Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions

1924 ◽  
Vol 12 (168) ◽  
pp. 27
Author(s):  
T. P. Nunn ◽  
E. H. Neville
Keyword(s):  
Euclidean Space ◽  
Three Dimensions ◽  
Analytical Geometry

scholarly journals Trading spaces: building three-dimensional nets from two-dimensional tilings

2012 ◽  
Vol 2 (5) ◽  
pp. 555-566 ◽  
Author(s):  
Toen Castle ◽  
Myfanwy E. Evans ◽  
Stephen T. Hyde ◽  
Stuart Ramsden ◽  
Vanessa Robins
Keyword(s):  
Euclidean Space ◽  
Ab Initio ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional Euclidean Space ◽  
Two Dimensional ◽  
Complex Patterns ◽  
Geometrically Complex

We construct some examples of finite and infinite crystalline three-dimensional nets derived from symmetric reticulations of homogeneous two-dimensional spaces: elliptic ( S 2 ), Euclidean ( E 2 ) and hyperbolic ( H 2 ) space. Those reticulations are edges and vertices of simple spherical, planar and hyperbolic tilings. We show that various projections of the simplest symmetric tilings of those spaces into three-dimensional Euclidean space lead to topologically and geometrically complex patterns, including multiple interwoven nets and tangled nets that are otherwise difficult to generate ab initio in three dimensions.

Analytical Geometry of Three Dimensions

1934 ◽  
Vol 18 (231) ◽  
pp. 340
Author(s):  
H. W. Turnbull ◽  
D. M. Y. Sommerville
Keyword(s):  
Three Dimensions ◽  
Analytical Geometry

scholarly journals XII. On the Equations of Loci traced upon the Surface of the Sphere, as expressed by Spherical Co-ordinates

1834 ◽  
Vol 12 (1) ◽  
pp. 259-362
Author(s):  
Thomas Stephens Davies
Keyword(s):  
Three Dimensions ◽  
Orthogonal Projections ◽  
Analytical Geometry ◽  
Ground Plan ◽  
Modern System

The modern system of analytical geometry of three dimensions originated with Clairault, and received its final form from the hands of Monge. Descartes, it is true, had remarked, that the orthogonal projections of a curve anyhow situated in space, upon two given rectangular planes, determined the magnitude, species, and position of that curve; but this is, in fact, only an appropriation to scientific purposes of a principle which must have been employed from the earliest period of architectural delineation—the orthography and ichnography, or the ground-plan and section of the system of represented lines. Had Descartes, however, done more than make the suggestion—had he pointed out the particular aspect under which it could have been rendered available to geometrical research—had he furnished a suitable notation and methods of investigation—and, finally, had he given a few examples, calculated to render his analytical processes intelligible to other mathematicians;—then, indeed, this branch of science would have owed him deeper obligations than it can now be said to do.

ON THE COUPLER CURVE OF 3R2C LINKAGES

1998 ◽  
Vol 22 (3) ◽  
pp. 251-267
Author(s):  
H.S. Yan ◽  
W.H. Hsieh
Keyword(s):  
Algebraic Curve ◽  
Three Dimensions ◽  
Critical Properties ◽  
Dimensional Synthesis ◽  
Analytical Geometry ◽  
Loop Closure ◽  
Input And Output ◽  
Closure Equation

The purpose of this paper is to investigate the properties of the coupler curves generated by all 3R2C linkages. First, the 3x3 matrix with dual elements is used to derive the loop closure equation, the displacement equations are derived, and all joints variables are expressed in terms of input and output variables. Then, the parametric equations of the coupler curve are found by the D-H matrix. Finally, homogeneous coordinate is introduced to those displacement equations, and the order and some critical properties of the coupler curve are investigated based on the theories of algebraic curve and analytical geometry of three dimensions. In addition, RCRCR and RRCCR linkages are used as examples for illustration. Moreover, the results on the application of dimensional synthesis are discussed.

scholarly journals A bilinear transformation

1956 ◽  
Vol 40 ◽  
pp. 1-7
Author(s):  
G. N. Watson
Keyword(s):  
Three Dimensions ◽  
Bilinear Transformation ◽  
Analytical Geometry

The problem which I enunciate and solve in this paper seems to have originated in the study of properties of polyhedral functions. It is a problem of elementary analytical geometry of three dimensions, and the solution which I give, though somewhat tedious, is both elementary and direct. There are several comments which I have to make about current solutions, but I reserve these until the end of the paper since they will be more easily appreciated when it is possible to compare the current solutions with my solution.

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