Linear Algebra Application: Computer Graphics

1. Linear Algebra Application: Computer Graphics By: Gabrien Clark Math 2700.002 May 5th, 2010

2. 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.

3. 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:

4. 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:

5. 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.

6. 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:

7. Basic Transformations

8. 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.

9. 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: = =

10. 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. XZ-plane XY-plane YZ-plane

11. 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: = =

12. 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.)

13. 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.

14. 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.

15. 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.

16. Rotation in 2-Dimensions • Counter-Clockwise Rotation Matrix: • Clockwise Rotation Matrix:

17. Rotation in 3-Dimensions • Rotation about the x-axis: • Rotation about the y-axis: • Rotation about the z-axis:

18. 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.

19. 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.

20. Works Cited cont. • http://www.swagelok.com/images/cmi/imageLibrary/CAD%20image.jpg • http://primaryvisualcortex.files.wordpress.com/2008/10/necker_cube.png • http://wordnetweb.princeton.edu/perl/webwn?s=translation • http://www.art.unt.edu/ntieva/pages/about/newsletters/vol_14/no_1/TranslationLG.jpg • http://wordnetweb.princeton.edu/perl/webwn?s=rotation • http://homepages.inf.ed.ac.uk/rbf/HIPR2/rotateb.gif