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 h0 is the Hessian matrix of the phase function evaluated at the barycenter p0 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 bi which samples the function at the vertexes, thus obtaining
in which ui = u0 + h0 · (vi − p0). Again formula (10) can be interpreted as a quadrature rule with sampling points at triangle vertexes vi and weights Ci(u0, h0) now depending on both the phase progression rate u0 and the phase curvature h0. Since the integrand in (11) presents the quadratic phase (12), from an asymptotic point of view, the main contributions to each integral Ci are due to three different kind of critical points of different asymptotic order (Figure 3): (1) one stationary phase point ps = p0 − h0−1 · u0, if present; (2) three partial stationary phase points pse,ℓ = vℓ − lℓ[(vℓ − ps) · h0 · lℓ]/(lℓ · h0 · lℓ), with ℓ = 1, 2, 3, if present; and (3) three vertexes vq, with q = 1, 2, 3. It is worth noting that, as the three ϕi(12) associated to the three respective terms in (10) all differ only by a constant, the three integrals Ci share the same critical points which are therefore independent of i. Consequently, each integral Ci can be exactly decomposed in [Carluccio et al., 2010]
By defining the quantities
the stationary phase point contribution of Ci is given by
where δ = 1 if the eigenvalues of the Hessian matrix h0 are both positive, δ = −1 if they are both negative, and δ = j if they have opposite sign. In (13), Us is the Heaviside step function, that is equal to 1 if ps ∈ A, or Us = 0 otherwise; in other words, this contribution has to be accounted for only if the stationary phase point ps lies inside the triangle A.
 The partial stationary phase point contributions in (13) are
where ηi,ℓ = γi, −γi −βi, βi, 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); use,ℓ = u0 + h0 · (pse,ℓ −p0) is the gradient of the phase function at pse,ℓ, while is the UTD wedge transition function [Kouyoumjian and Pathak, 1974],
with arg ∈ (−3/4π, π/4), whose argument
is the measure of the phase distance between the stationary phase point ps and the partial stationary phase point pse,ℓ on the ℓth edge. In (13), Uℓse is the Heaviside step function, that is equal to 1 if pse,ℓ ∂A, or Uℓse = 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 Ci,qv are given by
and contain the UTD vertex transition function [Hill, 1990; Carluccio et al., 2010],
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 qth vertex and the ℓth side of the triangle, respectively. The xq,ℓ parameters are a measure of the phase distance between the partial stationary phase point on the ℓth side, pse,ℓ, and the qth vertex vq. The wq parameter is defined as
In (22) and (23) the branch of the square root is chosen so that −3/4π < arg() < π/4.
 As described by Carluccio et al. , 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.