Transcript Linear Algebra Application: Computer Graphics

By: Gabrien Clark
Math 2700.002
May 5th, 2010
Introduction
 In the simplest sense
computer graphics are images
viewable on a computer
screen. The images are
generated using computers
and likewise, are manipulated
by computers. Underlying the
representation of the images
on the computer screen is the
mathematics of Linear
Algebra.
2-Dimensional Graphics
 Examples of computer graphics are those of which belong
to 2 dimensions. Common 2D graphics include text. For
example the vertices of the letter H can be represented by
the following data matrix D:
3-Dimensional Graphics
 3-Dimensional graphics live in R3 versus 2-Dimensional graphics which
live in R2. 3-Dimensional graphics have a vast deal more applications in
comparison to 2-Dimensional graphics, and are, likewise, more
complicated. We will now work with the variable Z, in addition to X
and Y, to fully represent coordinates on the X, Y, and Z axes, or simply
space. For example we can represent a cube with the following data
matrix D:
Homogeneous Coordinates
 Homogeneous coordinates are a system of coordinates
used in projective geometry.
 They have the advantage that the coordinates of a
point, even those at infinity, can be represented using
finite coordinates. Often formulas involving
homogeneous coordinates are simpler and more
symmetric than their Cartesian counterparts.
Homogeneous Coordinates cont.
 Each point (x, y) that lives in R2 has homogeneous
coordinates (x, y, 1)
 Each point (x, y, z) that lives in R3 has homogeneous
coordinates (x, y, z, 1)
 (X, Y, H) are homogeneous coordinates for (x, y) and
(X, Y, Z, H) are coordinates for (x, y, z)
 So:
Basic Transformations
Scaling
Translation
Rotation
Scaling
 A point P with coordinates (x, y, z) is moved to a new
point P’ with coordinates (x’, y’, x’) which, in turn, is
equivalent to (C1x, C2y, C3z) where the Ci’s are scalars.
 What we end up seeing is either an enlargement or
diminishment of the original image.
Scaling in 2-Dimensions
 The scaling transformation is given by the matrix
S=
 The transformation is given by the multiplication of
the matrices S and A:
=
=
Scaling in 3-Dimensions
 In 3-Dimensions, scaling moves the coordinates (X,Y,Z) to new
coordinates (C1, C2, C3) where the Ci’s are scalars. Scaling in 3Dimensions is exactly like scaling in 2-Dimensions, except that the
scaling occurs along 3 axes, rather than 2.
 Note that if we view strictly from the XY-plane the scaling in the Zdirection can not be seen, if we view strictly from the XZ-plane the
scaling in the Y-direction can not be seen, and if we view strictly from
the YZ-plane then the scaling in the X-direction can not be seen.
XZ-plane
XY-plane
YZ-plane
Scaling in 3-Dimensions cont.
 The scaling transformation is given by the matrix
S=
 The transformation is given by the multiplication of
the matrices S and A:
=
=
Translation
 Translation is moving every point a constant distance
in a specified direction.
 The origin of the coordinate system is moved to
another position but the direction of each axis remains
the same. (There is no rotation or reflection.)
Translation in 2-Dimensions
 Mathematically speaking translation in 2-Dimensons is
represented by:
 Where e1 and e2 are the first two columns of the Identity
Matrix, and X0 and Y0 are the coordinates of the translation
vector T.
Translation in 3-Dimensions
 Mathematically speaking we can represent the 3-
Dimensional translation transformation with:
 Where e1, e2, and e3 are the first three columns of
the Identity Matrix, and X0,Y0, & Z0 are the
coordinates of the translation vector T.
Rotation
 A more complex transformation, rotation changes the
orientation of the image about some axis.
 The coordinate axes are rotated by a fixed angle θ
about the origin.
 The post-rotational coordinates of an image can be
obtained by multiplying the rotation matrix by the
data matrix containing the original coordinates of the
image.
Rotation in 2-Dimensions
 Counter-Clockwise Rotation Matrix:
 Clockwise Rotation Matrix:
Rotation in 3-Dimensions
 Rotation about the x-axis:
 Rotation about the y-axis:
 Rotation about the z-axis:
Composite Transformations
 The movement of images on a computer screen require
two or more basic transformations, such as scaling,
translating, and rotating.
 The mathematics responsible for this movement
corresponds to matrix multiplication of the
transformation matrices and the data matrix of the
homogeneous coordinates.
Works Cited
 Lay, David C. Linear Algebra and Its Applications. Boston: AddisonWesley,
2003. Print.
 Anton, Howard. Elementary Linear Algebra. New York: John Wiley, 1994.
657-65. Print.
 Wikipedia contributors. "Computer graphics." Wikipedia, The Free
Encyclopedia.
 Wikipedia, The Free Encyclopedia, 3 May. 2010. Web. 4 May.
2010.
 Wikipedia contributors. "Rotation matrix." Wikipedia, The Free Encyclopedia.
Wikipedia, The Free Encyclopedia, 2 May. 2010. Web. 4 May. 2010.
 Jordon, H. Rep. Web. Apr.-May 2010.
http://math.illinoisstate.edu/akmanf/newwebsite/linearalgebra/computergrap
hics.pdf.
 Wikipedia contributors. "Homogeneous coordinates." Wikipedia, The Free
Encyclopedia. Wikipedia, The Free Encyclopedia, 5 May. 2010. Web. 5 May.
2010.
Works Cited cont.
 http://www.swagelok.com/images/cmi/imageLibrary/CAD
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http://primaryvisualcortex.files.wordpress.com/2008/10/n
ecker_cube.png
http://wordnetweb.princeton.edu/perl/webwn?s=translati
on
http://www.art.unt.edu/ntieva/pages/about/newsletters/v
ol_14/no_1/TranslationLG.jpg
http://wordnetweb.princeton.edu/perl/webwn?s=rotation
http://homepages.inf.ed.ac.uk/rbf/HIPR2/rotateb.gif