Frames
In 3D space, a sequence of 3 linearly independent vectors
v
1
,
v
2
,
v
3
is called a
frame, since it gives a coordinate system (a frame of reference). Any vector
v
can
be written as a linear combination
v
=
x
v
1
+
y
v
2
+
z
v
3
of vectors in the frame and
x,y,z
are called the coordinates of
v
in this frame.
In computations involving three dimensional space it is often necessary to ﬁnd
the coordinates of vectors in many diﬀerent frames. When we have two frames
there are two diﬀerent sets of coordinates for each point and there is a matrix to
change coordinates between the frames. Here we discuss some of the basic theory
of frames and coordinate changes.
A frame called the Frenet frame is useful in the study of curves. It is called a
moving frame because there is one at each point on the curve, and the points are
considered as a function of a parameter
t
often thought of as time. The idea of a
Frenet Frame can be adapted to study the shape of of long molecules such as DNA
and proteins, as will be discussed in a later chapter.
3.1.
Basic deﬁnitions.
A sequence of three vectors
v
1
,
v
2
,
v
3
can be made into
the columns of a 3
×
3 matrix denoted (
v
1
,
v
2
,
v
3
). If the determinant of this
matrix is not 0, then the sequence of vectors is called a
frame
and the vectors are
linearly independent
. We will not distinguish between the frame and the matrix.
The vectors
e
1
=
1
0
0
e
2
=
0
1
0
e
3
=
0
0
1
form a frame (
e
1
,
e
2
,
e
3
) which is the identity matrix
I
. This is called the
standard
basis
or
lab frame
.
The determinant of the matrix (
v
1
,
v
2
,
v
3
) is equal to the
scalar triple product
,
v
1
·
(
v
2
×
v
3
) = det(
v
1
,
v
2
,
v
3
)
.
If the determinant is positive, the frame is said to be
right-handed
, and if the
determinant is negative, the frame is said to be
left-handed
.
3.2.
Frames and gram matrices.
If
F
= (
v
1
,
v
2
,
v
3
) is a matrix, the entry in
row
i
, column
j
of the matrix
F
t
F
is the dot product
v
i
·
v
j
. This matrix is called
the
gram matrix
of the vectors
v
1
,
v
2
,
v
3
.
The frame is said to be an
orthogonal frame
if the vectors are mutually perpen-
dicular. If all of the vectors are of length 1 in an orthogonal frame, it is called an
orthonormal frame
. The condition that
F
is an orthonormal frame can be written
as
v
i
·
v
j
=
(
1 if
i
=
j
0 if
i
6
=
j.
This is equivalent to the gram matrix being the identity,