Computer Graphics CC416 Lecture 04: Bresenham Line Algorithm & Mid-point circle algorithm Dr. Manal Helal – Fall 2014

Output primitives Graphics programming packages provide functions to describe a scene in terms of basic geometric structures referred to as Output Primitives Output primitives include: –Points –Lines –Circles –Spline curves

Output primitives can be used to present more complex objects

A point is plotted by setting the bit value corresponding to a specified screen position within the frame buffer to 1 In RGB system, the frame buffer is loaded with the color codes for intensities to be displayed

Line drawing: –Is accomplished by calculating intermediate positions along the line path between two specified endpoint positions –Digital devices display a straight line segment by plotting discrete points between the two endpoints

Discrete coordinate positions along the line path are calculated from the equation of the line – Screen locations are referenced with integer values, so plotted positions may only approximate actual Line positions –A computed line position of (10.48,20.51), for example, would be converted to pixel position (10,21) –This rounding of coordinate values to integers causes lines to be displayed with a stair-step appearance

Line Drawing Algorithms The straight line equation –m represents the slope –b as the y intercept Given the end points of a line (x 1, y 1 ) and (x 2, y 2 ), we can determine the values of m and b

For any given x interval Δx along a line, we can compute the corresponding y interval Δy Similarly, we can obtain the x interval Δ x corresponding to a specified Δ y as

These equations form the basis for determining deflection voltages in analog devices –For lines with slope magnitudes ImI < 1, Δx can be set proportional to a small horizontal deflection voltage and the corresponding vertical deflection is then set proportional to Δy using: –For lines whose slopes have magnitudes ImI > 1, Δy can be set proportional to a small vertical deflection voltage with the corresponding horizontal deflection voltage set proportional to Δx using:

–For lines with m = 1, Δx = Δy and the horizontal and vertical deflections voltages are equal On raster systems, lines are plotted with pixels –Step sizes in the horizontal and vertical directions are constrained by pixel separations. –That is, we must "sample" a line at discrete positions and determine the nearest pixel to the line at each sampled position.

Bresenham's Line Algorithm It decides which of two possible pixel positions is closer to the line path at each sample step This is done by testing the sign of an integer parameter, whose value is proportional to the difference between the separations of the two pixel positions from the actual line path. Is using incremental integer calculations

Pixel positions along a line path are determined by sampling at unit x intervals For point (x o, y o ) of a given line, we step to each successive column (x position) and plot the pixel whose y value is closest to the line path. –Assuming we have a pixel at (x k, y k ) is displayed, we next need to decide which pixel to plot in column x k+1 –The choices are the pixels at positions (x k +1, y k ) and (x k +1, y k +1).

We label vertical pixel separations from the mathematical line path as d 1, and d 2 The coordinate on the mathematical line at pixel column position x k +1 is calculated as shown in next slide

Then and

The difference between these two separations is At step k, a decision parameter p k is used The sign of p k is the sine of d 1 -d 2, since Δx > 0 for our example

Then c is independent of pixel position and will be eliminated in the recursive calculations for p k

If the pixel at y k is closer to the line path than the pixel at y k +1 (i.e. d 1 <d 2 ) then p k is negative so we plot lower pixel; otherwise we plot the upper pixel

At step k+1, the decision parameter will be Then But, x k+1 = x k +1 Where y k+1 -y k is either 0 or 1 depending on the sign of p k

This recursive calculation is done at each x position P 0 is calculated at the starting point (x 0, y 0 ) Since the constants 2Δy and 2Δy- 2Δx are calculated once Then the calculations include only subtraction and addition

Bresenham algorithm for |m|<1 1.Input the two line endpoints and store the left endpoint in (x 0, y 0 ) 2.Load (x 0, y 0 ) into the frame buffer; that is, plot the first point. 3.Calculate constants Δx, Δy, 2Δy, and 2Δy- 2Δx, and obtain the starting value for the decision parameter as

4.At each x k along the line, starting at k = 0, perform the following test: –If P k < 0, the next point to plot is (x k +1,y k )and –Otherwise, the next point to plot is (x k +1, y k +1) and 5.Repeat step 4, Δx times

Example, a line from (20, 10) to (30, 18) – Δx = 10, Δy = 8, then m=0.8 –2Δy = 16, 2Δy-2Δx = 4 –P 0 = 2Δy - Δx = 6 –Plot initial point (20, 10) –Repeat applying one of the following:

MID-POINT CIRCLE ALGORITHM

Circle Generation Algorithm The circle is a frequently used component in pictures and graphs A procedure for generating either full circles or circular arcs is included in most graphics packages.

Properties of Circles A circle is defined as the set of points that are all at a given distance r from a center position (x c, y c ) This distance relationship is expressed by the Pythagorean theorem in Cartesian coordinates as

We could use this equation to calculate the position of points on a circle circumference by stepping along the x axis in unit steps from x c - r to x c + r and calculating the corresponding y values at each position as

This is not the best method for generating a circle for the following reasons: 1.It involves considerable computation at each step 2.The spacing between plotted pixel positions is not uniform

Another way to eliminate the unequal spacing is to calculate points along the circular boundary using polar coordinates r and ɵ

Notes: 1.The step size chosen for ɵ depends on the application and the display device 2.Computation can be reduced by considering the symmetry of circles

3.The Cartesian equation involves multiplications and square root calculations 4.The polar equations contain multiplications and trigonometric calculations. 5.More efficient circle algorithms are based on incremental calculation of decision parameters, as in the Bresenham line algorithm, which involves only simple integer operations

Solution: –To get a direct distance comparison, test the halfway position between two pixels to determine if this midpoint is inside or outside the circle boundary

Midpoint Circle Algorithm Sample at unit intervals and determine the closest pixel position to the specified circle path at each step First step –calculate pixel positions around a circle path centered at the coordinate origin (0,0). –Then each calculated position (x, y) is moved to its proper screen position by adding x c, to x and y c, to y

we can take unit steps in the positive x direction over one octant, then use the symmetry To apply the midpoint method, we define a circle function:

the circle function is the decision parameter in the midpoint algorithm, and we can set up incremental calculations for this function as we did in the line algorithm. Assuming we have just plotted the pixel at (x k, y k ) –we next need to determine whether the pixel at position (x k + 1, y k ) or the one at position (x k + 1, y k - 1) is closer to the circle.

The relative position of any point (x, y) can be determined by checking the sign of the circle function

Our decision parameter is the circle function evaluated at the midpoint between these two pixels: –If p k < 0, this midpoint is inside the circle and the pixel on scan line y k is closer to the circle boundary. –Otherwise, the mid-position is outside or on the circle boundary, and we select the pixel on scan line y k - 1.

We obtain a recursive expression for the next decision parameter by evaluating the circle function at sampling position x k = x k + 2: or

where y k+1 is either y k or y k-1 depending on the sign of p k. increments for obtaining p k+1, are either –2x k (if p k is negative) –or 2x k – 2y k+1

Evaluation of the terms 2x k+1, and 2y k+1, can also be done incrementally as

The initial decision parameter is obtained by evaluating the circle function at the start position (x 0, y 0 ) = (0, r) or If the radius r is specified as an integer, we can simply round p 0 to

Midpoint Circle Algorithm 1.Input radius r and circle center (x c, y c ), and obtain the first point on the circumference of a circle centered on the origin as : (x 0, y 0 ) = (0, r) 2.calculate the initial value of the decision parameter as:

3.At each x k position, starting at k = 0, perform the following test: –If p k < 0, the next point along the circle centered on (0,0) is (x k+1, y k ) and –Otherwise, the next point along the circle is (x k + 1, y k - 1) and –Where

4.Determine symmetry points in the other seven octants. 5.Move each calculated pixel position (x, y) onto the circular path centered on (x c, y c ) and plot the coordinate values: 6.Repeat steps 3 through 5 until x >= y.

Example –Given a circle radius r = 10 –Consider the first quadrant from x = 0 to x = y. –The initial value of the decision parameter is p 0 = 1 - r = -9 –For the circle centered on the coordinate origin, the initial point is (x 0, y 0 ) = (0, 10) –initial increment terms for calculating the decision parameters are 2x 0 = 0, 2y 0 = 20

Successive decision parameter values and positions along the circle path are calculated as: (x k+1, y k ) (x k + 1, y k - 1)