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Line Drawing and Computer Graphics
Typology: Exams
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Circle Drawing Using Polar Coordinates
Polar Coordinates:
Radius r Angle
calculate points along the circular boundary
using polar coordinates r and
Expressing the circle equation in parametric polar form yields the pair of equations:
x = xc + r cos y = yc + r sin
Using above equation circle can be plotted by calculating x and y coordinates as takes values from 0 to 360 degrees or 0 to 2 radians.
The step size for depends on:
. application and . display device
Larger angular separations along the circumference can be connected with straight-line segments to approximate the circular path.
Step size at 1/r gives continuous boundary
This plots pixel positions that are approximately one unit apart.
Circle2 (xcenter, ycenter, radius)
for = 0 to 2 step 1/r x = xc + r * cos y = yc + r * sin drawPixel (x, y) Drawbacks/ Shortcomings
This works, but …
. is inefficient . multiplications & square root . large gaps in the circle for values of x close to r
DDA Algorithm
DDA abbreviated for digital differential analyzer has a very simple technique.
Find difference, dx and dy as:
dy = y2 – y
dx = x2 – x
if |dx| > |dy| then
step = |dx| else step = |dy| Now very simple to say that step is the total number of pixels required for a line.
Next step is to find xIncrement and yIncrement:
xIncrement = dx/step
yIncrement = dy/step
Next a loop is required that will run ‘step’ times.
In the loop drawPixel and add xIncrement to x1 and yIncrement to y1.
Now sum-up all above in the algorithm:
DDA_Line (Point p1, Point p2)
Now, using these three inputs there are a number of ways to draw an ellipse.
To use polar coordinates r and , for that we have parametric equations:
x = xc + rx cos y = yc + ry sin