Computer graphics PART 2

[aayush_005]

Incremental Method:

Incremental methods come into their own when not only the final result but the intermediate results are of use. When we plot lines incrementally, we start at one end and finish by computing the coordinates of the other end. In between, the incremental technique generates the coordinates of all the dots that lie on the line. Thus one iterative process generates many useful results. The Incremental method is explained briefly below with an example.

Ex: The newcomer to a city often finds his way about by an Incremental method. If a person is at 203 house number in Main Street and is looking for number 735, where houses are numbered according their date of construction.


Incremental computing techniques are a form of iterative
computation, in which each iterative step is simplified by maintaining a small amount of state, or memory, about the progress of the computation. The visitor looking for the house number 735 needs only three pieces of state information: the direction in which he is going, the number of the house he has just passed, and the number of house he is looking for. If he reaches a house whose number lies outside the range of these two house numbers, he has to move in reverse direction.
The following algorithm is an example of a simple Incremental method to solve the above problem. The algorithm is in PASCAL mode.

Algorithm:
FindHouse (HR : integer)
Arguments HR is an integer for house required.
Local HP is an integer for house passed.
D is an integer for direction.
T is an integer for reading house number.
begin
D := 1 ;
HP := ReadHouseNumber ;
while (HP <> HR ) do
begin
MoveToNextHouse(D);
T := ReadHouseNumber;
If (T > max (HR , HP)) or (T < min (HR , Hp)) then
D := - D;
HP := T;
end
end;

Line Drawing Algorithms:

The straight-line segments are used a great deal in computer-generated pictures. They occur in block diagrams, bar charts and graphs. The curves can be approximated quite effectively by sequences of short straight-line segments. Since straight lines are so useful, it is worth taking care that they are well drawn. The criteria is applied to computer-generated lines is as follows, to ensure that they start and finish in exactly the right places, to maintain constant blackness, or density, along the length of each line, and to makes sure that lines have matching density where appropriate.



Criteria:

1. Lines should appear straight.
A line segment, through no other addressable points, may happen to pass through no other addressable points in between. In these cases we must approximate the line by choosing addressable points close to it. If we choose well, the line will appear straight; if not, we shall produce crooked lines.

2. Lines should terminate accurately.
The effect is often seen as a small gap between the endpoint of one line and the starting point of the next or as a cumulative error. So the lines are terminating accurately, no gap between the endpoints.

3. Lines should have constant density.
With bright lines plotted on a dark background, line density is observed as brightness; when the line is black and the background light, it is seen as blackness. To maintain constant density, dots should be equally spaced. This can be achieved only in lines parallel or at 45° to the axes.

4. Line density should be independent of line length and angle.
Before plotting the line we must therefore determine its exact length. Also we must be able to control the rate, in terms of distance traveled, at which dots are plotted. Normally the best one is to use a line-generation algorithm that keeps line density constant and to compute an approximate line-length estimate.

5. Line should be drawn rapidly.
In interactive applications we would like lines to appear rapidly on the screen. This implies using the minimum of computation to draw the line.

Each method basically generates two sets of signals; these signals step the x and y coordinates of the point that traces out the line. Each of the Line Drawing Methods generates the same total number of x signals and y signals, since these signal totals must sum to the magnitudes of ∆x and ∆y, the displacements in x and y of the finishing point of the line from its starting point. The methods differ only in the order in which the signals are issued and in how they are generated.
The Line Drawing Methods are three: the Symmetrical DDA, the Simple DDA, and Bresenham’s algorithms.

The Symmetrical DDA:

The Digital Differential Analyzer (DDA) generates lines from their differential equations. We can build DDAs to draw curves as well as straight lines provided these curves can be defined by ordinary differential equations.
The equation of a straight line is particularly simple:

dy/dx = ∆y/∆x

The DDA works on the principle that we simultaneously increment x and y by small steps proportional to the first derivatives of x and y. in the case of a straight line the first derivatives are constant and are proportional to ∆x and ∆y. thus in the ideal case of an infinite-precision display we could generate a line by incrementing x and y by ∆x and ∆y, where  is some small quantity. In the real world of limited-precision displays we must generate only addressable points. This can be done by rounding to the nearest integer after each incremental step; after rounding we display a dot at the resultant x and y.

An alternative to rounding is the use of arithmetic overflow: x and y are kept in registers that have two parts, integer and fractional. The incrementing values, which are both less than unity, are repeatedly added to the fractional parts, and whenever the result overflows, the corresponding integer part is incremented. The integer parts of the x and y registers are used in plotting the line. This would normally have the effect of truncating rather than rounding, so we initialize the DDA with the value 0.5 in each of the fractional parts to achieve true rounding.

One advantage of this arrangement is that it allow us to detect changes in x and y and hence to avoid plotting the same point twice. The overflow indicator generated by the DDA produce the signals we need to reposition the point that traces out the line. The appearance of lines generated by the DDA depends on the value chosen for . In the case of the Symmetrical DDA we choose  = 2-n where

2 n-1 <= max ( |∆x|, | ∆y|) < 2n

The Symmetrical DDA generates accurate lines, since the displacement of a displayed dot from the true line is never greater than one-half a screen unit. Logically the Symmetrical DDA is simple; the use of a negative power of 2 for  means that he incrementing values can be determined by shifting the ∆x and ∆y registers rather than by a division. Each step in the line is computed with just two additions.

The Simple DDA:

The Symmetrical DDA uses a power of 2 as a line-length estimate, since this simplifies the logic. The principle of the DDA tells us that we may use any line-length estimate and any corresponding value of  provided neither ∆x nor ∆y exceeds unit magnitude.

For the simple DDA we choose a line-length estimate equal to the larger of the magnitude of ∆x and ∆y, so that either ∆x or ∆y is of unit magnitude. This allows us to replace one of the DDA’s adders with a simple counter. The simple DDA therefore generates unit steps in the direction of greatest motion. A following algorithm is the PASCAL implementation of the simple DDA.

Algorithm:
DDA (x1, y1, x2, y2: integer)

Arguments x1 and y1 are integers for representing x-coordinate and y-coordinate of one endpoint.
x2 and y2 are integers for representing x-coordinate and y-coordinate of another endpoint.
Local l is an integer for length.
i is an integer for counting purpose.
x and y are two real for representing x-coordinate and y-coordinate of a point.
xi and yi are two real for representing x and y incrementing values.
begin
l := abs(x2 – x2);
If abs(y2 – y1) > l then
l := abs(y2 – y1);
xi := (x2 – x1) / l;
yi := (y2 – y1) / l;
x := x1 + 0.5;
y := y1 + 0.5;
for i = 1 to l do
begin
plot (trunk(x), trunk(y));
x := x + xi;
y := y + yi;
end
end;
The simple DDA is as accurate as its symmetrical counterpart by generates a different sequence of dots because of its different method of estimating line length. Logically it is simpler, except for the need to perform an initial division to determine the incrementing value. The simple DDA is an ideal basis for a software line generator, but the need for division logic makes it less suited to hardware implementation.

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