Computer Graphics: Graphics Output Primitives Line Drawing Algorithms

Transcription

1 Computer Graphics: Graphics Output Primitives Line Drawing Algorithms By: A. H. Abdul Hafez 1

2 Outlines 1. Basic concept of lines in OpenGL 2. Line Equation 3. DDA Algorithm 4. DDA Algorithm implementation 5. End 2

3 Basic concept of lines in OpenGL To display the line on a raster monitor, the graphics system must first project the endpoints to integer screen coordinates Next, it determines the nearest pixel positions along the line path between the two endpoints. Then the line color is loaded into the frame buffer at the corresponding pixel coordinates. 3

4 Basic concept of lines in OpenGL Reading from the frame buffer, the video controller plots the screen pixels. This process digitizes the line into a set of discrete integer positions that, in general, only approximates the actual line path. This rounding of coordinate values to integers causes all but horizontal and vertical limes to be displayed with a stair-step appearance ("the jaggies"). 4

5 Line Equation The Cartesian slope-intercept equation for a straight line is we can determine values for the slope m and y intercept b with the following calculations: For any given x interval δx along a line, we can compute the corresponding y interval δy On raster systems, lines are plotted with pixels. That is, we must "sample" a line at discrete positions and determine the nearest pixel to the line at each sampled position. Discrete sample positions along the x axis is shown. 5

6 DDA Algorithm A line is sampled at unit intervals in one coordinate and the corresponding integer values nearest the line path are determined for the other coordinate. We consider first a line with positive slope; start is at left: 1. If the slope is less than or equal to 1, we sample at unit x intervals (δx = 1) and compute successive y values as 2. For lines with a positive slope greater than 1, we reverse the roles of x and y. That is, we sample at unity intervals (δy = 1) and calculate consecutive x values as In this case, each computed x value is rounded to the nearest pixel position along the current y scan line. 6

7 DDA Algorithm We consider second a line with positive slope; start is at right: If the slope is less than or equal to 1, we sample at unit x intervals (δx = -1) and compute successive y values as For lines with a slope greater than 1, we sample at unity intervals (δy = -1) and calculate consecutive x values as If the slop is negative, we consider similarly to the two above cases. See next slide for generalization. 7

8 DDA Algorithm. General case of slope If the absolute value of the-slope-is less than 1 and the starting endpoint is at the left, we set δx = 1 and calculate y values with When the starting endpoint is at the right (for the same slope), we set δx = -1 and obtain y positions using If the absolute value greater than 1 and the starting point at the left, we use δy = -1 and When the starting point is at the right, we use δy = 1 and 8

9 DDA Algorithm implementation This algorithm is summarized in the following procedure, which accepts as input two integer screen positions for the endpoints of a line segment. Horizontal and vertical differences between the endpoint positions are assigned to parameters dx and dy. The difference with the greater magnitude determines the value of parameter steps. Starting with pixel position (x 0, y 0 ), we determine the offset needed at each step to generate the next pixel position along the line path. We loop through this process steps times. 1. If the magnitude of dx is greater than the magnitude of dy and x 0 is less than x End, the values for the increments in the x and y directions are 1 and m=dy/dx, respectively. 2. If the greater change is in the x direction, but x 0 is greater than x End, then the decrements -1 and m=dy/dx are used to generate each new point on the line. 3. Otherwise, we use a unit increment (or decrement) in the y direction and an x increment (or decrement) of 1/m=dx/dy. 9

10 DDA Algorithm implementation 10

11 Bresenham's Line Algorithm Motivation: A section of the screen showing a pixel in column x k on scan line y k that is to be plotted along the path of a line segment with slope 0 < m < 1. Assuming we have determined that the pixel at (x k, y k ) is to be displayed, we next need to decide which pixel to plot in column x k+l = x k + 1. Our choices are the pixels at positions (x k + 1, y k ) and (x k + 1, y k +1). 11 March 6, 2016

12 Bresenham's Line Algorithm To determine which of the two pixels is closest to the line path, we can set up an efficient test that is based on the difference between the two pixel separations, and defining the decision parameter as Vertical distances between pixel positions and the line y coordinate at sampling position x k March 6, 2016

13 Bresenham's Line Algorithm 13

14 Bresenham's Line Algorithm EXAMPLE 3-1 from the text 14

15 Bresenham's Line Algorithm EXAMPLE 3-1 cont. 15

16 Bresenham's Line Algorithm Pixel positions along the line path between endpoints (20, 10) and (30, 18), plotted with Bresenham s line algorithm. 16

17 Bresenham's Line Algorithm Code 17

18 Bresenham's Line Algorithm Code 18

19 The end of the Lecture Thanks for your time Questions are welcome 19