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COMPUTER GRAPHICS:

1 COMPUTER GRAPHICS CHAPTER 3 2D GRAPHICS ALGORITHMS

2D Graphics Algorithms:

2 2D Graphics Algorithms Output Primitives Line Drawing Algorithms DDA Algorithm Midpoint Algorithm Bersenhem’s Algorithm Circle Drawing Algorithms Midpoint Circle Algorithm Antialising Fill-Area Algorithms

Output Primitives:

3 Output Primitives

Output Primitives:

4 Output Primitives The basic objects out of which a graphics display is created are called. Describes the geometry of objects and – typically referred to as geometric primitives . Examples: point, line, text, filled region, images, quadric surfaces, spline curves Each of the output primitives has its own set of attributes .

Output Primitives:

Output Primitives Points Attributes: Size, Color. glPointSize(p); glBegin(GL_POINTS); glVertex2d(x1, y1); glVertex2d(x2, y2); glVertex2d(x3, y3); glEnd()

Output Primitives:

Output Primitives Lines Attributes: Color, Thickness, Type glLineWidth(p); glBegin(GL_LINES); glVertex2d(x1, y1); glVertex2d(x2, y2); glVertex2d(x3, y3); glVertex2d(x4, y4); glEnd()

Output Primitives:

Output Primitives Polylines (open) A set of line segments joined end to end. Attributes: Color, Thickness, Type glLineWidth(p); glBegin(GL_LINE_STRIP); glVertex2d(x1, y1); glVertex2d(x2, y2); glVertex2d(x3, y3); glVertex2d(x4, y4); glEnd()

Output Primitives:

Output Primitives Polylines (closed) A polyline with the last point connected to the first point . Attributes: Color, Thickness, Type Note : A closed polyline cannot be filled. glBegin(GL_LINE_LOOP); glVertex2d(x1, y1); glVertex2d(x2, y2); glVertex2d(x3, y3); glVertex2d(x4, y4); glEnd()

Output Primitives:

Output Primitives Polygons A set of line segments joined end to end. Attributes: Fill color, Thickness, Fill pattern Note : Polygons can be filled. glBegin(GL_POLYGON); glVertex2d(x1, y1); glVertex2d(x2, y2); glVertex2d(x3, y3); glVertex2d(x4, y4); glEnd()

Output Primitives:

10 Output Primitives Text Attributes: Font, Color, Size, Spacing, Orientation. Font: Type (Helvetica, Times, Courier etc.) Size (10 pt, 14 pt etc.) Style (Bold, Italic, Underlined)

Output Primitives:

11 Output Primitives Images Attributes: Image Size, Image Type, Color Depth. Image Type: Binary (only two levels) Monochrome Color. Color Depth: Number of bits used to represent color.

Output Primitives:

12 Output Primitives Output Primitive Attributes Point Size Color Line Thickness (1pt, 2pt … ) Type (Dashed, Dotted, Solid) Color Text Font (Arial, Courier, Times Roman … ) Size (12pt, 16pt ..) Spacing Orientation (Slant angle) Style (Bold, Underlined, Double lined) Color Filled Region Fill Pattern Fill Type (Solid Fill, Gradient Fill) Fill Color Images Color Depth (Number of bits/pixel)

Line Drawing Algorithms:

13 Line Drawing Algorithms

Line Drawing:

Line Drawing Line drawing is fundamental to computer graphics. We must have fast and efficient line drawing functions. Rasterization Problem : Given only the two end points, how to compute the intermediate pixels, so that the set of pixels closely approximate the ideal line.

Line Drawing - Analytical Method:

Line Drawing - Analytical Method y x y=mx+c a x b x A(a x ,a y ) B(b x ,b y )

Line Drawing - Analytical Method:

Directly based on the analytical equation of a line. Involves floating point multiplication and addition Requires round-off function. double m = (double)(by-ay)/(bx-ax); double c = ay - m*ax; double y; int iy; for (int x=ax ; x <=bx ; x++) { y = m*x + c; iy = round(y); setPixel (x, iy); } Line Drawing - Analytical Method

Incremental Algorithms:

Compute one point based on the previous point: (x 0 , y 0 )…….…………..(x k , y k ) (x k+1 , y k+1 ) ……. I have got a pixel on the line (Current Pixel). How do I get the next pixel on the line ? Next pixel on next column (when slope is small) Next pixel on next row (when slope is large) Incremental Algorithms

Incrementing along x:

Current Pixel (x k , y k ) To find (x k+1 , y k+! ): x k+1 = x k +1 y k+1 = ? (5,2) (6,1) (6,2) (6,3) Assumes that the next pixel to be set is on the next column of pixels (Incrementing the value of x !) Not valid if slope of the line is large. Incrementing along x

Line Drawing - DDA:

Digital Differential Analyzer Algorithm is an incremental algorithm. Assumption: Slope is less than 1 (Increment along x). Current Pixel = (x k , y k ). (x k , y k ) lies on the given line. y k = m.x k + c Next pixel is on next column. x k+1 = x k +1 Next point (x k+1 , y k+1 ) on the line y k+1 = m.x k+1 + c = m (x k +1) +c = y k + m Given a point (x k , y k ) on a line, the next point is given by x k+1 = x k +1 y k+1 = y k + m Line Drawing - DDA

Line Drawing - DDA:

Does not involve any floating point multiplication. Involves floating point addition. Requires round-off function Line Drawing - DDA double m = (double) (by-ay)/(bx-ax); double y = ay; int iy; for (int x=ax ; x <=bx ; x++) { iy = round(y); setPixel (x, iy); y+ = m; }

Midpoint Algorithm:

x k+1 = x k +1 y k+1 = Either y k or y k +1 Midpoint algorithm is an incremental algorithm Midpoint Algorithm Assumption: Slope < 1 Current Pixel

Midpoint Algorithm - Notations:

Candidate Pixels Current Pixel ( x k , y k ) Midpoint Line Coordinates of Midpoint = ( x k +1, y k +(1/2) ) ( x k +1, y k ) ( x k +1, y k +1) Midpoint Algorithm - Notations

Midpoint Algorithm: Choice of the next pixel:

Midpoint Below Line Midpoint Above Line Midpoint Algorithm: Choice of the next pixel If the midpoint is below the line, then the next pixel is (x k +1, y k +1). If the midpoint is above the line, then the next pixel is (x k +1, y k ).

Equation of a line revisited.:

A(a x ,a y ) B(b x ,b y ) Equation of a line revisited. Let w = b x  a x , and h = b y  a y . Then, h ( x  a x )  w ( y  a y ) = 0. ( h , w , a x , a y are all integers). In other words, every point ( x , y ) on the line satisfies the equation F ( x , y ) =0, where F ( x , y ) = h ( x  a x )  w ( y  a y ). Equation of the line:

Midpoint Algorithm: Regions below and above the line.:

Midpoint Algorithm: Regions below and above the line. F (x,y) > 0 (for any point below line) F(x,y) < 0 (for any point above line) F(x,y) = 0

Midpoint Algorithm Decision Criteria:

F(MP) > 0 Midpoint below line F(MP) < 0 Midpoint above line Midpoint Algorithm Decision Criteria

Midpoint Algorithm Decision Criteria:

Midpoint Algorithm Decision Criteria F(MP) = F (x k +1, y k + ½) = F k (Notation) If F k < 0 : The midpoint is above the line. So the next pixel is (x k +1, y k ). If F k  0 : The midpoint is below or on the line. So the next pixel is (x k +1, y k +1). Decision Parameter

Midpoint Algorithm – Story so far.:

Midpoint Algorithm – Story so far. Midpoint Below Line Next pixel = (x k +1, y k +1) F k > 0 y k+1 = y k +1 Midpoint Above Line Next pixel = (x k +1, y k ) F k < 0 y k+1 = y k

Midpoint Algorithm Update Equation:

Midpoint Algorithm Update Equation F k = F (x k +1, y k + ½) = h (x k +1  a x )  w (y k +½  a y ) But, F k +1 = F k + h  w (y k+1­­­  y k ). (Refer notes) So, F k < 0 : y k+1 = y k . Hence, F k +1 = F k + h . F k  0 : y k+1 = y k +1. Hence, F k +1 = F k + h  w . F 0 = h  w /2. Update Equation

Midpoint Algorithm:

30 Midpoint Algorithm int h = by-ay; int w = bx-ax; float F=h-w/2; int x=ax, y=ay; for (x=ax; x<=bx; x++){ setPixel(x, y); if(F < 0) F+ = h; else{ F+ = h-w; y++; } }

Bresenham’s Algorithm:

31 Bresenham’s Algorithm int h = by-ay; int w = bx-ax; int F=2*h-w; int x=ax, y=ay; for (x=ax; x<=bx; x++){ setPixel(x, y); if(F < 0) F+ = 2*h; else{ F+ = 2*(h-w); y++; } }

Circle Drawing Algorithms:

32 Circle Drawing Algorithms

Midpoint Circle Drawing Algorithm:

33 To determine the closest pixel position to the specified circle path at each step. For given radius r and screen center position (x c , y c ), calculate pixel positions around a circle path centered at the coodinate origin (0,0) . Then, move each calculated position (x, y) to its proper screen position by adding x c to x and y c to y . Along the circle section from x=0 to x=y in the first quadrant , the gradient varies from 0 to -1. Midpoint Circle Drawing Algorithm

Midpoint Circle Drawing Algorithm:

34 Midpoint Circle Drawing Algorithm 8 segments of octants for a circle:

Midpoint Circle Drawing Algorithm:

35 Midpoint Circle Drawing Algorithm Circle function: f circle (x,y) = x 2 + y 2 –r 2 > 0, (x,y) outside the circle < 0, (x,y) inside the circle = 0, (x,y) is on the circle boundary { f circle (x,y) =

Midpoint Circle Drawing Algorithm:

36 Midpoint Circle Drawing Algorithm y k y k -1 midpoint Next pixel = (x k +1, y k ) F k < 0 y k+1 = y k y k y k -1 midpoint Next pixel = (x k +1, y k -1) F k >= 0 y k+1 = y k - 1

Midpoint Circle Drawing Algorithm:

37 Midpoint Circle Drawing Algorithm We know x k+1 = x k +1, F k = F (x k +1, y k - ½) F k = (x k +1) 2 + (y k - ½) 2 - r 2 -------- (1) F k+1 = F (x k +1, y k - ½) F k+1 = (x k +2) 2 + (y k+1 - ½) 2 - r 2 -------- (2) (2) – (1) F k+1 = F k + 2(x k +1) + (y 2 k+1 – y 2 k ) - (y k+1 – y k ) + 1 If F k < 0, F k+1 = F k + 2x k+1 +1 If F k >= 0, F k+1 = F k + 2x k+1 +1 – 2y k+1

Midpoint Circle Drawing Algorithm:

38 Midpoint Circle Drawing Algorithm For the initial point, (x 0 , y 0) = (0, r) f 0 = f circle (1, r- ½ ) = 1 + (r- ½ ) 2 – r 2 = 5 – r 4 ≈ 1 – r

Midpoint Circle Drawing Algorithm:

39 Midpoint Circle Drawing Algorithm Example: Given a circle radius = 10, determine the circle octant in the first quadrant from x=0 to x=y. Solution: f 0 = 5 – r 4 = 5 – 10 4 = -8.75 ≈ –9

Midpoint Circle Drawing Algorithm:

40 Midpoint Circle Drawing Algorithm Initial (x 0 , y 0 ) = (1,10) Decision parameters are: 2x 0 = 2, 2y 0 = 20 k F k x y 2x k+1 2y k+1 0 -9 1 10 2 20 1 -9+2+1=-6 2 10 4 20 2 -6+4+1=-1 3 10 6 20 3 -1+6+1=6 4 9 8 18 4 6+8+1-18=-3 5 9 10 18 5 -3+10+1=8 6 8 12 16 6 8+12+1-16=5 7 7 14 14

Midpoint Circle Drawing Algorithm:

41 Midpoint Circle Drawing Algorithm void circleMidpoint (int xCenter, int yCenter, int radius) { int x = 0; Int y = radius; int f = 1 – radius; circlePlotPoints (xCenter, yCenter, x, y); while (x < y) { x++; if (f < 0) f += 2*x+1; else { y--; f += 2*(x-y)+1; } } circlePlotPoints (xCenter, yCenter, x, y); }

Midpoint Circle Drawing Algorithm:

42 Midpoint Circle Drawing Algorithm void circlePlotPoints (int xCenter, int yCenter, int x, int y) { setPixel (xCenter + x, yCenter + y); setPixel (xCenter – x, yCenter + y); setPixel (xCenter + x, yCenter – y); setPixel (xCenter – x, yCenter – y); setPixel (xCenter + y , yCenter + x ); setPixel (xCenter – y , yCenter + x ); setPixel (xCenter + y , yCenter – x ); setPixel (xCenter – y , yCenter – x ); }

Antialiasing:

43 Antialiasing

Antialiasing:

44 Antialiasing Antialiasing is a technique used to diminish the jagged edges of an image or a line, so that the line appears to be smoother; by changing the pixels around the edges to intermediate colors or gray scales. Eg. Antialiasing disabled: Eg. Antialiasing enabled:

Antialiasing (OpenGL):

Antialiasing (OpenGL) Antialiasing disabled Antialiasing enabled Setting antialiasing option for lines: glEnable (GL_LINE_SMOOTH);

Fill Area Algorithms:

46 Fill Area Algorithms

Fill Area Algorithms:

47 Fill Area Algorithms Fill-Area algorithms are used to fill the interior of a polygonal shape. Many algorithms perform fill operations by first identifying the interior points, given the polygon boundary.

Basic Filling Algorithm:

48 The basic filling algorithm is commonly used in interactive graphics packages, where the user specifies an interior point of the region to be filled. Basic Filling Algorithm 4-connected pixels

Basic Filling Algorithm:

49 [1] Set the user specified point. [2] Store the four neighboring pixels in a stack. [3] Remove a pixel from the stack. [4] If the pixel is not set, Set the pixel Push its four neighboring pixels into the stack [5] Go to step 3 [6] Repeat till the stack is empty. Basic Filling Algorithm

Basic Filling Algorithm (Code):

50 void fill(int x, int y) { if(getPixel(x,y)==0){ setPixel(x,y); fill(x+1,y); fill(x-1,y); fill(x,y+1); fill(x,y-1); } } Basic Filling Algorithm (Code)

Basic Filling Algorithm:

51 Requires an interior point. Involves considerable amount of stack operations. The boundary has to be closed. Not suitable for self-intersecting polygons Basic Filling Algorithm

Types of Basic Filling Algorithms:

52 Boundary Fill Algorithm For filling a region with a single boundary color. Condition for setting pixels: Color is not the same as border color Color is not the same as fill color Flood Fill Algorithm For filling a region with multiple boundary colors. Condition for setting pixels: Color is same as the old interior color Types of Basic Filling Algorithms

Boundary Fill Algorithm (Code):

53 void boundaryFill(int x, int y, int fillColor, int borderColor) { getPixel(x, y, color); if ((color != borderColor) && (color != fillColor)) { setPixel(x,y); boundaryFill(x+1,y,fillColor,borderColor); boundaryFill(x-1,y,fillColor,borderColor); boundaryFill(x,y+1,fillColor,borderColor); boundaryFill(x,y-1,fillColor,borderColor); } } Boundary Fill Algorithm (Code)

Flood Fill Algorithm (Code):

54 void floodFill(int x, int y, int fillColor, int oldColor) { getPixel(x, y, color); if (color != oldColor) { setPixel(x,y); floodFill(x+1, y, fillColor, oldColor); floodFill(x-1, y, fillColor, oldColor); floodFill(x, y+1, fillColor, oldColor); floodFill(x, y-1, fillColor, oldColor); } } Flood Fill Algorithm (Code)

Filling Polygons (OpenGL):

Filling Polygons (OpenGL) Enabling polygon fill (Default): glPolygonMode(GL_FRONT_AND_BACK, GL_FILL); Disabling polygon fill: glPolygonMode(GL_FRONT_AND_BACK, GL_LINE);

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