to which the accumulation points of the circle map coincide with the maximum or the minimum of the RR intervals at Φ = 1/4 (corresponding to a maximum) and Φ = 5/4 (corresponding to a minimum).
The winding number (W) is commonly used to describe the average number of RR intervals in one respiratory cycle. The notion of a winding number results from viewing the combined angular dynamics of the two oscillators, heart and respiration, as a motion on a torus. W is the ratio of the average number of faster oscillations (windings) per single cycle of the slower oscillator.
Note that all these measures describing the coupling between respiration and heart rate can be determined without measuring respiration. They can be obtained directly from the experimental time series, as described in methods, and also from the piecewise linear model, e.g., from a reconstruction of time series based on the model. Another useful feature of the model is that all the characteristic measures have simple dependence on the model parameters (see ). Figure 8 shows a comparison of both approaches, the direct determination from the empirical time series and the approximation based on the model parameters. The standard deviation of the direct measures was obtained by subdividing the time range into three intervals. Therefore, it can be taken as a measure to describe the variance in time.
The agreement between both approaches is reasonably good forW. The disagreement for the other measures results from imperfections of the model fit as well as from the analytic approximation described in the . Both of these are of the same order of magnitude. However, despite this disagreement, both measures allow a clear distinction of the frequency dependence of the nextangle map of the four subjects. The differences are particularly pronounced for the degrees of clustering (Fig.9).
The following characteristic changes inC _{1} andC _{2} are observed. For respiratory cycle lengths shorter than 8 s, both measures show similar changes. They increase monotonically with almost the same slope. W shows a similar behavior. For long cycle lengths, all subjects show a saturation effect. However, the subjects differ in their saturation level: subject 4 shows a high clustering for short heartbeat intervals, subject 2 has a high degree of clustering for long heartbeat intervals, andsubjects 1 and3 have both measures nearly equal. Because of their discriminatory properties and stabilities,C _{1} andC _{2} might serve as robust characteristics for a subject, in addition to theasymmetry index 2 (2).
Together, these seven measures encode the systematic information about the pattern of heart rate variability during slowpaced respiration. In the validity range of the model, these measures are guaranteed to be time invariants. Furthermore, for respiratory cycle lengths shorter than 8 s, these measures are still invariant in time.
DISCUSSION
The main result of the present study is that heart rate variability during paced respiration at longer cycle lengths (8–12 s) obeys a dynamic rule that can be expressed by a onedimensional, nonlinear circle map (nextangle map). This map describes the rotation dynamics in the twodimensional embedding space spanned by two successive heartbeat intervals. These dynamics represent properties of the characteristic pattern of RR intervals within one respiratory cycle. The existence of this law of motion implies that the relative partitioning of the respiratory cycle by heartbeat intervals is invariant in time compared with the amplitude of the RSA, for which such an invariance could not be demonstrated (28).
Data presented by the empirically obtained circle maps for four healthy adult volunteers breathing at 10 different respiratory cycle lengths led us to define a piecewise linear, phenomenological model that captures essential features of heart rate dynamics. The description of heart rate variability by this nonlinear model was contrasted with its description by a linear autoregressive model with the same number of parameters. The nonlinear model provides a significantly better fit to the empirical data than to the surrogate data. This is further proof that the fluctuations of the empirical heart rate data have essential nonlinear dynamic features (5, 16, 20). The parameters of the piecewise linear model were related to characteristic time invariants. Two of them, A _{1} andA _{2}, are direct measures of the nonlinearity of the model.
These characteristic invariants, quantifying properties of the pattern of heartbeat intervals within one respiratory cycle, can also be obtained directly from the RR interval time series without using the model. Their algebraic relationship to the model parameters guarantees their invariance in time. This time invariance holds not only for the longer respiratory cycle lengths, for which the onedimensional deterministic process is identified, but also for cycle lengths in the range of spontaneous breathing, for which no similar deterministic onedimensional structure can be detected (14).
The observation of onedimensional dynamics for slow pacing of the cardiorespiratory system is indeed surprising, because this system is known to be influenced by physiological processes on many different timescales, including longer timescales such as circadian rhythm or hormonal cycles with cycle lengths ranging from minutes to hours. A transition to onedimensional dynamics can only be expected for periodically forced systems with a maximal time constant of all internal feedback mechanisms that is either shorter than, or of the same order of magnitude as, the cycle length of the driving system (entrainment of faster internal degrees of freedom). For nonlinear coupled oscillators, the frequency range of the entrainment (1:1 Arnold tongue) can be relatively broad. The fact that such a transition to onedimensional dynamics has only been found for pacing cycle lengths longer than 7 s should be seen as further proof that the cardiorespiratory system has at least one important degree of freedom that is not entrained by the respiration for shorter cycle lengths. Therefore, this additional degree of freedom (oscillator) must have a cycle length distinctly longer than 7 s. A similar transition at the same cycle length has been found using symbolic dynamics (24).
The present analysis leads to the hypothesis that this intrinsic degree of freedom is identical to the socalled lowfrequency component or Mayer wave of the cardiorespiratory system, which is a wellknown oscillatory phenomenon with a cycle length of ∼10 s (8) that can preferentially be identified in blood pressure data.
There is no doubt that the pattern of the RSA is influenced by dynamic properties of the baroreflex feedback loop and of the intracentral cardiorespiratory feedback mechanisms. The pronounced interindividual variation of the onedimensional map at fixed cycle length and of the respiratory frequencydependent transition to it underlines the expectation that the analysis of the structure of circle maps might be a good probe for the characterization of the dynamic properties of the baroreflex and of its interaction with different hypothesized cardiorespiratory oscillators.
The onedimensional nonlinear model is able to reproduce features of the cardiorespiratory system that cannot be expressed by linear autoregressive models. Thus we expect the characteristic invariants to express additional physiological features of the cardiorespiratory system that are inaccessible using the standard measures. Therefore, the measures based on the pattern of heartbeat intervals within one respiratory cycle supplement the standard measures based on amplitudes in the time or frequency domain. The latter are known to be good indicators for the strength of the parasympatheticmediated baroreflex feedback (8). In addition, the quantitative results from nonlinear time series analyses may be useful in estimating parameters of mechanistic models. The detailed physiological interpretation of the newly defined measures should be explored in further studies.
Acknowledgments
The authors thank R. Engbert (Potsdam, Germany), A. Hilgers (London, UK), N. Stollenwerk (Jülich, Germany), and the members of the Interdisciplinary Hellersen Meetings, initiated by H. P. Koepchen, for valuable comments.
Footnotes

Address for reprint requests: F. Drepper, ZEL, Forschungszentrum Jülich, 52425 Jülich, Germany.

Present address of K. Suder: Institut für Physiologie, Abteilung Neurophysiologie, RuhrUniversität, 44780 Bochum, Germany (Email: suder{at}neurop2.ruhrunibochum.de).
 Copyright © 1998 the American Physiological Society
Appendix
Obtaining characteristic measures from the model.
On the basis of the piecewise linear map with parameters as defined in Fig. 6, an analytic approximation of the characteristic measures can be given. Three of the invariants may be defined in terms of the average number of iterations falling into the different line segmentsi. The iteration numbersk _{i} can be approximated analytically (see Ref. 28 for further details).
By starting from the assumption that the first injection of a coordinate happens predominantly at the right end of a line segment, an upper bound for the number of iterations can be derived. First, one needs the relationship between two successive coordinates (angles)y_{k}
,y
_{k}
_{−1}
This relationship is an approximation, because it does not take into account the exact distribution of the angles, which is not uniform as can be seen in the empirical data.
By construction,a _{3} anda _{6} are equal to 1. The correspondingk_{i} values depend on the length of line i and the distance to the bisector:k _{3} = (x _{4} −x _{3})/h _{3}and k _{6} = (x _{1} + 2 −x _{6})/h _{6}.
W is the sum of the sixk_{i}
values