[5] **viXra:1405.0324 [pdf]**
*submitted on 2014-05-26 14:33:21*

**Authors:** Jan Hakenberg

**Comments:** 31 Pages.

The formula for the volume enclosed by subdivision surfaces has been identified only recently. We present example meshes with cycles of edges defined as sharp creases, and state the volume enclosed by their limit surface defined by Catmull-Clark, and Loop subdivision. The article can serve as a reference for future implementations of the volume formula.

**Category:** Geometry

[4] **viXra:1405.0260 [pdf]**
*submitted on 2014-05-18 00:57:53*

**Authors:** S.Kalimuthu

**Comments:** 5 Pages. NA

Great circle triangles and its related trigonometry are wider applications in astronomy, astrophysics, cosmology, engineering fields, space travel, sea voyages, electronics, architecture etc. Maxwell’s electromagnetic theory showed that light is an electromagnetic wave, Dirac’s equation revealed the existence and generation of anti particles and Einstein’s filed equations predicted bending of light rays near a massive body, gravitational time dilation, gravitational waves , gravitational lenses, black holes, dark matter, dark energy and big bang singularity. All these findings have been experimentally established except gravitational waves. In this short work, the author finds a peculiar phenomenon in great circle triangles / Euler triangles

**Category:** Geometry

[3] **viXra:1405.0246 [pdf]**
*submitted on 2014-05-15 04:16:44*

**Authors:** Jan Hakenberg

**Comments:** 28 Pages.

Simple meshes such as the cube, tetrahedron, and tripod frequently appear in the literature to illustrate the concept of subdivision. The formula for the volume enclosed by subdivision surfaces has only recently been identified. We specify simple meshes and state the volume enclosed by the corresponding limit surfaces. We consider the subdivision schemes Doo-Sabin, Midedge, Catmull-Clark, and Loop.

**Category:** Geometry

[2] **viXra:1405.0215 [pdf]**
*submitted on 2014-05-11 19:02:40*

**Authors:** Ion Patrascu

**Comments:** 2 Pages.

O geometrie Smarandache este o geometrie în care cel puțin o axiomă este fie validată și invalidată, sau numai invalidată dar în multiple feluri (în cadrul aceluiași spațiu geometric).
Vom construi un model de geometrie Smarandache în care axioma paralelelor este validată pentru unele drepte și puncte, și invalidată pentru alte drepte și puncte.

**Category:** Geometry

[1] **viXra:1405.0012 [pdf]**
*submitted on 2014-05-02 10:39:37*

**Authors:** Jan Hakenberg, Ulrich Reif, Scott Schaefer, Joe Warren

**Comments:** 15 Pages.

We present a framework to derive the coefficients of trilinear forms that compute the exact volume enclosed by subdivision surfaces. The coefficients depend only on the local mesh topology, such as the valence of a vertex, and the subdivision rules. The input to the trilinear form are the initial control points of the mesh.

Our framework allows us to explicitly state volume formulas for surfaces generated by the popular subdivision algorithms Doo-Sabin, Catmull-Clark, and Loop. The trilinear forms grow in complexity as the vertex valence increases. In practice, the explicit formulas are restricted to meshes with a certain maximum valence of a vertex.

The approach extends to higher order momentums such as the center of gravity, and the inertia of the volume enclosed by subdivision surfaces.

**Category:** Geometry