3D printing dimensional calibration shape: Clebsch Cubic

3D printing and other layer manufacturing processes are challenged by dimensional accuracy. Several techniques are used to validate and calibrate dimensional accuracy through the complete building envelope. The validation process involves the growing and measuring of a shape with known parameters. The measured result is compared with the intended digital model. Processes with the risk of deformation after time or post processing may find this technique beneficial. We propose to use objects from algebraic geometry as test shapes. A cubic surface is given as the zero set of a 3rd degree polynomial with 3 variables. A class of cubics in real 3D space contains exactly 27 real lines. We provide a library for the computer algebra system Singular which, from 6 given points in the plane, constructs a cubic and the lines on it. A surface shape derived from a cubic offers simplicity to the dimensional comparison process, in that the straight lines and many other features can be analytically determined and easily measured using non-digital equipment. For example, the surface contains so-called Eckardt points, in each of which three of the lines intersect, and also other intersection points of pairs of lines. Distances between these intersection points can easily be measured, since the points are connected by straight lines. At all intersection points of lines, angles can be verified. Hence, many features distributed over the build volume are known analytically, and can be used for the validation process. Due to the thin shape geometry the material required to produce an algebraic surface is minimal. This paper is the first in a series that proposes the process chain to first define a cubic with a configuration of lines in a given print volume and then to develop the point cloud for the final manufacturing. Simple measuring techniques are recommended.Comment: 8 pages, 1 figure, 1 tabl

Low omega-6 vs. low omega-6 plus high omega-3 dietary intervention for Chronic Daily Headache: Protocol for a randomized clinical trial

<p>Abstract</p> <p>Background</p> <p>Targeted analgesic dietary interventions are a promising strategy for alleviating pain and improving quality of life in patients with persistent pain syndromes, such as chronic daily headache (CDH). High intakes of the omega-6 (n-6) polyunsaturated fatty acids (PUFAs), linoleic acid (LA) and arachidonic acid (AA) may promote physical pain by increasing the abundance, and subsequent metabolism, of LA and AA in immune and nervous system tissues. Here we describe methodology for an ongoing randomized clinical trial comparing the metabolic and clinical effects of a low n-6, average n-3 PUFA diet, to the effects of a low n-6 plus high n-3 PUFA diet, in patients with CDH. Our primary aim is to determine if: A) both diets reduce n-6 PUFAs in plasma and erythrocyte lipid pools, compared to baseline; and B) the low n-6 plus high n-3 diet produces a greater decline in n-6 PUFAs, compared to the low n-6 diet alone. Secondary clinical outcomes include headache-specific quality-of-life, and headache frequency and intensity.</p> <p>Methods</p> <p>Adults meeting the International Classification of Headache Disorders criteria for CDH are included. After a 6-week baseline phase, participants are randomized to a low n-6 diet, or a low n-6 plus high n-3 diet, for 12 weeks. Foods meeting nutrient intake targets are provided for 2 meals and 2 snacks per day. A research dietitian provides intensive dietary counseling at 2-week intervals. Web-based intervention materials complement dietitian advice. Blood and clinical outcome data are collected every 4 weeks.</p> <p>Results</p> <p>Subject recruitment and retention has been excellent; 35 of 40 randomized participants completed the 12-week intervention. Preliminary blinded analysis of composite data from the first 20 participants found significant reductions in erythrocyte n-6 LA, AA and %n-6 in HUFA, and increases in n-3 EPA, DHA and the omega-3 index, indicating adherence.</p> <p>Trial Registration</p> <p>ClinicalTrials.gov <a href="http://www.clinicaltrials.gov/ct2/show/(NCT01157208)">(NCT01157208)</a></p

A note on the intersection of Veronese surfaces

The main purpose of this note it to prove the following Theorem 0.1 Any two Veronese surfaces in P 5 whose intersection is zerodimensiona