Digital differential analyzer (graphics algorithm): Difference between revisions

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{{short description|Hardware or software used for interpolation of variables over an interval}}

{{About|a graphics algorithm|the digital implementation of a Differentialdifferential Analyzeranalyzer|Digital Differentialdifferential Analyzeranalyzer}}

In [[computer graphics]], a '''[[digital differential analyzer]]''' ('''DDA''') is hardware or software used for [[linear interpolation]] of [[Variable (computer science)|variables]] over an [[Interval (mathematics)|interval]] between start and end point. DDAs are used for [[rasterization]] of lines, triangles and polygons. InThey itscan simplestbe implementation,extended theto DDAnon algorithmlinear interpolatesfunctions, valuessuch inas interval[[texture mapping#Perspective bycorrectness|perspective computingcorrect fortexture each x_i the equations x_i = x_i−1+1/mmapping]], y_i[[quadratic = y_i−1 + mcurves]], where Δx = x_end − x_start and traversing Δy = y_end − y_start and m = Δy/Δx[[voxels]].

In its simplest implementation for linear cases such as [[Line (geometry)|line]]s, the DDA algorithm interpolates values in interval by computing for each x_i the equations x_i = x_i−1 + 1, y_i = y_i−1 + m, where m is the [[slope]] of the line. This slope can be expressed in DDA as follows:

== Performance ==

:<math>m = \frac{y_{\rm end} -y_{\rm start}}{x_{\rm end}-x_{\rm start}}</math>

The DDA method can be implemented using [[floating-point]] or [[integer]] arithmetic. The native floating-point implementation requires one addition and one rounding operation per interpolated value (e.g. coordinate x, y, depth, color component etc.) and output result. This process is only efficient when an [[Floating-point unit|FPU]] with fast add and rounding operation is available.

where ''m'' represents the slope of the line and ''c'' is the y intercept. In fact any two consecutive point(x,y)points lying on this line segment should satisfy the equation.

== Performance ==

The DDA method can be implemented using [[floating-point]] or [[integer]] arithmetic. The native floating-point implementation requires one addition and one rounding operation per interpolated value (e.g. coordinate x, y, depth, color component etc.) and output result. This process is only efficient when an [[Floating-point unit|FPU]] with fast add and rounding operation iswill be available.

The [[Fixed-point arithmetic|fixed-point]] integer operation requires two additions per output cycle, and in case of fractional part overflow, one additional increment and subtraction. The probability of fractional part overflows is proportional to the ratio m of the interpolated start/end values.

DDAs are well suited for hardware implementation and can be pipelined for maximized throughput.

This slope can be expressed in DDA

as

:<math>m = \frac{y_{end} -y_{start}}{x_{end}-x_{start}}</math>

where ''m'' represents the slope of the line and ''c'' is the y intercept. In fact any two consecutive point(x,y) lying on this line segment should satisfy the equation.

== Algorithm ==

TheA linear DDA starts by calculating the smaller of dy or dx for a unit increment of the other... A line is then sampled at unit intervals in one coordinate and corresponding integer values nearest the line path are determined for the other coordinate.

Considering a line with positive slope, if the slope is less than or equal to 1, we sample at unit x intervals (dx=1) and compute successive y values as

: <math>y_{k+1} = y_k + m</math>

: <math>x_{k+1} = x_k + 1</math>

Subscript k takes integer values starting from 0, for the 1st point and increases by 1 until endpoint is reached.

y value is rounded off to nearest integer to correspond to a screen pixel.

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For lines with slope greater than 1, we reverse the role of x and y i.e. we sample at dy=1 and calculate consecutive x values as

: <math>x_{k+1} = x_k + \frac{1}{m}</math>

: <math>y_{k+1} = y_k + 1</math>

Similar calculations are carried out to determine pixel positions along a line with negative slope. Thus, if the absolute value of the slope is less than 1, we set dx=1 if <math> x_{\rm start}<x_{\rm end}</math> i.e. the starting extreme point is at the left.

== Program ==

DDA algorithm program in [[C++]]:

<syntaxhighlight lang="cpp" line="1">

#include <graphics.h>

#include <iostream.h>

#include <math.h>

#include <dos.h>

#include <conio.h>

void main()

{

float x,

float y,

float x1, y1,

float x2, y2, dx, dy, step;

int i, gd = DETECT, gm;

initgraph(&gd, &gm, "C:\\TURBOC3\\BGI");

cout << "Enter the value of x1 and y1: ";

cin >> x1 >> y1;

cout << "Enter the value of x2 and y2: ";

cin >> x2 >> y2;

dx = (x2 - x1);

dy = (y2 - y1);

if (abs(dx) >= abs(dy))

step = abs(dx);

else

step = abs(dy);

dx = dx / step;

dy = dy / step;

x = x1;

y = y1;

i = 0;

while (i <= step) {

putpixel(round(x), round(y), 5);

x = x + dx;

y = y + dy;

i = i + 1;

delay(100);

}

getch();

closegraph();

}

</syntaxhighlight>

== See also ==

* [[Bresenham's line algorithm]] is an algorithm for line rendering.

* [[Incremental error algorithm]]

* [[Xiaolin Wu's line algorithm]] is an algorithm for line anti-aliasing

== References ==

http://www.museth.org/Ken/Publications_files/Museth_SIG14.pdf

{{no footnotes|date=June 2011}}

<references/>

* Alan Watt: ''3D Computer Graphics'', 3rd edition 2000, p.&nbsp;184 (Rasterizing edges). {{ISBN |0-201-39855-9}}

[[Category:Computer graphics algorithms]]

[[Category:Digital geometry]]

[[Category:Articles with example C++ code]]