Some New Geometries for Which Laplace's Equation in Three Dimensions is Soluble

1964 ◽  
Vol 43 (1-4) ◽  
pp. 144-157 ◽  
Author(s):  
R. Ball ◽  
D. Bartlett ◽  
R. Bayer ◽  
F. Partovi
Keyword(s):  
Three Dimensions ◽  
Laplace’S Equation ◽  
Laplace's Equation

scholarly journals The Cauchy Problem for Laplace's Equation in Three Dimensions

1935 ◽  
Vol 42 (2) ◽  
pp. 65-74
Author(s):  
L. H. Johnson
Keyword(s):  
Cauchy Problem ◽  
Three Dimensions ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
The Cauchy Problem

The Numerical Solution of Laplace’s Equation in Three Dimensions

1982 ◽  
Vol 19 (2) ◽  
pp. 263-274 ◽  
Author(s):  
K. E. Atkinson
Keyword(s):  
Numerical Solution ◽  
Three Dimensions ◽  
Laplace’S Equation ◽  
Laplace's Equation

scholarly journals The Cauchy Problem for Laplace's Equation in Three Dimensions

1935 ◽  
Vol 42 (2) ◽  
pp. 65
Author(s):  
L. H. Johnson
Keyword(s):  
Cauchy Problem ◽  
Three Dimensions ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
The Cauchy Problem

An alǵebraic technique for the solution of Laplace's equation in three dimensions

1970 ◽  
Vol 67 (2) ◽  
pp. 383-389 ◽  
Author(s):  
K. S. Kunz
Keyword(s):  
Conformal Mapping ◽  
Analytic Functions ◽  
Euclidean Plane ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Complex Numbers ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Real Linear ◽  
Algebraic Technique

In obtaining a solution of Laplace's equation in two dimensions by the method of conformal mapping, one first maps the points (x, y) of the Euclidean plane R2 into the algebra of complex numbers C by means of the real-linear function g: R2→C using the prescription g(x, y) = x + iy ≡ z. One then obtains solutions of Laplace's equation by allowing those mappings of C into itself that are expressed by analytic functions.

scholarly journals Unfolding the hippocampus: an intrinsic coordinate system for subfield segmentations and quantitative mapping

2017 ◽  
Author(s):  
Jordan DeKraker ◽  
Kayla M. Ferko ◽  
Jonathan C. Lau ◽  
Stefan Köhler ◽  
Ali R. Khan
Keyword(s):  
Coordinate System ◽  
Grey Matter ◽  
Computational Method ◽  
Three Dimensions ◽  
Stratum Radiatum ◽  
Coordinate Space ◽  
Surgical Patient ◽  
List Type ◽  
Laplace’S Equation ◽  
Laplace's Equation

AbstractThe hippocampus, like the neocortex, has a morphological structure that is complex and variable in its folding pattern, especially in the hippocampal head. The current study presents a computational method to unfold hippocampal grey matter, with a particular focus on the hippocampal head where complexity is highest due to medial curving of the structure and the variable presence of digitations. This unfolding was performed on segmentations from high-resolution, T2-weighted 7T MRI data from 12 healthy participants and one surgical patient with epilepsy whose resected hippocampal tissue was used for histological validation. We traced a critical hippocampal component, the hippocampal sulcus and stratum radiatum, lacunosum moleculaire, (SRLM) in these images, then employed user-guided semi-automated techniques to detect and subsequently unfold the surrounding hippocampal grey matter. This unfolding was performed by solving Laplace’s equation in three dimensions of interest (long-axis, proximal-distal, and laminar). The resulting ‘unfolded coordinate space’ provides an intuitive way of mapping the hippocampal subfields in 2D space (long-axis and proximal-distal), such that similar borders can be applied in the head, body, and tail of the hippocampus independently of variability in folding. This unfolded coordinate space was employed to map intracortical myelin and thickness in relation to subfield borders, which revealed intracortical myelin differences that closely follow the subfield borders used here. Examination of a histological sample from a patient with epilepsy reveals that our unfolded coordinate system shows biological validity, and that subfield segmentations applied in this space are able to capture features not seen in manual tracing protocols.Research highlightsSRLM in hippocampal head consistently detected with 7T, T2 isotropic MRIHippocampal grey matter unfolded using Laplace’s equation in 3DIntracortical myelin and thickness mapped in unfolded coordinate spaceUnfolded subfields capture critical structural regularities and agree with histology

Algorithms for Multilayer Conformal Additive Manufacturing

2015 ◽  
Author(s):  
Joshua D. Davis ◽  
Michael D. Kutzer ◽  
Gregory S. Chirikjian
Keyword(s):  
Additive Manufacturing ◽  
Three Dimensions ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
New Methods ◽  
Rapid Advance ◽  
Natural Geometry ◽  
Offset Curves ◽  
Convex Geometries ◽  
Manufacturing Technologies

Despite the rapid advance of additive manufacturing technologies in recent years, methods to fully encase objects with multilayer, thick features are still undeveloped. This issue can be overcome by printing layers conformally about an object’s natural geometry, as opposed to current methods that utilize planar layering. With this mindset, two new methods are derived to generate uniformly distributed layers between initial and desired geometries in both two and three dimensions. The first method is based on variable offset curves and can only be applied to convex or star-convex geometries. The second method is based on manipulated solutions to Laplace’s equation and is applicable to all geometries. Using each method, we present examples of layer generation for several geometries of varying convexities. Results are compared, and the respective advantages and limitations of each method are discussed.

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