Authors: Morio Kikuchi
Comments: 15 Pages.
In equidistant curve coordinate system, the two expressions of the length between two points in disk and upper half-plane are the same.
Authors: Marcos Georgallides
Comments: 21 Pages.
Article < The Six , Triple Concurrency Points , Line > is an extension of two Fundamental branches of geometry that of Perspectivity ( Desargues`s theorem , where 3 concurrency Points in a center of Perspectivity and 3 concurrency points on a line of Perspectivity , per two sides ) and that of Projective geometry ( Pascal`s theorem , with the 3 concurrency Points on a line , per two sides ) . Analyzing Extremum Principle ( Extrema ) on lines and Points , it was found that in any triangle ( three points only , which form a Plane ) and on the circumcircle exist one Inscribed and one Circumscribed , Extrema Triangle , such that on the six Extrema lines ( with a common concurrency point ) , both Perspectivity and Projective geometry concurrence on Common points on Extrema Lines . i.e. 18 lines concurrence in Six Points , per three , on a line , six triple concurrency points line.
This Compact logic of Extrema exists on Points and in lines of Euclidean geometry.
Article < Energy Laws follow Properties of Euclidean geometry > , is the deeper concept of Pythagoras theorem , where Conservation laws , referred to Physics and Mechanics , follow Euclidean moulds because these Principles belong to geometry as Points and Spaces ( geometry ) create Quantities and Qualities . Analyzing Euclid Spaces , it was found that on any two Equal and perpendicular , One dimensional Units , exists a Plane Formation ( A changeable and constant Tensor ) of constructing Squares , such that the Sum of Areas of the two Changeable Squares ( the Sum of the Squares of sides) is constant and equal to that of the circumscribed Square .The same also exists in Space Formation , where then ,
The Total Resultant Volume (cube of Resultant Sphere ) is the Sum of Changeable Volumes ( the Sum of the Cubes of Spheres of sides ). In Space Formation Changeable Volumes are Perpendicular each other , meaning that Conservation in Space ( Solid geometry ) occurs on Perpendiculars since first dimensional Units are Vectors .
This geometrical mould of Conservation , is followed by Energy in Mechanics and Physics . i.e.
The referred Energy Conservation laws in Mechanics and Physics , follow the Principle ( mould ) of Conserved Areas for Pythagoras` theorem on the moving machine of the two changeable Squares , and Conserved Perpendicular Volumes for Spaces on the Three Changeable Spheres .
Authors: David Proffitt
Comments: 3 Pages.
A reformulation of the area of a planar two-dimensional object in the frequency domain allows for the computation of the true area of a band-limited boundary to be calculated.