The paper proposes a method for rapid prototyping (RP) fractal geometry represented objects. RP technology has made possible the physical fabrication of solid freeform objects. However, contemporary CAD/CAM/RP systems are developed for Euclidean geometry and are incapable of handling self-similar fractal objects (e.g. jewelry products). To address the problem, a Radial-Annular Tree (RAT) data s...

We attach the degenerate signature (n,0,1) to the dual Grassmann algebra of projective space to obtain a real Clifford algebra which provides a powerful, efficient model for euclidean geometry. We avoid problems with the degenerate metric by constructing an algebra isomorphism between the Grassmann algebra and its dual that yields non-metric meet and join operators. We focus on the cases of n=2...

This paper gives a brief overview of FG, a formal system for doing Euclidean geometry whose basic syntactic elements are geometric diagrams, and which has been implimentented as the computer system CDEG. The computational complexity of determining whether or not a given diagram is satisfiable is also briefly discussed. 1 A Diagrammatic Formal System To begin with, consider Euclid’s first propos...

In two articles published in 1868 and 1869, Eugenio Beltrami provided three models in Euclidean plane (or space) for non-Euclidean geometry. Our main aim here is giving an extensive account of the two articles’ content. We will also try to understand how the way Beltrami, especially in the first article, develops his theory depends on a changing attitude with regards to the definition of surfac...

The goal of this handout is to discuss models of Hyperbolic and Euclidean Geometry, and the consistency of Hyperbolic Geometry. 1. Some concepts from Euclidean Geometry We will use circles in Euclidean Geometry to build up models for Hyperbolic Geometry. So we will review some basic facts about circles. (Mostly without proofs). In this section assume that we are working in Euclidean Geometry. D...

This paper briefly describes CDEG 2.0, a computerized formal system for giving diagrammatic proofs in Euclidean geometry.

We calculate the values of the trigonometric functions for angles: π3 and π 6 , by [16]. After defining some trigonometric identities, we demonstrate conventional trigonometric formulas in the triangle, and the geometric property, by [14], of the triangle inscribed in a semicircle, by the proposition 3.31 in [15]. Then we define the diameter of the circumscribed circle of a triangle using the d...

Deformation principle as foundation of physical geometry and its application to space-time geometry. Abstract Physical geometry studies mutual disposition of geometrical objects and points in space, or space-time, which is described by the distance function d, or by the world function σ = d 2 /2. One suggests a new general method of the physical geometry construction. The proper Euclidean geome...

Deformation principle as foundation of physical geometry and its application to space-time geometry. Abstract Physical geometry studies mutual disposition of geometrical objects and points in space, or space-time, which is described by the distance function d, or by the world function σ = d 2 /2. One suggests a new general method of the physical geometry construction. The proper Euclidean geome...

The deformation principle admits one to obtain a very broad class of nonuniform geometries as a result of deformation of the proper Euclidean geometry. The Riemannian geometry is also obtained by means of a deformation of the Euclidean geometry. Application of the deformation principle appears to be not consecutive, and the Riemannian geometry appears to be not completely consistent. Two differ...