will be the same as from bto a; and the distance between the points does not change if
they are totally shifted over in one direction (such a sliding over of the points is called
translation; for more on this, see elsewhere in this chapter). In addition, the
Pythagorean theorem is valid for three points that are the vertices (the intersection
points of the sides of an angle) of a right triangle.
What are some of the basic “building blocks”of geometry?
There are several basic “building blocks” of geometry, all of which have to do with the
objects we often see in geometry. A zero-dimensional object that is specifically located
in n-dimensional space using ncoordinates is called a point.The idea of a point may
be obvious to most people, but for mathematicians, describing and dealing with points
is not straightforward. For example, Euclid once gave a vague definition of a point as
“that which has no part.”
Euclid also called the linea “breadthless length,” and further called a straight line
one that “lies evenly with the points on itself.” Modern mathematicians define lines as
one-dimensional objects (although they may be part of a higher-dimensional space).
They are mathematically defined as a theoretical course of a moving point that is
thought to have length but no other dimension. They are often called straight lines, or
by the archaic term, a right line, to emphasize the fact that there are no curves any-
where along the entire length. It is interesting to note that when geometry is used in
an axiomatic system, a line is usually considered an undefined term (for more about
axiomatic systems, see “Foundations of Mathematics”). In analytic geometry, a line is 171
GEOMETRY AND TRIGONOMETRY
How is a curve defined in geometry?
A
curve is a continuous collection of points drawn from one-dimensional
space to n-dimensional space; it is also considered an object that can be cre-
ated by moving a point. But note: Our usual use of the word “curve” does not
mean a straight line, but in mathematics, a line or triangle is often referred to as
a curve.Different forms of geometry define curves in various ways. Analytic geome-
try uses plane curves—such as circles, ellipses, hyperbolas, and parabolas—
which are usually considered as the graph of an equation or function. The prop-
erties of these curves are largely dependent on the degree of the equation in the
case of algebraic curves (curves with algebraic equations) or on the particular
function, as in the case of transcendental curves (curves whose equations are not
algebraic). Even more complex are space curves, all of which require special
techniques used only in differential geometry.