Computer Graphics. Lecture 3 Graphics Output Primitives. Somsak Walairacht, Computer Engineering, KMITL
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1 Computer Graphics Lecture 3 Graphics Output Primitives Somsa Walairacht, Computer Engineering, KMITL
2 Outline Line Drawing Algorithms Circle-, Ellipse-Generating Algorithms Fill-Area Primitives Polgon Fill Area Piel-Arra Primitives Character Primitives Computer Graphics
3 Line Drawing Algorithms Line Properties DDA Algorithm Bresenham s Line Algorithm Curve Properties Midpoint Algorithm Computer Graphics 3
4 A Straight-Line Segment Defined b the coordinate positions for the endpoints of the segment Graphics sstem projects the endpoints to integer screen coordinates and determine the nearest piel positions along the line path between the two endpoints A computed line position of (0.48, 0.5 is converted to piel position (0, It causes all but horizontal and vertical lines to be displaed with a stair-step appearance ( the jaggies Computer Graphics 4
5 OpenGL output functions glbegin(gl_lines; // Specif what to draw, // here lines // Geometric info via vertices: glverte*(; // glverte*(; //... //... glend; * glverte[34][isfd] [34]: D, 3D, 4D [isfd]: integer, short, float, double For instance: glvertei(00, 5; H&B 4-4:8-8
6 Point and Line primitives 3 4 GL_POINTS: sequence of points GL_LINES: sequence of line segments GL_LINE_STRIP: polline GL_LINE_LOOP: closed polline H&B 4-4:8-8
7 . Line Properties Line Equations Cartesian slope-intercept equation = m + b With specific to a line with ( 0, 0 and ( end, end m Given interval d, interval d can be found b Similarl end end d md d d m 0 0 b 0 m 0 Computer Graphics 7
8 Lines & Slopes The slope of a line (m is defined b its start and end coordinates The diagram below shows some eamples of lines and their slopes m = -4 m = - m = m = 4 m = m = - m = m = - / m = - / 3 m = / m = / 3 m = 0 m = 0 Computer Graphics 8
9 Scan-conversion Process for Straight Lines For lines with slope magnitude < : d can be set proportional to horizontal position, d can be calculated from d=m(d For lines with slope magnitude > : d can be set proportional to vertical position, d can be calculated from d=d/m Slope magnitude = : d=d m = m = m = m = / 3 m = 0 Computer Graphics 9
10 Eample 4 5 (7, 5 First wor out m and b: 3 (, m b b 0 m 0 Now for each value wor out the value: 3 4 ( ( ( ( Computer Graphics 0
11 Eample (cont Now just round off the results and turn on these piels to draw our line ( 3 5 ( ( ( Computer Graphics
12 . DDA Algorithm Digital Differential Analzer (DDA Scan-conversion line algorithm Based on calculating either d or d A line is sampled at unit intervals in one coordinate and the corresponding values nearest to line path are determined for the other coordinate Computer Graphics
13 DDA ( ความช นของเส นตรงม ค าเป นบวก The line is processed from left endpoint to right endpoint Slope is less than or equal to d =, + = + m Slope is greater than d =, + = + /m The line is processed from right endpoint to left endpoint Slope is less than or equal to d = -, + = -m Slope is greater than d = -, + = -/m ความช นของเส นตรงม ค าเป นลบ The line is processed from left endpoint to right endpoint Abs of slope is less than d = +, + = - m Abs of slope is greater than d = -, + = + / m The line is processed from right endpoint to left endpoint Abs of slope is less than d = -, + = + m Abs of slope is greater than d = +, + = - / m Computer Graphics 3
14 Sample Code inline int round (const float a { return int (a + 0.5; } void linedda (int 0, int 0, int End, int End { int d = End - 0, d = End - 0, steps, ; float Increment, Increment, = 0, = 0; if (fabs (d > fabs (d steps = fabs (d; else steps = fabs (d; Increment = float (d / float (steps; Increment = float (d / float (steps; setpiel (round (, round (; for ( = 0; < steps; ++ { += Increment; += Increment; setpiel (round (, round (; } } Computer Graphics 4
15 DDA (3 Fast method Accumulation of round-off errors can cause piel position to drift awa from the true line path for long segment Rounding operation and floating-point arithmetic are epensive Computer Graphics 5
16 Integer DDA ให d = end - start, d = end - start ให = abs(d, = abs(d, และ error = - พ จารณาได 4 กรณ 0<m< และ d, d เป นบวก error เป นลบ เล อกจ ด E แล ว error = error+ error เป นบวก หร อ=ศ นย เล อกจ ด NE แล ว error = error+- m> และ d, d เป นบวก error เป นลบ หร อ=ศ นย เล อกจ ด NE แล ว error = error+- error เป นบวก เล อกจ ด N แล ว error = error- -<m<0 และ d เป นลบ d เป นบวก error เป นลบ เล อกจ ด W แล ว error = error+ error เป นบวก หร อ=ศ นย เล อกจ ด NW แล ว error = error+- m<- และ d เป นลบ d เป นบวก error เป นลบ หร อ=ศ นย เล อกจ ด NW แล ว error = error+- error เป นบวก เล อกจ ด N แล ว error = error- Computer Graphics 6
17 Integer DDA ( Eample (,=(, (,=(0, ล าด บท Error เด ม จ ดป จจ บ น จ ดถ ดไป Error ใหม,,3-4=-3-3,3 3, =- 3-3,4 4, =- 4-4,5 5, = ,6 6, = 6 6,7 6,8-4= ,8 7, =- 8-7,9 8, =- 9-8,0 9, -+5-4= , 0, 0+5-4= Computer Graphics 7
18 4. Bresenham s Line Algorithm Accurate and efficient raster-line generating algorithm Onl use integer for calculations Determine the nearest piel position to the line B using the sign of integer parameter Parameter is proportional to the difference between the vertical separation of the two piel positions from the actual line path Computer Graphics 8
19 Bresenham s Line Algorithm ( Reference for Bresenham s Line Algorithm Computer Graphics 9
20 Bresenham s Line Algorithm (3 Proving Bresenham s Line Algorithm Find distance between the line to the upper and lower piels d d lower upper ( m( b m( b Find the difference between two piel, to determine which one is closest to the path d lower d upper m( b Computer Graphics 0
21 Computer Graphics Bresenham s Line Algorithm (4 To convert the parameter to an integer For and + Since + = + Starting point ( 0, 0 and m=d/d ( ( p p ( p p p p ( ( ( ( c d d p upper lower ( ( ( ( b c 0 0 b b m ] ( ( [ ( ( c c p p
22 Bresenham s Line Algorithm (5 p p ( Note: For a line with positive slope >.0, interchange the roles of the and directions Computer Graphics
23 Bresenham s Line Algorithm (6 Computer Graphics 3
24 OpenGL Curve Functions Routines for generating basic curves are not included in OpenGL core librar GLU has routines for 3D Quadrics, such as spheres, clinders, B-splines Simple curve can be approimated using a polline Computer Graphics 4
25 Circle-Generating Algorithms Properties of Circles Pthagorean Theorem ( c ( c r Stepping along from c -r to c +r and calculate Epressing circle equation in polar form r cos r cos Smmetr of circle c c c r ( c Computer Graphics 5
26 Computer Graphics 6 Curves Properties Properties of Ellipses Ellipse equation Epressing ellipse equation in polar form Other curves Parabolic trajector Hperbola ( ( o o o b a ( ( c c r r cos cos r r c c ( ( r r
27 A Simple Circle Drawing Algorithm c r ( c Computer Graphics 7
28 Midpoint Circle Algorithm Midpoint Algorithm Useful in drawing curves Determine the closest piel to the curve path b using the midpoint between two candidate piels Eample, midpoint at + along a circular path Computer Graphics 8
29 Midpoint Circle Algorithm ( Eample: Midpoint Circle Algorithm Define circle function as f circ (, r B putting a point ( o, o into the circle function ( o, o is in the circle boundar, if f circ < 0 ( o, o is on the circle boundar, if f circ = 0 ( o, o is outside the circle boundar, if f circ > 0 Decision parameter can be evaluated from the circle function at the midpoint p fcirc(, Computer Graphics 9
30 Computer Graphics 30 Midpoint Circle Algorithm (3 Successive decision parameters, + + = + + is either or - depending on the sign of p When =+, if p < 0, plot ( +, and use Eq. if p > 0, plot ( +, - and use Eq. ( ( (, ( circ p f p...(...( p p p p
31 Midpoint Circle Algorithm (4 Computer Graphics 3
32 Midpoint Circle Algorithm (5 Given a circle radius r=0 Computer Graphics 3
33 Fill-Area Primitives Area filled with solid color or pattern Most graphics require a fill area be a polgon Curved surfaces can be approimated with a set of polgon patches Sometimes called as surface tessellation Computer Graphics 38
34 Polgon A polgon is a plane figure specified b a set of three or more coordinate positions, called vertices A polgon must have all its vertices within a single plane and there can be no edge crossings To avoid ambiguous object references, polgon refers onl to those planar shapes that have a closed-polline boundar and no edge crossings Computer Graphics 39
35 Conve vs. Concave Conve : All interior angles < 80 degree, and All line segments between interior points in polgon, and All points at the same side of line through edge, and From each interior point complete boundar visible Concave : not conve H&B 4-7:84-94
36 Identifing Concave Polgons A concave polgon has at least one interior angle greater than 80 Etension of some edges intersect other edges Line segment b a pair of interior points intersects the polgon boundar The cross product of adjacent edges is used to test for concavit Computer Graphics 4
37 Splitting Concave Polgons Using edge vectors and edge cross products If an cross product has a negative z, we can split it along the line of the first edge vector in the cross-product pair Computer Graphics 4
38 Splitting Concave Polgons ( Using a rotational method Proceeding counterclocwise around the polgon edges, shift the position of the polgon so that each verte V in turn is at the coordinate origin Rotate the polgon about the origin in a clocwise direction so that the net verte V + is on the -ais Computer Graphics 43
39 Splitting Concave Polgons (3 If the following verte, V +, is below the -ais, then split the polgon along the - ais Steps are repeated until all vertices in the polgon list are tested Computer Graphics 44
40 Splitting a Conve Polgon into a Set of Triangles Defining an sequence of 3 consecutive vertices to be a new polgon (a triangle The middle triangle verte is then deleted from the original verte list Repeat until the original polgon is reduced to just three vertices Computer Graphics 45
41 Inside-Outside Tests Odd-even rule, also called the odd-parit rule or the even-odd rule Identifing interior areas of a plane figure Drawing a line from an position P to a distant point outside the coordinate etents of the closed polline Then count the number of line-segment crossings along this line Odd=interior, otherwise = eterior Computer Graphics 46
42 Inside-Outside Tests ( Nonzero winding-number rule Counts the number of times the boundar of an object winds around a particular point in the counterclocwise direction The interior points of a D object can be defined b having a nonzero value for the winding number Nonzero = interior, otherwise = eterior Computer Graphics 47
43 Nonzero Winding-Number Rule Mechanism Initializing the winding number to 0 A line drawn from an position P to a distant point beond the coordinate etents of the object Count the number of object line segments that cross the reference line in each direction Add when intersects a segment that crosses from right to left (counterclocwise Subtract when intersects a segment that crosses from left to right (clocwise Computer Graphics 48
44 Variations of the Nonzero Winding-Number Rule Border direction Winding number Add (counterclocwise Subtract (clocwise Computer Graphics 49
45 Polgon Tables Objects in scene are described as a set of polgon surface facets Geometric data for objects in scene are arranged in Verte table, store the coordinate value of verte Edge table, contains pointers bac into the verte table to identif the vertices for each polgon edge Surface-facet table, contains pointers bac into the edge table to identif the edge for each polgon edge Edge table can be epanded to include forward pointers into the surface-facet table Computer Graphics 50
46 Polgon Tables ( Polgon table for two adjacent polgon surface facets Convenient reference Object can be displaed efficientl Computer Graphics 5
47 Polgon Tables (3 Tables can be organized into two groups: geometric tables and attribute tables Geometric tables contain verte coordinates and parameters to identif the spatial orientation of the polgon surfaces Attribute tables specif the degree of transparenc of the object and its surface reflectivit and teture characteristics Somsa Computer Walairacht, Graphics Computer Engineering, KMITL 5
48 Computer Graphics 59 Plane Properties General equation of a plane Use Cramer s rule to compute polgon facets data Distance from a point to a plane Front and bac polgon faces Bac face, side of polgon faces into the object interior Front face, the visible or outward side 0 d cz b a ,,, z z z D D d c z z z D d b z z z D d a Distance,, ( 0, c b a d cz b a z p d cz b a d cz b a d cz b a d cz b a
49 Plane Properties ( Identif a point behind or in front of a polgon surface Put the point into the plane equation if A + B + Cz +D < 0, the point (,,z is behind the plane if A + B + Cz +D > 0, the point (,,z is in front of the plane Orientation of a polgon surface in space, described b normal vector Computer Graphics 60
50 End of Lecture 3 Computer Graphics 8
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