Cardiopulmonary exercise testing provides a predictive tool for early and late outcomes in abdominal aortic aneurysm patients

Thompson AR, Peters N, Lovegrove RE et al. Cardiopulmonary exercise testing provides a predictive tool for early and late outcomes in abdominal aortic aneurysm patients. Ann R Coll Surg Engl 2011; 93: 474–481.


Introduction
In a recent paper, Hernández-Saldaña [1] presents an analysis of the locus formed by the set of maxima in the trajectories of a projectile launched in a linear resisting medium. In doing so Hernández-Saldaña writes the locus in polar form in terms of the now familiar Lambert W function [2] and goes on to characterize the locus by analysing its curvature. Angles of maximum curvature as a function of the dimensionless drag parameter ε are given and compared to a number of other important maximal angles for the problem 1 .
In this short comment we show that a remark made by Hernández-Saldaña concerning the validity of an expression first presented by us in [3] for one of these maximal angles is in error. We also draw attention to an earlier treatment [4] of the locus of apexes for a projectile in a linear resisting medium. We see that the locus, when expressed in Cartesian form, can be written in terms of the now familiar though less common secondary real branch of the Lambert W function.

Maximal angles
One such angle is that which maximizes the range of the projectile. When written in closed form it can be expressed in terms of the Lambert W function (see (22) of [1]). Another such angle arises from an important effect a resisting medium has on the motion of a projectile, the introduction of asymmetries into the trajectory paths. Since the horizontal component of the velocity of the projectile decreases as it moves through the resisting medium, on ascent the trajectory is shallower and longer, while on descent it is steeper and shorter. This results in the paths taking on a forwardly skewed appearance, and it is natural to ask at what angle to the horizontal should a projectile be launched for its trajectory to take on its greatest forward skew.
As Hernández-Saldaña notes, such a question was answered in [3]. Briefly, the horizontal distance travelled by a projectile on ascent to its apex is (see (5) Here ρ = v 2 0 g is a normalized length. If we let the optimal angle of projection for greatest forward skew be α * max , an expression for this angle can be found. It will be the angle satisfying dx a /dα = 0. Differentiating (1) with respect to α, stationary points in the derivative are found to occur when the launch angle satisfies the cubic equation Here we have set u = sin α * max . The three roots to (2) are completely characterized by its discriminant [6], D = (27ε 2 − 32)/(108ε 4 ). Three different cases for the solution arise: (i) for three distinct real roots, D < 0 and corresponds to 0 < ε < √ 32/27, (ii) for two real roots, one of which is a double root, D = 0 and corresponds to ε = √ 32/27, and (iii) for one real root and two complex roots which are conjugates, D > 0 and corresponds to ε > √ 32/27. In a linearly resisting medium as ε is positive, Descartes' sign rule [5] ensures that the maximum number of positive real roots for (2) is at most 1. Thus there will be only one angle α * max ∈ (0, π/2) for which x a is maximized. If the Cardano-Tartaglia formula is used, we find for the positive real root α * max = arcsin 1 3ε where D ± = ±3ε 3(27ε 2 − 32) + 27ε 2 − 16.
In a footnote at the bottom of page 1325 of [1], Hernández-Saldaña erroneously maintains that (3) is only valid for ε > √ 32/27 and, in particular, no longer valid for the special case of ε = 1. This is not the case. While it is true that D ± becomes negative when 0 < ε < √ 32/27, this is nothing more than an example of the famous casus irreducibilis (irreducible case) first identified by the Italian mathematician Girolamo Cardano (1501-76). In this case, producing the positive real root requires a detour through the complex plane.
What initially makes the special case of ε = 1 particularly interesting is an apparent connection with the golden ratio φ = (1 + √ 5)/2. On setting ε equal to unity in (3), The golden ratio connection, while intriguing, is not intrinsic to the problem itself. Recall that the golden ratio and its reciprocal φ −1 = ( √ 5 − 1)/2 = φ − 1 are the two roots to the quadratic equation u 2 − u − 1 = 0. As Essén and Apazidis [7] have rightly pointed out, any physical problem that leads to a quadratic equation with adjustable parameters will always result in the golden ratio if those parameters are chosen so as to give u 2 − u − 1 = 0. In our case, when ε is set equal to unity, a simple factorization of (2) leads to the required quadratic equation for an apparent golden ratio connection.
It is possible to rewrite (3) in terms of an expression which remains real for all positive ε using the so-called Viète form for the solution of a cubic equation. For the positive real root, the result is

Peak trajectory locus
In [4] the locus of apexes in the trajectories for a projectile fired in a linearly resisting medium is given in Cartesian form. The result is where Here y m is the vertical height of the projectile above ground level at its apex, while x m is the corresponding horizontal coordinate at this maximum height. The locus of apexes C m is thus defined by (x m , y m ).
What is particularly interesting here about the locus of apexes is, when expressed in Cartesian form, that it requires the secondary real branch for the Lambert W function W −1 (x). While it is well known that the linear drag problem utilizes the Lambert W function in an essential way, it is almost always the principal branch for the Lambert W function which arises.
Finally, in the weak damping limit, provided the initial launch speed is not too large, ε → 0 + . Expanding the Lambert W function term appearing in C m about b = 0, to leading order in b, the Cartesian form for C m in the limit of weak damping becomes In (8) the approximate elliptic-like form in C m is apparent, it being a perturbation of the wellknown unresisted elliptic property for parabolic trajectories, a result recovered on setting the drag coefficient b equal to zero.

Conclusion
We have shown that a remark made by Hernández-Saldaña in [1] concerning the validity of an expression first presented by us in [3] for the optimal angle of projection for greatest forward skew in the trajectory of a projectile launched in a linear resisting medium is in error. The error possibly stems from the irreducible case arising in the solution of a cubic equation. When written in terms of nested radials, the expression for the optimal angle can result in a detour through the complex plane. To avoid this unnecessary complication, we rewrote it in terms of an expression that remains real for all positive ε. We have also drawn attention to an earlier treatment of the locus of apexes for a projectile in a linear resisting medium where, when expressed in Cartesian form, it was shown that it could be written in terms of the familiar though less common secondary real branch of the Lambert W function.