Computer Graphics Lecture 06 Circle Drawing Techniques Taqdees A. Siddiqi

What is Circle A circle is the set of points in a plane that are equidistant from a given point O. The distance r from the center is called the radius, and the point O is called the center. Twice the radius is known as the diameter. The angle a circle subtends from its center is a full angle, equal to 360° or 2  radians.

Circle Cont..

A circle has the maximum possible area for a given perimeter, and the minimum possible perimeter for a given area. The perimeter C of a circle is called the circumference, and is given by C = 2  r

Circle Drawing Techniques - Input for circle drawing. one center point (x c, y c ) and. radius r Now, using these two inputs there are a number of ways to draw a circle.

Circle Drawing Techniques These techniques have: - Understanding curve. very simple to. complex - Time complexity. inefficient to. efficient

Circle Drawing Using Cartesian Coordinates This technique uses the equation for a circle with radius r centered at (0,0): x 2 + y 2 = r 2, an obvious choice is to plot y = ± r 2 - x 2 against different values of x.

Circle Drawing Using Cartesian Coordinates Using above equation a circle can be easily drawn. The value of x varies from r-x c to r+x c, and y is calculated using above formula. Using this technique a simple algorithm will be:

Circle Drawing Using Cartesian Coordinates for x= radius-x center to radius+x center y = y c + r 2 – ( x - x c ) 2 drawPixel (x, y) y = y c - r 2 – ( x - x c ) 2 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 (as shown in the next figure)

Drawbacks/ Shortcomings

Circle Drawing Using Polar Coordinates Polar Coordinates: Radiusr 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:

Circle Drawing Using Polar Coordinates Cont… x = x c + r cos  y = y c + 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.

Circle Drawing Using Polar Coordinates Cont… 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.

Polar Coordinates Algorithm Circle2 (xcenter, ycenter, radius) for  = 0 to 2  step 1/r x = x c + r * cos  y = y c + r * sin  drawPixel (x, y)

Discussion - very simple technique - solves problem of unequal space - is inefficient in terms of calculations - involves floating point calculations

Reduction in Calculations  Symmetry in octants  Upper half circle symmetric to lower half circle  Left half circle symmetric to right half circle  Finally two halves of the same quarters are symmetric to each other

Eight Octants Symmetry

Optimizing the Algorithm Now this algorithm can be optimized by using symmetric octants as: Circle2 (xcenter, ycenter, radius) for  = 0 to  /4 step 1/r x = x c + r * cos  y = y c + r * sin  DrawSymmetricPoints(xcenter, ycenter, x,y)

Optimized Algorithm DrawSymmeticPoints (xcenter, ycenter, x, y) Plot (x + xcenter, y + ycenter) Plot (y + xcenter, x + ycenter) Plot (y + xcenter, -x + ycenter) Plot (x + xcenter, -y + ycenter) Plot (-x + xcenter, -y + ycenter) Plot (-y + xcenter, -x + ycenter) Plot (-y + xcenter, x + ycenter) Plot (-x + xcenter, y + ycenter)

Inefficiency Still Prevails  Reduction in calculations by exploiting symmetric octants but…  Floating point calculations still involved

Midpoint Circle Algorithm  Derivation of decision parameter  Decrement decision in the y coordinate against increment of x coordinate

Midpoint Circle Algorithm Consider only the first octant of a circle of radius r centered on the origin. We begin by plotting point (0, r) and end when x = y.

Midpoint Circle Algorithm The decision at each step is whether to choose : the pixel to the right of the current pixel or the pixel which is to the right and below the current pixel (8-way stepping)

Midpoint Circle Algorithm Assume: P = (x k, y k ) is the current pixel. Q = (x k +1, y k ) is the pixel to the right R = (x k +1, y k -1) is the pixel to the right and below. X = Y X 2- Y 2 - r 2 =0

Midpoint Circle Algorithm To apply the midpoint method, we define a circle function: f circle (x, y) = x 2 + y 2 – r 2 The following relations can be observed: f circle (x, y) < 0,if (x, y) is inside the circle boundary f circle (x, y) = 0,if (x, y) is on the circle boundary f circle (x, y) > 0,if (x, y) is outside the circle boundary

Midpoint Circle Algorithm Circle function tests are performed for the midpoints between pixels near the circle path at each sampling step.

Midpoint Circle Algorithm X k +1 X 2+ Y 2 -R 2 =0 k k k

Midpoint Circle Algorithm Figure given in previous slide shows the midpoint between the two candidate pixels at sampling position x k +1. Our decision parameter is the circle function evaluated at the midpoint between these two pixels:

Decision Parameter P k = f circle ( x k + 1, y k - ½ ) P k = ( x k + 1 ) 2 + ( y k - ½ ) 2 – r 2 ……...(1)

Decision Parameter 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.

Decision Parameter Successive decision parameters are obtained using incremental calculations. We obtain a recursive expression for the next decision parameter by evaluating the circle function at sampling position x k+1 +1 = x k +2:

Decision Parameter P k+1 = f circle ( x k+1 + 1, y k+1 - ½ ) P k+1 = [ ( x k + 1 ) + 1 ] 2 + ( y k+1 - ½ ) 2 – r 2.……… (2)

Decision Parameter Subtracting (1) from (2), we get: P k+1 - P k = [ ( x k + 1 ) + 1 ] 2 + ( y k+1 - ½ ) 2 – r 2 – ( x k + 1 ) 2 - ( y k - ½ ) 2 + r 2 or P k+1 = P k + 2( x k + 1 ) + ( y 2 k+1 - y 2 k ) – ( y k+1 - y k ) + 1

Decision Parameter Where y k+1 is either y k or y k -1, depending on the sign of P k. Therefore, if P k < 0 or negative then y k+1 will be y k and the formula to calculate P k+1 will be:

Decision Parameter P k+1 = P k + 2( x k + 1 ) + ( y 2 k - y 2 k ) – ( y k - y k ) + 1 P k+1 = P k + 2( x k + 1 ) + 1

Decision Parameter Otherwise, if P k > 0 or positive then y k+1 will be y k -1 and the formula to calculate P k+1 will be:

Decision Parameter P k+1 = P k + 2( x k + 1 ) + [ (y k -1) 2 - y 2 k ] – (y k -1- y k ) + 1 P k+1 = P k + 2(x k + 1) + (y 2 k - 2 y k +1 - y 2 k ) – ( y k -1- y k ) + 1 P k+1 = P k + 2( x k + 1 ) - 2 y k P k+1 = P k + 2( x k + 1 ) - 2 y k P k+1 = P k + 2( x k + 1 ) - 2 ( y k – 1 ) + 1

Decision Parameter Now a similar case that we observed in line algorithm is that how would starting P k be calculated. For this, the starting pixel position will be (0, r). Therefore, putting this value in equation, we get

Decision Parameter P 0 = ( ) 2 + ( r - ½ ) 2 – r 2 P 0 = 1 + r 2 - r + ¼ – r 2 P 0 = 5/4 – r If radius r is specified as an integer, we can simply round p 0 to: P 0 = 1 – r Since all increments are integer. Finally the algorithm can be summed up as :

Algorithm MidpointCircle (xcenter, ycenter, radius) x = 0; y = r; p = 1 - r; do DrawSymmetricPoints (xcenter, ycenter, x, y) x = x + 1 If p < 0 Then p = p + 2 * ( y + 1 ) + 1

Algorithm Cont… else y = y - 1 p = p + 2 * ( y + 1) –2 ( x – 1 ) + 1 while ( x < y ) Example to calculate first octant of the circle using above algorithm.

Example center ( 0, 0 )radius =10

Example Cont… Behaviour of calculated points around the circle is observable

Computer Graphics Lecture 06 Circle Drawing Techniques Taqdees A. Siddiqi