In the construction and analysis of a planar Pythagorean–hodograph (PH) quintic curve r(t), t∈[0,1] using the complex representation, it is convenient to invoke a translation/rotation/scaling transformation so r(t) is in canonical form with r(0)=0, r(1)=1 and possesses just two complex degrees of freedom. By choosing two of the five control–polygon legs of a quintic PH curve as these free complex parameters, the remaining three control–polygon legs can be expressed in terms of them and the roots of a quadratic or quartic equation. Consequently, depending on the chosen two control–polygon legs, there exist either two or four distinct quintic PH curves that are consistent with them. A comprehensive analysis of all possible pairs of chosen control polygon legs is developed, and examples are provided to illustrate this control–polygon paradigm for the construction of planar quintic PH curves.

The construction of tensor-product surface patches with a family of Pythagorean-hodograph (PH) isoparametric curves is investigated. The simplest nontrivial instances, interpolating four prescribed patch boundary curves, involve degree (5,4) tensor-product surface patches x(u,v) whose v=constant isoparametric curves are all spatial PH quintics. It is shown that the construction can be reduced to solving a novel type of quadratic quaternion equation, in which the quaternion unknown and its conjugate exhibit left and right coefficients, while the quadratic term has a coefficient interposed between them. A closed-form solution for this type of equation is derived, and conditions for the existence of solutions are identified. The surfaces incorporate three residual scalar freedoms which can be exploited to improve the interior shape of the patch. The implementation of the method is illustrated through a selection of computed examples.

The blending or filleting of sharp corners is a common requirement in geometric design applications — motivated by aesthetic, ergonomic, kinematic, or mechanical stress considerations. Corner blending curves are usually required to exhibit a specified order of geometric continuity with the segments they connect, and to satisfy specific constraints on their curvature profiles and the extremum deviation from the original corner. The free parameters of polynomial corner curves of degree ≤6 and continuity up to G3 are exploited to solve a convex optimization problem, that minimizes a weighted sum of dimensionless measures of the mid-point curvature, maximum deviation, and the uniformity of parametric speed. It is found that large mid-point curvature weights result in undesirable bimodal curvature profiles, but emphasizing the parametric speed uniformity typically yields good corner shapes (since the curvature is strongly dependent upon parametric speed variation). A constrained optimization problem, wherein a particular value of the corner curve deviation is specified, is also addressed. Finally, the shape of Pythagorean-hodograph corner curves is compared with that of the optimized “ordinary” polynomial corner curves.

The clothoid is a planar curve with the intuitive geometrical property of a linear variation of the curvature with arc length, a feature that is important in many geometric design applications. However, the exact parameterization of the clothoid is defined in terms of the irreducible Fresnel integrals, which are computationally expensive to evaluate and incompatible with the polynomial/rational representations employed in computer aided geometric design. Consequently, applications that seek to exploit the simple curvature variation of the clothoid must rely on approximations that satisfy a prescribed tolerance. In the present study, we investigate the use of planar Pythagorean-hodograph (PH) curves as polynomial approximants to monotone clothoid segments, based on geometric Hermite interpolation of end points, tangents, and curvatures, and precise matching of the clothoid segment arc length. The construction, employing PH curves of degree 7, involves iterative solution of a system of five algebraic equations in five real unknowns. This is achieved by exploiting a closed-form solution to the problem of interpolating the specified data (except the curvatures) using quintic PH curves, to determine starting values that ensure rapid and accurate convergence to the desired solution.

Seasonal influenza A and B viruses represent a global concern. Antiviral drugs are crucial to treat severe influenza in high-risk patients and prevent virus spread in case of a pandemic. The emergence of viruses showing drug resistance, in particular for the recently licensed polymerase inhibitor baloxavir marboxil, drives the need for developing alternative antivirals. The endonuclease activity residing in the N-terminal domain of the polymerase acidic protein (PA_N) is crucial for viral RNA synthesis and a validated target for drug design. Its function can be impaired by molecules bearing a metal-binding pharmacophore (MBP) able to coordinate the two divalent metal ions in the active site. In the present work, the 2,3-dihydro-6,7-dihydroxy-1H-isoindol-1-one scaffold is explored for the inhibition of influenza virus PA endonuclease. The structure-activity relationship was analysed by modifying the substituents on the lipophilic moiety linked to the MBP. The new compounds exhibited nanomolar inhibitory activity in a FRET-based enzymatic assay, and a few compounds (15-17, 21) offered inhibition in the micromolar range, in a cell-based influenza virus polymerase assay. When investigated against a panel of PA-mutant forms, compound 17 was shown to retain full activity against the baloxavir-resistant I38T mutant. This was corroborated by docking studies providing insight into the binding mode of this novel class of PA inhibitors.

Objective

Melanopsin retinal ganglion cells (mRGCs) are photoreceptors driving circadian photoentrainment, and circadian dysfunction characterizes Alzheimer disease (AD). We investigated mRGCs in AD, hypothesizing that they contribute to circadian dysfunction.

Methods

We assessed retinal nerve fiber layer (RNFL) thickness by optical coherence tomography (OCT) in 21 mild-moderate AD patients, and in a subgroup of 16 we evaluated rest-activity circadian rhythm by actigraphy. We studied postmortem mRGCs by immunohistochemistry in retinas, and axons in optic nerve cross-sections of 14 neuropathologically confirmed AD patients. We coimmunostained for retinal amyloid β (Aβ) deposition and melanopsin to locate mRGCs. All AD cohorts were compared with age-matched controls.

Results

We demonstrated an age-related optic neuropathy in AD by OCT, with a significant reduction of RNFL thickness (p = 0.038), more evident in the superior quadrant (p = 0.006). Axonal loss was confirmed in postmortem AD optic nerves. Abnormal circadian function characterized only a subgroup of AD patients. Sleep efficiency was significantly reduced in AD patients (p = 0.001). We also found a significant loss of mRGCs in postmortem AD retinal specimens (p = 0.003) across all ages and abnormal mRGC dendritic morphology and size (p = 0.003). In flat-mounted AD retinas, Aβ accumulation was remarkably evident inside and around mRGCs.

Interpretation

We show variable degrees of rest-activity circadian dysfunction in AD patients. We also demonstrate age-related loss of optic nerve axons and specifically mRGC loss and pathology in postmortem AD retinal specimens, associated with Aβ deposition. These results all support the concept that mRGC degeneration is a contributor to circadian rhythm dysfunction in AD.