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- Circle Drawing Asst. Prof. Dr. Ahmet Sayar Kocaeli University Computer Engineering Advanced Computer Graphics Spring 2012.

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Circle Drawing Asst. Prof. Dr. Ahmet Sayar Kocaeli University Computer Engineering Advanced Computer Graphics Spring 2012 Slide 2 A circle is a set of points that are all at a given distance r from a center position ( x c, y c ). For any circle point ( x, y ), 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 Properties of Circle Slide 3 A Simple Circle Drawing Algorithm The equation for a circle is: where r is the radius of the circle So, we can write a simple circle drawing algorithm by solving the equation for y at unit x intervals using: Slide 4 Brute Force Method Computational cost: floating point multiplications, subtraction, square-root, rounding Slide 5 Issues on Brute Force Method Problems in the brute force method Involves considerable computations at each steps. Lots of float- point calculations Spacing between plotted pixel positions is not uniform Eliminating unequal spacing Expressing the circle equation in parametric polar form x= x c + r cos(alpha), y= y c + r sin(alpha) for 0 o < alpha < 360 o For a more continuous boundary, set the step size at 1/r Reducing computation cost Considering the symmetry of circles Still require a good deal of computation time for a octant Slide 6 Common approach However, unsurprisingly this is not a brilliant solution! Firstly, the resulting circle has large gaps where the slope approaches the vertical Secondly, the calculations are not very efficient The square (multiply) operations The square root operation try really hard to avoid these! We need a more efficient, more accurate solution Slide 7 Midpoint Circle Algorithm (1/4) Only need to draw the arc between (0, r) and (r/sqrt(2),r/sqrt(2)) The coordinates of the other pixels can be implied from the 8- way symmetry rule. 45 o Slide 8 Eight-Way Symmetry The first thing we can notice to make our circle drawing algorithm more efficient is that circles centred at (0, 0) have eight-way symmetry (x, y) (y, x) (y, -x) (x, -y)(-x, -y) (-y, -x) (y, -x) (-x, y) Slide 9 Mid-Point Circle Algorithm Similarly to the case with lines, there is an incremental algorithm for drawing circles the mid-point circle algorithm In the mid-point circle algorithm we use eight-way symmetry so only ever calculate the points for the top right eighth of a circle, and then use symmetry to get the rest of the points Slide 10 Midpoint Circle Algorithm E SE M Choose E as the next pixel if M lies inside the circle, and SE otherwise. d = d0:Select SE d new = d + (2x p 2y p +5) xpxp x p +1 y-1/2 d r 2 = x 2 +y 2 d= F(x,y) = x 2 +y 2 -r 2 Slide 11 Choosing the Next Pixel M (x, y)(x+1, y) (x+1, y-1) E SE choose SE choose E decision variable d Slide 12 Change of d when E is chosen M old (x, y)(x+1, y) (x+1, y-1) E SE (x+2, y) (x+2, y-1) M new Slide 13 Change of d when SE is chosen M old (x, y)(x+1, y) E SE (x+1, y-2)(x+2, y-2) M new Slide 14 Initial value of d (0,R) M0M0 (1,R) (1,R-1) Remember f(x+1,y+1/2) when we talked about midpoint line drawing Slide 15 Midpoint Circle Algorithm (3/4) Start with P (x = 0, y = R). x = 0; y = R; d = 5/4 R; /* real */ While (x < y) { If (d >= 0) // SE is chosen y = y 1 d = d + 2 * (x y) + 5 else // E is chosen d = d + 2 * x + 3 x = x+1; WritePixel(x, y) } Where is the arch drawn? Slide 16 New Decision Variable Our circle algorithm requires arithmetic with real numbers. Lets create a new decision variable kd kd=d-1/4 Substitute kd+1/4 for d in the code. Note kd > -1/4 can be replaced with h > 0 since kd will always have an integer value. Slide 17 Midpoint Circle Algorithm Start with P (x = 0, y = R). x = 0; y = R; kd = 1 R; While (x < y) { If (kd >= 0) // SE is chosen y = y 1 kd = kd + 2 * (x y) + 5 else // E is chosen kd = kd + 2 * x + 3 x = x+1; WritePixel(x, y) } Slide 18 LINE DRAWING with Midpoint Algorithm x=x L y=y L d=x H - x L c=y L - y H sum=2c+d //initial value // decision param sum is multiplied with 2, act val (c+d/2) draw(x,y) while ( x < x H ) // iterate until reaching the end point if ( sum < 0 ) // below the line and choose NE sum += 2d y++ x++ sum += 2c draw(x,y) Compare this with the algorithm presented in the previous slide EXPLAIN ! Slide 19 Initial value of decision param: sum What about starting value? (xL+1,yL+1/2) is on the line! Sum = 2c + d Slide 20 Drawing circle in OpenGL OpenGL does not have any primitives for drawing curves or circles. However we can approximate a circle using a triangle fan like this: glBegin(GL_TRIANGLE_FAN) glVertex2f(x1, y1) for angle# = 0 to 360 step 5 glVertex2f(x1 + sind(angle#) * radius#, y1 + cosd(angle#) * radius#) next glEnd() Slide 21 If you want to use mid-point to draw circle If you calculate the points on circle through the given mid-point algorithm, then use glBegin(GL_POINTS) glVertex2f(x1, y1) glEnd()