Pictures play a vital role in human communication, in robotic manufacturing, and in digital imaging. In a typical application of digital imaging, a CCD camera records a digital picture frame that is read into the memory of a digital computer. The digital computer then manipulates this frame (or array) of data in order to crop, enlarge or reduce, enhance or smooth, translateor rotate the original picture. These procedures are called digital picture processing or computer graphics. When a sequence of picture frames is processed and displayed at video frame rates (30 frames per second), then we have an animated picture.
In this chapter we use the linear algebra we developed in The chapter on Linear Algebra
to develop a rudimentary set of tools for doing computer graphics on line
drawings. We begin with an example: the rotation of a single point in the
(x,y)(x,y) plane.
Point
P
P has coordinates (3,1)(3,1) in the (x,y)(x,y) plane as
shown in Figure 1. Find the coordinates of the point
P
'
P', which is rotated
π6π6 radians from
P
P.
To solve this problem, we can begin by converting the point
P
P from
rectangular coordinates to polar coordinates. We have
r
=
x
2
+
y
2
=
10
θ
=
tan

1
(
y
x
)
≈
0
.
3217
r
a
d
i
a
n
.
r
=
x
2
+
y
2
=
10
θ
=
tan

1
(
y
x
)
≈
0
.
3217
r
a
d
i
a
n
.
(1)The rotated point
P
'
P' has the same radius
r
r, and its angle is θ+π6θ+π6. We now
convert back to rectangular coordinates to find
x
'
x' and
y
'
y' for point
P
'
P':
x
'
=
r
c
o
s
(
θ
+
π
6
)
≈
10
cos
(
0
.
8453
)
≈
2
.
10
y
'
=
r
s
i
n
(
θ
+
π
6
)
≈
10
sin
(
0
.
8453
)
≈
2
.
37
.
x
'
=
r
c
o
s
(
θ
+
π
6
)
≈
10
cos
(
0
.
8453
)
≈
2
.
10
y
'
=
r
s
i
n
(
θ
+
π
6
)
≈
10
sin
(
0
.
8453
)
≈
2
.
37
.
(2)So the rotated point P
'
P' has coordinates (2.10, 2.37).
Now imagine trying to rotate the graphical image of some complex
object like an airplane. You could try to rotate all 10,000 (or so) points in the
same way as the single point was just rotated. However, a much easier way
to rotate all the points together is provided by linear algebra. In fact, with a
single linear algebraic operation we can rotate and scale an entire object and
project it from three dimensions to two for display on a flat screen or sheet
of paper.
In this chapter we study vector graphics, a linear algebraic method
of storing and manipulating computer images. Vector graphics is especially
suited to moving, rotating, and scaling (enlarging and reducing) images and
objects within images. Cropping is often necessary too, although it is a little
more difficult with vector graphics. Vector graphics also allows us to store
objects in three dimensions and then view the objects from various locations
in space by using projections.
In vector graphics, pictures are drawn from straight lines. A curve
can be approximated as closely as desired by a series of short, straight lines.
Clearly some pictures are better suited to representation by straight lines
than are others. For example, we can achieve a fairly good representation of
a building or an airplane in vector graphics, while a photograph of a forest
would be extremely difficult to convert to straight lines. Many computer
aided design (CAD) programs use vector graphics to manipulate mechanical
drawings.
When the time comes to actually display a vector graphics image, it
may be necessary to alter the representation to match the display device. Personal computer display screens are divided into thousands of tiny rectangles
called picture elements, or pixels. Each pixel is either off (black) or on (perhaps with variable intensity and/or color). With a CRT display, the electron
beam scans the rows of pixels in a raster pattern. To draw a line on a pixel
display device, we must first convert the line into a list of pixels to be illuminated. Dot matrix and laser printers are also pixel display devices, while
pen plotters and a few specialized CRT devices can display vector graphics
directly. We will let MATLAB do the conversion to pixels and automatically
handle cropping when necessary.
We begin our study of vector graphics by representing each point in
an image by a vector. These vectors are arranged sidebyside into a matrix
G
G containing all the points in the image. Other matrices will be used as
operators to perform the desired transformations on the image points. For
example, we will find a matrix
R
R, which functions as a rotation: the matrix
product RGRG represents a rotated version of the original image
G
G.
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