1. 2M050: Computer Graphics
Jack van Wijk
HG6.71

2. Aims Introduce basic graphics concepts and terminology
Base for development of 2D interactive computer graphics programmes (OGO 2.3)

3. Literature
Computer Graphics - C Version Donald Hearn - M. Pauline Baker 2nd edition - international edition Prentice Hall

4. Overview
Introduction
Geometry
Interaction
Raster graphics

5. Introduction What is Computer Graphics? Applications
Computer Graphics in Eindhoven
Raster/vector graphics
Hardware

6. Computer Graphics Computer Graphics is ubiquitous:
Visual system is most important sense:
High bandwidth
Natural communication
Fast developments in
Hardware
Software

7. Computer Graphics Image Analysis Mathematical Model Image
(pattern recognition)
Mathematical
Model
Image
Image Synthesis (Rendering)
Modeling
Image processing

8. Supporting Disciplines
Computer science (algorithms, data structures, software engineering, …)
Mathematics (geometry, numerical, …)
Physics (Optics, mechanics, …)
Psychology (Colour, perception)
Art and design

9. Applications Computer Aided Design (CAD)
Computer Aided Geometric Design (CAGD)
Entertainment (animation, games, …)
Geographic Information Systems (GIS)
Visualization (Scientific Vis., Inform. Vis.)
Medical Visualization …

10. Computer Graphics Eindhoven
Current:
Information visualisation
Interactive 3D design
Virtual reality
Past: Rasterization, Animation
Jack van Wijk
Kees Huizing
Arjan Kok
Robert van Liere
Wim Nuij
Alex Telea
Huub van de Wetering

11. Interactive Computer Graphics
User
input
Application
Screen
image

12. Graphics pipeline
User
Edit
Model
Map
Geometry, colour
Screen
Display

13. Representations in graphics
Vector Graphics
Image is represented by continuous geometric objects: lines, curves, etc.
Raster Graphics
Image is represented as an rectangular grid of coloured squares

14. Vector graphics Graphics objects: geometry + colour
Complexity ~ O(number of objects)
Geometric transformation possible without loss of information (zoom, rotate, …)
Diagrams, schemes, ...
Examples: PowerPoint, CorelDraw, ...

15. Raster graphics Generic Image processing techniques
Geometric Transformation: loss of information
Complexity ~ O(number of pixels)
Jagged edges, anti-aliasing
Realistic images, textures, ...
Examples: Paint, PhotoShop, ...

16. Rasterization, Pattern recognition
Conversion
Vector graphics
Rasterization, Pattern recognition
Scan conversion
Raster graphics

17. Hardware Vector graphics Raster graphics Colour lookup table
3D rendering hardware

18. Vector Graphics Hardware
move 10 20
line 20 40
...
char O
char R
Display list
continuous & smooth lines
no filled objects
random scan
refresh speed depends on complexity of the scene
V E C T O R
Display Controller

19. Raster Graphics Hardware7 6
Frame buffer
Video Controller
R A S T E R
jaggies (stair casing)
filled objects
(anti)aliasing
refresh speed independent of scene complexity
pixel
scan conversion
resolution
bit planes

20. Colour Lookup Table
R G B 1 2 4 7
...
colour index 7 6
Frame buffer
CLUT:
pixel = code
True colour:
pixel = R,G,B

21. 3D rendering hardware Geometric representation: Triangles
Viewing: Transformation
Hidden surface removal: z-buffer
Lighting and illumination: Gouraud shading
Realism: texture mapping
Special effects: transparency, antialiasing

22. 2D geometric modelling Coordinates Transformations
Parametric and implicit representations
Algorithms

23. Coordinates Point: position on plane p = (px , py) x = (x, y)
x = (x1 , x2)
x = x1 e1 + x2 e2, e1 = (1, 0), e2 = (0, 1)
Vector: direction and magnitude
v = (vx , vy), etc. y p v x

24. Vector arithmetic Addition of two vectors: v + w = (vx + wx , vy + wy)
Multiplication vector-scalar:
av = (avx , avy) y
v+w w v x y 2v v x

25. Coordinate systems
world
train
wheel
image

26. Why transformations? Model of objects Viewing
world coordinates: km, mm, etc.
hierarchical models:
human = torso + arm + arm + head + leg + leg
arm = upperarm + lowerarm + hand …
Viewing
zoom in, move drawing, etc.

27. Transformation types Translate according to vector v:
t = p + v
Scale with factor s:
s = sp
Rotate over angle a:
rx = cos(a)px - sin(a)py
ry = sin(a)px + cos(a)py y s t r v p a x

28. Homogeneous coordinates
Unified representation of rotation, scaling, translation
Unified representation of points and vectors
Compact representation for sequences of transformations
Here: convenient notation, much more to it

29. Homogeneous coordinates
Extra coordinate added:
p = (px , py , pw) or
x = (x, y, w)
Cartesian coordinates: divide by w
x = (x/w, y/w)
Here: for a point w = 1, for a vector w = 0

30. Matrices for transformation

31. Direct interpretationy’
(x, y) y
(x’,y’) x b a t x’

32. Translation matrix

33. Scaling matrix

34. Rotation matrix

35. Sequences of transformations
y’’’
x’’’ x’ y’ y
x’’
y’’ x
Sequences of transformations can be described with a single transformation matrix, which is the result of concatenation of all transformations.

36. Order of transformations
x’’
y’’
x’’
y’’ x’ y’ x’ y’ x y x y
Matrix multiplication is not commutative. Different orders of multiplication give different results.

37. Order of transformations
Pre-multiplication:
x’ = M n M n-1…M 2 M 1 x
Transformation M n in global coordinates
Post-multiplication:
x’ = M 1 M 2…M n-1 M n x
Transformation M n in local coordinates, i.e., the
coordinate system that results from application of
M 1 M 2…M n-1

38. Window and viewport Viewport: Area on screen to be used for drawing.
Unit: pixels (screen coordinates)
Note: y-axis often points down
Window:
Virtual area to be used by application
Unit: km, mm,… (world coordinates)
(800,600)
(600,400)
(200,200)
(0,0)
(2,1)
(-2,-1)

39. Window/viewport transform
Determine a matrix M, such that the window (wx1, wx2, wy1, wy2) is mapped on the viewport (vx1, vx2, vy1, vy2):
A = T(-wx1, -wy1)
B = S(1/(wx2-wx1), 1/(wy2-wy1)) A
C = S(vx2-vx1 ,vy2-vy1)B
M = T(vx1, vy1) C

40. Forward and backward Drawing: (meters to pixels) Use x’ = Mx
(vx2, vy2)
Viewport
x’:screen coordinates
Drawing: (meters to pixels)
Use x’ = Mx
Picking:(pixels to meters)
Use x = M-1x’
(vx1, vy1)
Drawing
Picking
(wx2, wy2)
Window:
x: user coordinates
(wx1, wy1)

41. Implementation example
Suppose, basic library supports two functions:
MoveTo(x, y: integer);
LineTo(x, y: integer);
x and y in pixels.
How to make life easier?

42. State variables Define state variables:
Viewport: array[1..2, 1..2] of integer;
Window: array:[1..2, 1..2] of real;
Mwv, Mobject: array[1..3, 1..3] of real;
Mwv: transformation from world to view
Mobject: extra object transformation

43. Procedures Define coordinate system: SetViewPort(x1, x2, y1, y2):
Update Viewport and Mwv
SetWindow(x1, x2, y1, y2):
Update Window and Mwv

44. Procedures (continued)
Define object transformation:
ResetTrans:
Mobject := IdentityMatrix
Translate(tx, ty):
Mobject := T(tx,ty)* Mobject
Rotate(alpha):
Mobject := R(tx,ty)* Mobject
Scale(sx, sy):
Mobject := S(sx, sy)* Mobject

45. Procedures (continued)
Handling hierarchical models:
PushMatrix();
Push an object transformation on a stack;
PopMatrix()
Pop an object transformation from the stack.
Or:
GetMatrix(M);
SetMatrix(M);

46. Procedures (continued)
Drawing procedures:
MyMoveTo(x, y):
(x’, y’) = Mwv*Mobject*(x,y);
MoveTo(x’, y’)
MyLineTo(x,y):
LineTo(x’, y’)

47. Application DrawUnitSquare: Main program: MyMoveTo(0, 0); Initialize;
MyLineTo(1, 0);
MyLineTo(1, 1);
MyLineTo(0, 1);
MyLineTo(0, 0);
Initialize:
SetViewPort(0, 100, 0, 100);
SetWindow(0, 1, 0, 1);
Main program:
Initialize;
Translate(-0.5, -0.5);
for i := 1 to 10 do
begin
Rotate(pi/20);
Scale(0.9, 0.9);
DrawUnitSquare;
end;

48. Puzzles
Modify the window/viewport transform for a display y-axis pointing downwards.
How to maintain aspect-ratio world->view? Which state variables?
Define a transformation that transforms a unit square into a “wybertje”, centred around the origin with width w and height h.

49. Geometry
Dot product, determinant
Representations
Line
Ellipse
Polygon

50. Good and bad Good: symmetric in x and y Good: matrices, vectors
Bad: y = f(x)
Good: dot product, determinant
Bad: arcsin, arccos

51. Dot product w q v
|w| cos q

52. Dot product properties

53. Determinant Det(v, w): signed area of parallellogramq v v q w
Det(v, w): signed area of parallellogram
Det(v, w) = 0 iff v and w are parallel

54. Curve representations
Parametric: x(t) = (x(t), y(t))
Implicit: f(x) = 0

55. Parametric line representation
Given point p and vector v:
x(t) = p + vt
Given two points p and q:
x(t) = p + (q-p)t , or
= pt + q(1-t) y t q p x v

56. Parametric representation
x(t) = (x(t), y(t))
Trace out curve:
MoveTo(x(0));
for i := 1 to N do LineTo(x(i*Dt));
Define segment: tmin £ t £ tmax

57. Implicit line representation
(x-p).n = 0
with n.v = 0
n is normal vector:
n = [-vy , vx]
Also:
ax+by+c=0 y p n x v

58. Implicit representation
f = 0
f =-1
f = 1
f =-2
f >0
f <0

59. Circle y r x

60. Ellipse y b a x

61. Generic ellipse a b y c x

62. Some standard puzzles Conversion of line representation
Projection of point on line
Line/Line intersection
Position points/line
Line/Circle intersection

63. Conversion line representationsu a

64. Projection point on lineq u q’
q-a s q w a

65. Intersection of line segmentsu v a b

66. Solving for s and t s t u v a b

67. Position points/line u a s c b

68. Line/circle intersectiony u r x t a

69. Polygons
Sequence of points pi, i = 1,…, N, connected by straight lines
Index arithmetic: modulo N
p0 = pN , pN+1 = p1 , etc. p1 p2 pN pi

70. Regular N-gon
triangle square pentagon hexagon octagon

71. Convex and concave Convex: Concave:
each line between two arbitrary points inside the polygon does not cross its boundary
Concave:
not convex

72. Convexity test
pi+1
pi+1 pi pi
pi-1
pi-1
Convex
Concave

73. Polygon area and orientationpi
pi+1 c

74. Point/polygon test 2 1 3

75. Point/polygon test (cntd.)
Beware of special cases:
Point at boundary
v parallel to edge
c + vt through vertex

76. Puzzles Define a procedure to clip a line segment against a rectangle.
Define a procedure to calculate the intersection of two polygons.
Define a procedure to draw a star.
Same, with the constraint that the edges pi-1 pi and pi+2 pi+3 are parallel.

77. Introduction Curve Modelling
Jack van Wijk
TU Eindhoven

78. Overview curve modelling
Parametric curves
Requirements
Concepts
Lagrangian interpolation
Bézier curve
B-spline
Cubic splines

79. Parametric curves t
p(t)
p’(t)

80. Tangent line to curve t s
q(s)
p(t)
p’(t)

81. Example
q(s) s
p’(t) t
p(t)

82. Curve modelling Problem: How to define a smooth curve? Solution:
Specify a sequence of points pi , i = 1,…, N, (control-points);
Generate a smooth curve that interpolates or approximates these control-points.

83. Requirements Smooth Local control Intuitive and easy to use
no discontinuities in direction and curvature;
Local control
Change of a point should have only local effect;
Intuitive and easy to use
no oscillations, variation diminishing
Approximate or interpolate?

84. Parametric interpolation

85. Linear interpolation of two points

86. Linear interpolation of N points1 wi 1 2 3 4 5 t

87. Linear interpolation Not Smooth Local control
Intuitive and easy to use
Interpolate

88. Lagrangian interpolation - 1

89. Lagrangian interpolation - 2

90. Lagrangian interpolation - 3

91. Lagrangian interpolation - 4
Smooth
no discontinuities in direction and curvature;
NO local control
Wild oscillations, not variation diminishing!
Interpolating

92. Bézier curve - 1
Puzzle: Define a smooth curve that interpolates the first and last point and approximates the others. p4 p3 p1 p2

93. Bézier curve - 2
Solution for N=3: p2
p(t) q2 q1 p1 p3

94. Bézier curve - 3
Solution for N=4: p2 p3
p(t) p1 p4

95. Bézier curve - 4

96. Bézier curve - 5

97. Bézier curve - 6 General Bézier curve: Degree = #points-1 Smooth
no discontinuities in direction and curvature;
NO local control
Variation diminishing, convex hull property
Interpolates first and last, further approximating

98. Convex hull property

99. Convex hull property example
Curve outside
convex hull
Curve inside
convex hull

100. B-splines Piecewise polynomial, locally non-zero
Degree: user definable
Continuity: degree-1
first degree: continuous in position
second degree: continuous in tangent
third degree: continuous in curvature

101. Quadratic B-spline

102. Cubic B-spline

103. B-splines - 3 General B-spline: Degree: from 1 to N-1
Smooth (if degree > 1)
Local control
Variation diminishing, convex hull property
Approximating

104. Cubic splines - 1 Most popular in Computer Graphics: powerful simple
inflection points, continuity
simple
low degree polynomials
local control
Many versions:
Bézier, B-spline, Catmull-Rom,...

105. Cubic splines - 2

106. Cubic splines - 2

107. Cubic splines - 3

108. Tangent vector
p’(t)
p(t) t

109. Significant values p(0) : startpoint segment
p’(0) : tangent through startpoint
p(1) : endpoint segment
p’(1) : tangent through endpoint
p’(0)
p(1) t
p’(1)
p(0)

110. Cubic Bézier curve p1 p2 p0 p3

111. Joining two Bezier segments
Positional continuity:
p3 = q0
Tangential continuity:
p3- p2 // q1- q0 p1 p2 p3 p0 q0 q3 q1 q2

112. Cubic B-spline curve p1 p2 p0 p3

113. Conversion - 1

114. Conversion - 2

115. Puzzle 1: Hermite spline

116. Puzzle 2: Limit on interpolation
Find out why a curve that
interpolates the control points,
stays within the convex hull,
and is smooth
cannot exist, both graphically and
mathematically.

117. Puzzle 3: Split Bezier curve
Find a recipe to split a cubic Bezier curve segment into two segments: p1 p2 q2 q3 q4 q1 q5 p0 p3 q0 q6

118. Finally... Curves: interpolation of points Demo program:
Interpolation is generally applicable, f.i. surfaces: interpolation of curves
Demo program:

119. B-splines - 2