Linear Algebra Application: Computer Graphics

By: Gabrien ClarkMath 2700.002May 5th, 2010

IntroductionIn 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 GraphicsExamples 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 Graphics3-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 CoordinatesHomogeneous 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 TransformationsScaling

Translation

Rotation

ScalingA 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-DimensionsThe 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 3-Dimensions 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 Z-direction 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.

XY-plane

XZ-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:

= =

TranslationTranslation 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-DimensionsMathematically 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-DimensionsMathematically 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.

RotationA 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-DimensionsCounter-Clockwise Rotation Matrix:

Clockwise Rotation Matrix:

Rotation in 3-DimensionsRotation about the x-axis:

Rotation about the y-axis:

Rotation about the z-axis:

Composite TransformationsThe 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/computergraphics.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/imageL

ibrary/CAD%20image.jpghttp://primaryvisualcortex.files.wordpress.co

m/2008/10/necker_cube.pnghttp://wordnetweb.princeton.edu/perl/webwn

?s=translationhttp://www.art.unt.edu/ntieva/pages/about/ne

wsletters/vol_14/no_1/TranslationLG.jpghttp://wordnetweb.princeton.edu/perl/webwn

?s=rotationhttp://homepages.inf.ed.ac.uk/rbf/HIPR2/rotat

eb.gif