文档介绍：DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES
Course notes written by A.-L. Mare
- Fall 2006 -
1
2
1. Curves in the Plane
. Points, Vectors, and Their Coordinates. Points and vectors are fundamental
objects in Geometry. The notion of point is intuitive and clear to everyone. The notion
of vector is a bit more delicate. In fact, rather than saying what a vector is, we prefer
to say what a vector has, namely: direction, sense, and length (or magnitude). It can be
represented by an arrow, and the main idea is that two arrows represent the same vector if
they have the same direction, sense, and length. An arrow representing a vector has a tail
and a tip. From the (rough) definition from above, we deduce that in order to represent (if
you want, to draw) a given vector as an arrow, it is necessary and sufficient to prescribe its
tail.
a
c
b b
a
c
a a
b
P
Figure 1. We see four copies of the vector a, three of
the vector b, and two of the vector c. We also see a point
P .
An important instrument in handling points, vectors, and (consequently) many other
geometric objects is the Cartesian coordinate system in the plane. This consists of a point
O, called the origin, and two perpendicular lines going through O, called coordinate axes.
Each line has a positive direction, indicated by an arrow (see Figure 2). We denote by R the
a
P
y
a
y
x
O O
x
Figure 2. The point P has coordinates x, y. The vector
a has also coordinates x, y.
set of all real numbers and by R2 the set of all pairs of numbers, of the form (x, y), where
x, y are in R. Points are identified with elements R2, as follows: to each point P corresponds
the pair (x, y) consisting of the coordinates of the projections of P on the two axes. We say
that P has coordinates (x, y). Also vectors are identified with elements of R2, as follows: if
3
a is a vector, we move its tail to the origin O, and we take the coordinates of its tip. We say
that a has coo