Chapter 3: Graphics Output Primitives. OpenGL Line Functions. OpenGL Point Functions. Line Drawing Algorithms

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1 Chater : Grahics Outut Primitives Primitives: functions in grahics acage that we use to describe icture element Points and straight lines are the simlest rimitives Some acages include circles, conic sections, quadric surfaces, sline curves and surfaces, and olygon color area Two-Dimensional World Coordinate Reference Frame in OenGL glmatrixmode(gl_projection); glloadidentity(); gluorthod (xmin, xmax, ymin, ymax); Video Screen Y max Dislay Window Y min X min X max OenGL Point Functions OenGL Line Functions OenGL function to state coordinates values for oint glvertex* (); where (*) suffix code required A glvertexmust be laced between glbeginand glend glbegin(gl_points); glvertex* (); glbegin(gl_points); glvertexi (, ); glvertexi (, ); glvertexi (, ); intnt [] = {, ; intnt [] = {, ; intnt[] = {, ; glbegin(gl_points); glvertexiv (nt); glvertexiv (nt); glvertexiv (nt); Straight line segments between each successive endoints glbegin (GL_LINES); glvertexiv ( ); glvertexiv ( ); glvertexiv ( ); glvertexiv ( ); OenGL Line Functions A sequence of connected line segments between first and last endoints glbegin (GL_LINE_STRIP); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); Last coordinate is connected to the first endoint glbegin (GL_LINE_LOOP); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); Line Drawing Algorithms Programmer secifies (x, y) values of end ixels Need algorithm to figure out which intermediate ixels are on line ath Pixel (x, y) values constrained to integer values Actual comuted intermediate line values may be floats Rounding may be required. E.g. comuted oint (.,.) rounded to (, ) Rounded ixel value is off actual line ath (jaggy!!) Sloed lines end u having jaggies Vertical, horizontal lines, no jaggies

2 Line Drawing Algorithms Line Drawing Algorithms Line: (, ) (, ) Which intermediate ixels to turn on? Sloe-intercet line equation y = m. x + b m : sloe of the line, b : y intercet Given two end oints (x, y ) and (x end, y end ), how to comute m and b? m = (y end -y ) / (x end -x ) b = y -m. x y end y Examle: Find the sloe of (, ) and (, ) m = ( - ) / ( ) = / =. x x end DDA (Digital Differential Analyzer) DDA (Digital Differential Analyzer) A line is samled at unit intervals in one coordinates and the corresonding integer values nearest the line ath are determined for the other coordinate Case (m <= ) : samle at unit x intervals ( x = ), comuter successive y values as y + = y + m Case (m > ) : samle at unit y intervals ( y = ), comuter successive x values as x + = x + / m m> m= m <, y + = y + m x = x + y = y + m x = x + y = y + m m >, x + = x + / m y = y + x = x + / m y = y + x = x + / m m< inline int round (const float a) { return int (a +.); DDA code void linedda (int x, int y, int xend, int yend) { int dx = xend - x, dy = yend - y, stes; float xincrement, yincrement, x = x, y = y; if (fabs (dx) > fabs (dy)) stes = fabs (dx); else stes = fabs (dy); xincrement = float (dx) / float (stes); yincrement = float (dy) / float (stes); Bresenham s Line Algorithm Accurate, efficient, and uses only incremental integer calculations Samling at unit intervals, decide which of two ossible ixel ositions is closer to the line ath?? setpixel (round (x), round (y)); for (int i = ; i < stes; i++) { x += xincrement; y += yincrement; setpixel (round (x), round (y));

3 Bresenham s Line Algorithm Bresenham s Line Algorithm y + y + y + y x x + x + x + y + y y x + d uer d lower Consider ositive sloe less than. Samling at osition x + = x +, decide (x +, y ) or (x +, y + ) y= m (x + ) + b d lower = y -y = m (x + ) + b y d uer = (y + ) - y = y + - m (x + ) b Determine which ixel is closer based on the difference d lower -d uer = m (x + ) y + b Substitute m = y / x so it involves only integer calculations d lower -d uer = ( y/ x) (x + ) y + b x (d lower -d uer ) = y (x + ) x y + x (b ) Bresenham s Line Algorithm P is a decision arameter for the thste P = x (d lower - d uer ) P = y. x - x. y + y + x(b ) if P is negative then ixel y is closer that (y + ) At ste + P + = y. x + - x. y + + c P + -P = y (x + -x )- x(y + - y ) But x + = x + P + = P + y - x(y + -y ) y + y is either or, deending on the sign of P c Bresenham s Line Algorithm for m <.. Calculate the constants x, y, y, and y - x, and obtain the starting value for decision arameter as P = y - x. At each x along the line, starting at =, erform the following test. If P <, the next oint to lot is (x +, y ) and P + = P + y Otherwise, the next oint is (x +, y + ) and P + = P + y - x. Perform ste x times At the starting ixel osition (x, y ) P = y - x Bresenham s Examle Line with endoints (, ) and (, ) m =., x =, y =, P = y - x =, y =, y - x = - Plot initial oint (x, y ) = (, ), and determine successive ixel ositions P - - (x +, y +) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Circle Drawing Algorithms A circle is a set of oints that are a given distance r from center oint (x c, y c ) ( ) + ( ) = = ± ( ) The calculations are not very efficient. The square (multily) oerations To mae our circle drawing algorithm more efficient is that circles centred at (, ) have eightway symmetry y c (-y, x) (-y, -x) x c (-x, y) (-x, -y) r θ (x, y) (y, x) (y, -x) (x, -y)

4 const float DEGRAD =./; void drawcircle (float radius) { glbegin (GL_LINE_LOOP); Circle Drawing Algorithms Another way: use olar coordinates r and θto exress circle in arameter olar form = +cos y = +sin r (x, y) sin θ cos Assume that we have just lotted oint (x, y ) The next oint is a choice between (x +, y ) and (x +, y -) We would lie to choose the oint that is nearest to the actual circle So how do we mae this choice? (x, y ) (x +, y ) (x +, y -) for (int i=; i < ; i++) { float deginrad = i * DEGRAD; glvertexf (cos (deginrad) * radius, sin (deginrad) * radius); Let s rearrange the equation of the circle slightly to give us: f circ ( x, y) = x + y r The equation evaluates as follows: f circ <, if ( x, y) is inside the circle boundary ( x, y) =, if ( x, y) is on the circle boundary >, if ( x, y) is outside the circle boundary By evaluating this function at the midoint between the candidate ixels we can mae our decision Assuming we have just lotted the ixel at (x, y ) so we need to choose between (x +, y ) and (x +,y - ) Our decision variable can be defined as: = fcirc( x +, y ) = ( x + ) + ( y ) r If < the midoint is inside the circle and the ixel at y is closer to the circle Otherwise the midoint is outside and y - is closer To ensure things are as efficient as ossible we can do all of our calculations incrementally First consider: ( + = fcirc x + +, y + ) or: = [( x + ) + ] + ( y ) r + + = + ( x + ) + ( y + y ) ( y + y ) + where y + is either y or y - deending on the sign of The first decision variable at osition (x, y ) = (, r) is given as: = fcirc r (, ) = + r ( ) r = r Then if < then the next decision variable is given as: + = + x + + If > then the decision variable is: + = + x + + y+

5 The The. Inut radius rand circle centre (x c, y c ), then set the coordinates for the first oint on the circumference of a circle centred on the origin as: x, y ) = (, ) ( r. Calculate the initial value of the decision arameter as: = r. Starting with = at each osition x, erform the following test. If <, the next oint along the circle centred on (, ) is (x +, y ) and: + = + x + + Otherwise the next oint along the circle is (x +, y -) and: + = + x + + y +. Determine symmetry oints in the other seven octants. Move each calculated ixel osition (x, y) onto the circular ath centred at (x c, y c ) to lot the coordinate values: x + = x x c y = y + yc. Reeat stes to until x >= y Midoint Circle Examle Given radius r = Determine osition along the circle octant in the first quadrant from x = to x = y P = r = -, x =, y = y = x Polygon Polygon is a lane figure secified by set of three or more coordinate ositions, called vertices If all interior angles of a olygon are less than or equal to degrees, the olygon is convex A olygon that is not convex is called a concaveolygon P (x +, y +) (, ) (, ) (, ) (, ) (, ) (, ) (, ) x + y + < > Convex Concave OenGL Polygon Fill Area Functions glrect* (x, y, x, y); one corner is at coordinate osition (x, y) and the oosite corner is at osition (x, y) glrect(,,, );,, OenGL Polygon Fill Area Functions glbegin(gl_polygon); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glbegin(gl_triangles); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv ();

6 OenGL Polygon Fill Area Functions glbegin(gl_triangle_strip); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glbegin(gl_triangle_fan); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); N triangles N triangles OenGL Polygon Fill Area Functions glbegin(gl_quads); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glbegin(gl_quad_strip); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv (); glvertexiv ();