Greenwood press
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book-20600
Reflection in
Rotation of 90° Reflection in Rotation of the y-axis counterclockwise the line y = x 30° counterclockwise −1 0 0 1 0 −1 1 0 0 1 1 0 cos 30 − sin 30 sin 30 cos 30 Common transformations of the coordinate plane. Computer graphics use products of 4 × 4 geometric matrices to model the changes of position of moving objects in space (such as the space shuttle), trans- form them to eye coordinates, select the area of vision that would fit on the com- puter screen, and project the three-dimensional image onto the two dimensions of the video screen. The matrix products must be computed very rapidly to give the images realistic motion, so processors in high-end graphic computers embed the matrix operations in their circuits. Additional matrices compute light-and- shadow patterns that make the image look realistic. The same matrix operations used to provide entertaining graphics are built into medical instruments such as MRI machines and digital X-ray machines. Matrices such as incidence matrices and path matrices organize connections and distances between points. Airlines use these matrices on a daily basis to determine the most profitable way to assign planes to flights between different cities. The complexity of handling the different forms of rotation that are encoun- tered in movement requires computers that can process matrix computations very rapidly. The space shuttle, for example, is constantly being monitored by matrices that represent rotations in three-space. These matrix products control pitch, the rotation that causes the nose to go up or down, yaw, the rotation that causes the nose to rotate left or right, and roll, the rotation that causes the shuttle to roll over. Stochastic matrices are formed from probabilities. They can represent com- plex situations such as the probabilities of changes in weather, the probabilities of rental-car movements among cities, or more simple situations, such as the probabilities of color shifts in generations of roses. When the probabilities are dependent only on the prior state, the matrix represents a Markov chain. High powers of the matrix will converge on a set of probabilities that define a final, steady state for the situation. In population biology, for example, Markov chains show how arbitrary proportions of genes in one generation can produce variation in the immediately following generation, but that over the long term converge to a specific and stable distribution. Biologists have used Markov chains to describe population growth, molecular genetics, pharmacology, tumor growth, and epi- demics. Social scientists have used them to explain voting behavior, mobility and population of towns, changes in attitudes, deliberations of trial juries, and con- sumer choices. Albert Einstein used Markov theory to study the Brownian motion of molecules. Physicists have employed them in the theory of radioactive transformations. Astronomers have used Markov chains to analyze the fluctua- tions in the brightness of galaxies. Ratings of football teams can be done solely on the basis of the team’s sta- tistics. But more effective and comprehensive ratings of the teams use the statis- tics of opponents as well. Matrices provide a way of organizing corresponding information on the team and those it has played. Solving the matrix systems that result provides a power rating that integrates information on the strength of the opponents with the information on the team. Sport statisticians contend that the use of the data make their national ratings more reliable than those that use human judgment. Download 1.81 Mb. Do'stlaringiz bilan baham: |
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