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Bresenham's Line Algorithm in Computer Graphics

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Bresenham's Line Algorithm in Computer Graphics

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Questions and Answers

What is the value of the initial decision parameter `p0` in Bresenham's line drawing algorithm?

$2\Delta y - \Delta x$ (correct)

$\Delta y$

$\Delta x$

$2\Delta y$

What is the value of the y-intercept `b` in the line equation $y = mx + b$?

$x_0 - m \cdot y_0$

$y_0 + m \cdot x_0$

$y_0 - m \cdot x_0$ (correct)

$x_0 + m \cdot y_0$

What is the value of the slope `m` in the line equation $y = mx + b$?

$x_0 / y_0$

$\Delta x / \Delta y$

$y_0 / x_0$

$\Delta y / \Delta x$ (correct)

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 $p_{k+1} = p_k + 2\Delta y$ Signup and view all the answers

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 $p_{k+1} = p_k + 2\Delta y - 2\Delta x$ Signup and view all the answers

What is the value of the constant `2Δy - 2Δx` used in Bresenham's line drawing algorithm?

$2\Delta y - \Delta x$ Signup and view all the answers

What is the value of the constant `2Δy` used in Bresenham's line drawing algorithm?

$2\Delta y$ Signup and view all the answers

For the line with endpoints (20, 10) and (30, 18), what is the value of Δx?

10 Signup and view all the answers

For the line with endpoints (20, 10) and (30, 18), what is the value of Δy?

8 Signup and view all the answers

For the line with endpoints (20, 10) and (30, 18), what is the value of the initial decision parameter p0?

6 Signup and view all the answers

For the line with endpoints (20, 10) and (30, 18), what is the value of the slope m?

0.8 Signup and view all the answers

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.

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Description

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.