Spheroidal Domains and Geometric Analysis in Euclidean Space

Conformal Geometry, Euclidean Space and Geometric Algebra

Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to programming complicated geometrical operations. But there is a fundamental weakness in this approach — the Euclidean distance between points is not handled in a s...

Isospectral Domains in Euclidean 3-Space

The question as to whether the shape of a drum can be heard has existed for around fifty years. The simple answer is ‘no’ as shown through the construction of isospectral domains. Isospectral domains are non-isometric domains that display the same spectra of frequencies of sound. These frequencies, deduced from the eigenvalues of the Laplacian, are determined by solving the wave equation in a d...

Geometric and Fourier analytic questions in Euclidean space

We summarize the proof in [5] of a sharp decay estimate for the L2 circular means of the Fourier transform of a measure in R2 with a given fractal dimension. 3.1 Connecting distance sets and spherical means of Fourier transforms of measures Let E be a compact set in R. The corresponding distance set ∆(E) is defined as ∆(E) = {|x − y| : x, y ∈ E}. It is conjectured that if dimE > 1 then |∆(E)| >...

Vector and Tensor Analysis in Euclidean Space

On the Quaternionic Curves in the Semi-Euclidean Space E_4_2

In this study, we investigate the semi-real quaternionic curves in the semi-Euclidean space E_4_2. Firstly, we introduce algebraic properties of semi-real quaternions. Then, we give some characterizations of semi-real quaternionic involute-evolute curves in the semi-Euclidean space E42 . Finally, we give an example illustrated with Mathematica Programme.