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Saturday, 15 February 2014

Compter Graphics Notes - Lab 10 - Fundamental Problems in Geometry

FUNDAMENTAL PROBLEMS IN GEOMETRY
Trigonometry
The trigonometric functions of a general angle: Let 0 be an angle with its vertex at the origin and its initial side coincident with the x-axis, and let (x, y) be any point in the Cartesian plane distinct from the origin, and on the terminal side of the angle. The six trigonometric functions are as follows:

Polar Coordinates
We are used to the standard Cartesian coordinate system of graphing, but there are many other methods. One common and very useful method is the polar coordinate method. In this system, the origin is at the center of the plane and one specifies coordinates by giving an angle 0 and a radius r:






If you need to translate polar coordinates to Cartesian coordinates, use these formulas:

3-D Coordinate Systems
In a 3-dimensional Cartesian system, we specify 3 values to identify a point (x, y, z):




This is a right-handed system which is most often used in graphics, i.e., the thumb of the right hand points down the z-axis if we imagine grabbing the z-axis with the fingers of the right hand curling from the positive x-axis toward the positive y-axis.

Matrices
A matrix is a rectangular array of quantities, called the elements of a matrix. We identify matrices according to the number of rows X number of columns.


When number of rows = number of columns, we have a square matrix. A matrix with a single row or column represents a vector, and a larger matrix can be viewed as a collection of row or column vectors.


To multiply a matrix A by a scalar value s, we just multiply each element of the matrix by s. Matrix addition is defined only for matrices with the same number of rows and columns and is simply the sum of corresponding elements.

We can multiply an m X n matrix A by a p X q matrix B only if n = p. We obtain an m X q matrix C whose elements are calculated as in the following example:

Vector multiplication in matrix notation produces the same result as the inner product providing the first vector is a row vector and the second is a column vector. The transpose of a matrix AT is obtained by interchanging rows and columns.
               


The matrix B is the inverse of a matrix A if AB = I where I is the identity matrix, being all O's except for 1's on the diagonal.

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