[9] The second proposed integration rule is based on a linear amplitude approximation and on a quadratic phase approximation. We consider the quadratic Taylor expansion for the phase function

where **h**_{0} is the Hessian matrix of the phase function evaluated at the barycenter **p**_{0} of the triangle. Similarly to the previous case, after “demodulating” the integrand *F*(**p**) by the above quadratic phase expansion, the remaining slowly varying integrand is linearly interpolated through the basis functions *b*_{i} which samples the function at the vertexes, thus obtaining

where

with

in which **u**_{i} = **u**_{0} + **h**_{0} · (**v**_{i} − **p**_{0}). Again formula (10) can be interpreted as a quadrature rule with sampling points at triangle vertexes **v**_{i} and weights *C*_{i}(**u**_{0}, **h**_{0}) now depending on both the phase progression rate **u**_{0} and the phase curvature **h**_{0}. Since the integrand in (11) presents the quadratic phase (12), from an asymptotic point of view, the main contributions to each integral *C*_{i} are due to three different kind of critical points of different asymptotic order (Figure 3): (1) one stationary phase point **p**_{s} = **p**_{0} − **h**_{0}^{−1} · **u**_{0}, if present; (2) three partial stationary phase points **p**_{se,ℓ} = **v**_{ℓ} − **l**_{ℓ}[(**v**_{ℓ} − **p**_{s}) · **h**_{0} · **l**_{ℓ}]/(**l**_{ℓ} · **h**_{0} · **l**_{ℓ}), with ℓ = 1, 2, 3, if present; and (3) three vertexes **v**_{q}, with *q* = 1, 2, 3. It is worth noting that, as the three ϕ_{i}(12) associated to the three respective terms in (10) all differ only by a constant, the three integrals *C*_{i} share the same critical points which are therefore independent of *i*. Consequently, each integral *C*_{i} can be exactly decomposed in [*Carluccio et al.*, 2010]

By defining the quantities

the stationary phase point contribution of *C*_{i} is given by

where *δ* = 1 if the eigenvalues of the Hessian matrix **h**_{0} are both positive, *δ* = −1 if they are both negative, and *δ* = *j* if they have opposite sign. In (13), *U*^{s} is the Heaviside step function, that is equal to 1 if **p**_{s} ∈ *A*, or *U*^{s} = 0 otherwise; in other words, this contribution has to be accounted for only if the stationary phase point **p**_{s} lies inside the triangle *A*.

[10] The partial stationary phase point contributions in (13) are

where *η*_{i,ℓ} = *γ*_{i}, −*γ*_{i} −*β*_{i}, *β*_{i}, for ℓ = *i*, *i* + 1, *i* + 2 (again intended “mod 3”), respectively. Also, in (16)_{ℓ} and _{ℓ} denote the unit vectors along and normal (outgoing) to the ℓth side of the triangle, respectively (see Figure 4); **u**_{se,ℓ} = **u**_{0} + **h**_{0} · (**p**_{se,ℓ} −**p**_{0}) is the gradient of the phase function at **p**_{se,ℓ}, while is the UTD wedge transition function [*Kouyoumjian and Pathak*, 1974],

with arg ∈ (−3/4*π*, *π*/4), whose argument

is the measure of the phase distance between the stationary phase point **p**_{s} and the partial stationary phase point **p**_{se,ℓ} on the ℓth edge. In (13), *U*_{ℓ}^{se} is the Heaviside step function, that is equal to 1 if **p**_{se,ℓ} ∂*A*, or *U*_{ℓ}^{se} = 0 otherwise; accordingly, this type of contribution has to be considered only if a the partial stationary phase point is present on the ℓth side of the triangle. The vertex contributions *C*_{i,q}^{v } are given by

and contain the UTD vertex transition function [*Hill*, 1990; *Carluccio et al.*, 2010],

where

is the generalized Fresnel integral [*Capolino and Maci*, 1995]. The argument of the and transition functions in (19) are defined by

where the upper (lower) sign applies when ℓ = *q* (ℓ = *q* − 1), and *q* and ℓ indexes identify the *q*th vertex and the ℓth side of the triangle, respectively. The *x*_{q,ℓ} parameters are a measure of the phase distance between the partial stationary phase point on the ℓth side, **p**_{se,ℓ}, and the *q*th vertex **v**_{q}. The *w*_{q} parameter is defined as

In (22) and (23) the branch of the square root is chosen so that −3/4*π* < arg() < *π*/4.

[11] As described by *Carluccio et al.* [2010], the UTD transition function allows both the uniform description of the coalescence of a single partial stationary phase point and a vertex, and the uniform description of the simultaneous coalescence of the stationary phase point, the vertex, and the two partial stationary phase points, which lie on the two edges joining at the vertex.