Note for Computer Graphics - CG by rishi prasad

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UNIT I - 2D PRIMITIVES Output primitives – Line, Circle and Ellipse drawing algorithms - Attributes of output primitives – Two dimensional Geometric transformation - Two dimensional viewing – Line, Polygon, Curve and Text clipping algorithms Introduction A picture is completely specified by the set of intensities for the pixel positions in the display. Shapes and colors of the objects can be described internally with pixel arrays into the frame buffer or with the set of the basic geometric – structure such as straight line segments and polygon color areas. To describe structure of basic object is referred to as output primitives. Each output primitive is specified with input co-ordinate data and other information about the way that objects is to be displayed. Additional output primitives that can be used to constant a picture include circles and other conic sections, quadric surfaces, Spline curves and surfaces, polygon floor areas and character string. Points and Lines Point plotting is accomplished by converting a single coordinate position furnished by an application program into appropriate operations for the output device. With a CRT monitor, for example, the electron beam is turned on to illuminate the screen phosphor at the selected location Line drawing is accomplished by calculating intermediate positions along the line path between two specified end points positions. An output device is then directed to fill in these positions between the end points Digital devices display a straight line segment by plotting discrete points between the two end points. Discrete coordinate positions along the line path are calculated from the equation of the line. For a raster video display, the line color (intensity) is then loaded into the frame buffer at the corresponding pixel coordinates. Reading from the frame buffer, the video controller then plots “the screen pixels”. Pixel positions are referenced according to scan-line number and column number (pixel position across a scan line). Scan lines are numbered consecutively from 0, starting at the bottom of the screen; and pixel columns are numbered from 0, left to right across each scan line 1

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Figure : Pixel Postions reference by scan line number and column number To load an intensity value into the frame buffer at a position corresponding to column x along scan line y, setpixel (x, y) To retrieve the current frame buffer intensity setting for a specified location we use a low level function getpixel (x, y) Line Drawing Algorithms  Digital Differential Analyzer (DDA) Algorithm  Bresenham’s Line Algorithm  Parallel Line Algorithm The Cartesian slope-intercept equation for a straight line is y=m.x+b (1) Where m as slope of the line and b as the y intercept Given that the two endpoints of a line segment are specified at positions (x1,y1) and (x2,y2) as in figure we can determine the values for the slope m and y intercept b with the following calculations 2

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Figure : Line Path between endpoint positions (x1,y1) and (x2,y2) m = ∆y / ∆x = y2-y1 / x2 - x1 (2) b= y1 - m . x1 (3) For any given x interval ∆x along a line, we can compute the corresponding y interval ∆y ∆y= m ∆x (4) We can obtain the x interval ∆x corresponding to a specified ∆y as ∆ x = ∆ y/m (5) For lines with slope magnitudes |m| < 1, ∆x can be set proportional to a small horizontal deflection voltage and the corresponding vertical deflection is then set proportional to ∆y as calculated from Eq (4). For lines whose slopes have magnitudes |m | >1 , ∆y can be set proportional to a small vertical deflection voltage with the corresponding horizontal deflection voltage set proportional to ∆x, calculated from Eq (5) For lines with m = 1, are equal. ∆x = ∆y and the horizontal and vertical deflections voltage Figure : Straight line Segment with five sampling positions along the x axis between x1 and x2 3

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Digital Differential Analyzer (DDA) Algortihm The digital differential analyzer (DDA) is a scan-conversion line algorithm based on calculation either ∆y or ∆x The line at unit intervals in one coordinate and determine corresponding integer values nearest the line path for the other coordinate. A line with positive slop, if the slope is less than or equal to 1, at unit x intervals (∆x=1) and compute each successive y values as yk+1 = yk + m (6) Subscript k takes integer values starting from 1 for the first point and increases by 1 until the final endpoint is reached. m can be any real number between 0 and 1 and, the calculated y values must be rounded to the nearest integer For lines with a positive slope greater than 1 we reverse the roles of x and y, (∆y=1) and calculate each succeeding x value as xk+1 = xk + (1/m) (7) Equation (6) and (7) are based on the assumption that lines are to be processed from the left endpoint to the right endpoint. If this processing is reversed, ∆x=-1 that the starting endpoint is at the right yk+1 = yk – m (8) When the slope is greater than 1 and ∆y = -1 with xk+1 = xk-1(1/m) (9) If the absolute value of the slope is less than 1 and the start endpoint is at the left, we set ∆x = 1 and calculate y values with Eq. (6) When the start endpoint is at the right (for the same slope), we set ∆x = -1 and obtain y positions from Eq. (8). Similarly, when the absolute value of a negative slope is greater than 1, we use ∆y = -1 and Eq. (9) or we use ∆y = 1 and Eq. (7). 4