Solid Analytic Geometry

186 Pages·1949·6.524 MB·English

by  Adrian Albert

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OSMANIA UNIVERSITY LIBRARY
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In this new text the author presents an ex-
position of the analytic geometry of three-
dimensional space. The material covers the
standard topics of space analytic geometry but
provides a treatment of the subject which per-
mits immediate generalization to n dimensions.
This treatment ties the subjectto modernmath-
ematics, and, in particular, to modern algebra.
The use of the theory of vector spaces and ma-
trices permits a major simplification in the
proofs and in the exposition in general. Thus
the aim of the book is to provide a modern and
simpler treatment of the subject matter which
permits easy generalization and fits the sub-
ject into its proper place in modern mathe-
matics.
The early part of the text is simplified by the
uucste. Aofbtrhieefcbountceapdtesquoafteincnhearptaenrdonsctahlearthperoordy-
of matrices provides a full, clear exposition
of the principal axis transformation in the
n-dimensional case. An additional feature is
provided by the chapter on spherical coordi-
nates where the mathematics of the approxima-
tions used in actual physical measurements of '
direction and distance in rotated coordinate
systems is presented. This material is r</^ usu-
ally found in texts on solid analytic geometry.
Chapters I and II contain a treatment of the
equations of lines and planes. After a prelimi-
nary study of the linear operations in n-dimen-
sionalvectors andinnerproducts areinterpreted
geometrically, and from a consideration of
scalar products and axis translations, the para-
metric equations of a line are obtained. The
vector approach then yields a very simple
derivation of the normal form of an equation
of a plane, and the standard forms of plane and
line equations are rather immediate conse-
quences.
Chapter III presents classical elementary sur-
ufascuealantdrecautrmveentthoeforsyp.heCrhesa,ptaenrdICVhacpotnetarinVs tthhee
classical descriptions of quadric surfaces in
standard position. This chapter ends with a
rather novel classification of quadrics accord-
ing to certain invariants.
Chapter VI is an exposition of that part of the
theory of matrices needed for a complete de-
velopment oAf the so-called principal axis trans-
formation. full account of the orthogonal
reduction of a real quadratic form in n varia-
bles is given, and the theory is applied in
Chapter VII to the three-dimensional case of
A
quadric surfaces. discussion of the sym-
metries of "dric surfaces is included.
Chapter Vi spherical coordinates, contains
a discussion of some practical aspects of the
theory of rotations and translations of axes in
space.
The final chapter offers a brief presentation of
the elements of projectiye geometry. The ef-
fect on the theory of linear transformations
of the use of homogeneous coordinates is given
.and the chapter contains a rigorous uidtrix
proof of the invariance of the cross ratio under
protective transformations.

SOLID ANALYTIC GEOMETRY