11 Questions
What is the value of the initial decision parameter p0 in Bresenham's line drawing algorithm?
$2\Delta y - \Delta x$
What is the value of the y-intercept b in the line equation $y = mx + b$?
$y_0 - m \cdot x_0$
What is the value of the slope m in the line equation $y = mx + b$?
$\Delta y / \Delta x$
In Bresenham's line drawing algorithm, how is the next pixel to plot determined if the decision parameter p_k is less than 0?
The next pixel is $(x_k + 1, y_k)$ and $p_{k+1} = p_k + 2\Delta y$
In Bresenham's line drawing algorithm, how is the next pixel to plot determined if the decision parameter p_k is greater than or equal to 0?
The next pixel is $(x_k + 1, y_k + 1)$ and $p_{k+1} = p_k + 2\Delta y - 2\Delta x$
What is the value of the constant 2Δy - 2Δx used in Bresenham's line drawing algorithm?
$2\Delta y - \Delta x$
What is the value of the constant 2Δy used in Bresenham's line drawing algorithm?
$2\Delta y$
For the line with endpoints (20, 10) and (30, 18), what is the value of Δx?
10
For the line with endpoints (20, 10) and (30, 18), what is the value of Δy?
8
For the line with endpoints (20, 10) and (30, 18), what is the value of the initial decision parameter p0?
6
For the line with endpoints (20, 10) and (30, 18), what is the value of the slope m?
0.8
Study Notes
Bresenham's Line Algorithm
- Bresenham's Line Algorithm is an efficient raster line-generating algorithm that uses only incremental integer calculations.
- It is used for scan-converting lines with positive slope less than 1.0.
Calculating Pixel Separations
- At sampling position xk + 1, the y-coordinate on the mathematical line is calculated as y = m(xk + 1) + b.
- The vertical pixel separations from the mathematical line path are labeled as dlower and dupper.
- dlower = y - yk = m(xk + 1) + b - yk and dupper = (yk + 1) - y = (yk + 1) - m(xk + 1) - b.
Determining the Closest Pixel
- The difference between the two pixel separations is calculated as dlower - dupper = 2m(xk + 1) - 2yk + 2b - 1.
- The decision parameter pk is set up to determine which pixel is closest to the line path.
- pk = 2∆y * xk - 2∆x * yk + 2∆y + ∆x(2b - 1) = 2∆y * xk - 2∆x * yk + C.
- If pk is negative, the pixel at yk is "closer" to the line path, and the lower pixel is plotted; otherwise, the upper pixel is plotted.
Incremental Integer Calculations
- The decision parameter pk is calculated using incremental integer calculations.
- The sign of pk is the same as the sign of dlower - dupper.
- The values of successive decision parameters are obtained using incremental integer calculations.
- pk+1 = pk + 2∆y - 2∆x(yk+1 - yk).
Bresenham's Line-Drawing Algorithm
- The algorithm is used for |m| < 1.0.
- The steps involved in the algorithm are:
- Input the two line endpoints and store the left endpoint in (x0, y0).
- Set the color for frame-buffer position (x0, y0) and plot the first point.
- Calculate the constants ∆x, ∆y, 2∆y, and 2∆y - 2∆x, and obtain the starting value for the decision parameter as p0 = 2∆y - ∆x.
- At each xk along the line, starting at k = 0, perform the following test. If pk < 0, the next point to plot is (xk + 1, yk) and pk+1 = pk + 2∆y. Otherwise, the next point to plot is (xk + 1, yk + 1) and pk+1 = pk + 2∆y - 2∆x.
- Perform step 4 ∆x - 1 times.
Example
- For the line (x0, y0) = (20, 10) and (xend, yend) = (30, 18), the line has a slope of 0.8, ∆x = 10, and ∆y = 8.
- The initial decision parameter p0 = 2∆y - ∆x = 6.
- The increments for calculating successive decision parameters are 2∆y = 16 and 2∆y - 2∆x = -4.
Test your understanding of Bresenham's Line Algorithm, an efficient raster line-generating algorithm that uses incremental integer calculations. This quiz focuses on the scan-conversion process for lines with positive slope less than 1.0 and the decision-making process for plotting pixels.
Make Your Own Quizzes and Flashcards
Convert your notes into interactive study material.
Get started for free
