Chapter 3: Windows, viewports

1
Chapter 3 Windows, viewports

• World coordinates rather than screen coordinates.
• World window defines which part of world should
  be drawn, and which clipped away.
• Viewport defined in screen window
• mapping (scaling, shifting) between world window
  and viewport
• draw in world window automatically mapped to
  viewport

2
Chapter 3 Windows, viewports

• 3.2.1 Window to viewport mapping
• W, V rectangles left, right, top, bottom
• May have different ratios distortion
• A, B scale C, D translate

3
Chapter 3 Windows, viewports

• Read Example 3.2.1, exercise 3.2.1, example
  3.2.2
• Selfstudy
• Ex. 3.2.3 Drawing polylines from a file
• Ex. 3.2.4 Tiling a window with a motif
• Ex. 3.2.5 Clipping, zooming and roaming
• p. 91 smooth animations and double buffering
• Ex. 3.2.2 Whirling swirls

4
Chapter 3 Windows, viewports

• 3.2.2 Setting Window and Viewport automatically
• Selfstudy.
• 3.3 Clipping lines
• OpenGL automatically algorithms

5
Chapter 3 Windows, viewports

• 3.3.2 Cohen-Sutherland clipping
• Checks for trivial accept or trivial reject
• Inside-outside code word for each endpoint
• Trivial accept Both code words are FFFF
• Trivial reject Code words have T in same
  position

6
Chapter 3 Windows, viewports

• Chopping (no trivial accept/reject)
• Goal A.x, A.y?
• A.x W.right
• A.y?
• delx P2.x P1.x dely P2.y P1.y
• e P1.x W.right d/dely e/delx
• Therefore P1.y P1.y (W.right P1.x) dely
  /delx

7
Chapter 3 Windows, viewports

• The Canvas Class Selfstudy.
• 3.4 Developing the canvas class
• 3.5 Relative drawing
• 3.6 Figures based on regular polygons
• 3.7 Drawing circles and arcs
• 3.8 Parametric forms of curves
• Implicit Point on line if F(x,y) 0
  (inside-outside form)
• Parametric Position at time t is given by x(t),
  y(t)
• Finding implicit form for parametric form
• NB! Practice exercises p. 122-123 selfstudy
• Drawing parametric curves Trivial.

8
Chapter 3 Windows, viewports

• 3.8.3 Super-ellipses
• Implicit (x/W)n (y/H)n 1
• Parametric
• x(t) W cos(t) cos(t)2/n-1
• y(t) H sin(t) sin(t)2/n-1
• Also superhyperbola
• 3.8.4 Polar Coordinate Shapes
• x(t) r(t) cos (?(t))
• x(t) r(t) cos (?(t))
• Given point (r, ? ), Cartesian point (x,y)is
  given by x f (?) cos (?)y f (?) sin (?)

9
Chapter 3 Windows, viewports

• Note conic sections, logarithmic spiral
• 3.8.5 3D Curves
• Helix, toroidal spiral
• Read Case Studies pp. 130 142

10
Chapter 4 Vector Tools

• Vector arithmetic allows to express geometric
  concepts algebraically.
• 4.2 Review of vectors
• Vector is object with length and direction
• Think of vector as displacement
• The difference between two points is a vector
  v Q-P

11
Chapter 4 Vector Tools

• 4.2.1 Operations with vectors
• vector addition, scalar multiplication
• 4.2.2 Linear combination of vectors
• w a1v1 a2v2 ... amvm
• Affine combination a1a2...am 1

12
Chapter 4 Vector Tools

• Convex combination
• a1a2...am 1
• ai ? 0, for i 1, ..., m
• Set of all convex combinations of a vector v
• v (1-a) v1 av2 , for 0 ? a ? 1

13
Chapter 4 Vector Tools

• 4.2.3 Magnitude of a vector unit vectors
• w is distance from head to tail, so that w
  (w12w22...wn2)0.5
• Scaling vector to unit length known as
  normalizing and obtain unit vector w (w/w)
• 4.3 Dot product
• d v . w
• Properties
• a . b b . a
• (ac) . b a . b c . b
• (sa) . b s (a . b)
• b2 b . b

14
Chapter 4 Vector Tools

• 4.3.2 Angle between two vectors
• cos (?) (b/b).(c/c)The cosine between two
  vectors is the dot product of the normalized
  vectors.
• 4.3.3 The sign of b.c, and perpendicularity
• perpendicular normal orthogonal
• standard unit vectors

15
Chapter 4 Vector Tools

• 4.3.4 The 2D Perp Vector
• Let a(ax,ay). Then a? (-ay,ax) is the
  counterclockwise perpendicular to a (the perp).
• Selfstudy Practice exercises p. 157.
• 4.3.5 Orthogonal projections and distances
• How far? Where? Decompose?

16
Chapter 4 Vector Tools

• 4.3.6 Applications of projection Reflections
• Selfstudy.
• 4.4 The Cross Product of Two Vectors
• i j ka x b ax
  ay az bx by bz
• Examples practice exercises Selfstudy.

17
Chapter 3 Windows, Viewports

• Programming Task 2 Implement Case Study 3.6.1
  (Basic tilings), p. 138, in Hill.