At R = [infinity] and equal coefficients (9) the elliptic sphere transforms into the Euclidean plane
and Pythagoras' theorem begins to rule again (4).
The conical curves (circle, ellipse, hyperbola, parabola) considered on the Euclidean plane
are widely known and can also be found in the navigational applications.
After a historical introduction, he develops analytic projective geometry as an extension of the geometry of the euclidean plane
, sets out the axiomatic foundation of it, and introduces a metric into it.
12" is disclosed not to be the unitary object that its representation on the Cartesian, Euclidean plane
of the score suggests, but the locus of a complex of interacting perceptual frames.
A drawing of a graph G is a representation of G in the Euclidean plane
Just as a flat surface--like that of a sheet of paper--is a piece of the infinite mathematical surface known as the Euclidean plane
, a saddle-shaped surface can be thought of as a small piece of the hyperbolic plane.
A unit disk graph is associated with a set of unit disks in the Euclidean plane
That is, in the case of the Euclidean plane
(x, [phi]) wrapping over a cylinder we can identify the azimuthal parameter [phi] with the evolution parameter [tau].
Boroczky (Hungarian Academy of Sciences) builds from the foundation set by Toth (Regular Figures) and Rogers (Packing and Covering) by describing arrangements of congruent convex bodies that either form a packing in a convex container or cover a convex shape, covering arrangements in dimension two (including congruent domains in the Euclidean plane
, translative arrangements, parametric density, and packings and coverings of circular discs) and arrangements in higher dimensions, including packings and coverings by spherical and unit balls, and congruent convex bodies.
In this case a one-parameter homothetic motion in 2-dimensional Euclidean plane
defined by transformation