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Chapter 4 Vector Graphics. Multimedia Systems. Key Points. Points can be identified by coordinates . Lines and shapes can be described by equations . Approximating abstract shapes on a grid of finite pixels leads to `jaggies'. Anti-aliasing can offset this effect.

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Chapter 4 Vector Graphics

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chapter 4 vector graphics

Chapter 4 Vector Graphics

Multimedia Systems

key points
Key Points
  • Points can be identified by coordinates.Lines and shapes can be described by equations.
  • Approximating abstract shapes on a grid of finite pixels leads to `jaggies'.Anti-aliasing can offset this effect.
  • Bezier curves are drawn using four control points.
  • Bezier curves can be made to join together smoothly into paths.
  • Paths and shapes can be stroked and filled.
  • Geometrical transformations — translation, scaling, rotation, reflection and shearing — can be applied easily to vector shapes.
key points1
Key Points
  • Three approaches to 3-D modelling are: constructive solid geometry, free-form modelling and procedural modelling.
  • 3-D rendering models the effect of light and texture, as well as displaying the modelled objects in space.
  • Ray tracing and radiosity are computationally expensive rendering algorithms that can produce photo-realistic results.
  • Specialized 3-D applications, such as Bryce and Poser, are easier to use, and may be more efficient, than more general 3-D modelling and rendering systems.
  • Vector Graphics
    • Compact
    • Scaleable
    • Resolution independent
    • Easy to edit
    • Attractive for networked multimedia
  • The compactness of vector graphics makes them particular attractive for network multimedia, since the large sizes of the images files lend to excessive download times.
  • Absence of any standard format for vector graphics prevents it from popularization.
    • As SVG and SWF standards are adopted, this will change.
  • In vector graphics, images are built up using shapes that can easily be described mathematically.
  • Vector graphics has been eclipsed (衰退) in recent years by bitmap graphics for 2D images.
  • Vector graphics is mandatory (強制性) in 3D graphics, since processing voxels is still impractical in modern machines.
coordinates and vectors
Coordinates and Vectors
  • Image stored as a rectangular array of pixels.
  • Coordinates (x,y), Fig. 4.1
    • Integer
  • Real coordinate, (2.35, 2.9), Fig. 4.2
  • Drawing programs allow to display axes (ruler) along edges of your drawing
  • Vectors
  • Approximating a straight line, Fig. 4.4
coordinates and vectors1
Coordinates and Vectors
  • Points




anti aliasing
  • Approximating a straight line
  • Using intermediate grey values
  • Brightness is proportional to area of intersection
  • At expense of fuzziness
  • A simple mathematical representation
    • Stored compactly and rendered efficiently
  • Rectangles, squares, ellipses and circles, straight lines, polygons, Bezier curves
    • Spirals and stars, sometimes
  • Fills with color, pattern or gradients
  • Rectangles
  • Ellipses
  • Curves

Hermite Curves

  • Hermite parametric cubic curvesC = C(t) = a0 + a1t+ a2t2 + a3 t3
    • Four vectors a0 ,a1 ,a2 ,a3 (12 coefficients, 3D) are required to define the curve.
    • Usually these vectors can be specified by curve’s behavior at end points t=0 and t=1
  • Assume endpoints C(0), C(1) tangent vectors, C’(0), C’(1) are given, then

a0 = C(0)

a1 + a2 + a3 = C(1)

a1 = C’(0)

a1 +2 a2 + 3 a3 = C’(1)




Hermite Curves

  • a0 = C(0)a1 = C’(0)a2 = 3( C(1) - C(0)) - 2C’(0) - C’(1)a3 = 2 ( C(0) - C(1)) + C’(0) + C’(1)
  • C(t) = (1-3t2 + 2t3) C(0) + (3t2 -2t3) C(1) + (t - 2t2 + t3) C’(0) + (-t2 + t3) C’(1))
  • C(t) = [1 t t2 t3] 1 0 0 0 C (0) 0 0 1 0 C (1) -3 3 3 -2 C’(0) 2 -2 1 1 C’(1)

Bezier Curves

  • Given four control points b0, b1, b2 , b3, then the corresponding Bezier curve is given byC(t) = (1-t)3b0 + 3t(1-t)2b1 + 3t2(1-t)b2 + t3b3C’(t) = -3(1-t)2b0 + 3(1-4t+3t2)b1 + 3(2t-3t2)b2 + 3t2b3
    • C(0)=b0C(1)=b3C’(0)=3(b1-b0)C’(1)=3(b3-b2)

Bezier Curves

  • C(t) = [1 t t2 t3] 1 0 0 0 b0 -3 3 0 0 b1 3 -6 3 0 b2 -1 3 -3 1 b3





bezier curves
Bezier Curves
  • Four control points
    • Two endpoints, two direction points
    • Length of lines from each endpoint to its direction point representing the speed with which the curve sets off towards the direction point
    • Fig. 4.8, 4.9
bezier curves1
Bezier Curves
  • Constructing a Bezier curve
    • Fig. 4.10-13
    • Finding mid-points of lines
bezier curves2
Bezier Curves
  • Figs. 4.14-18
    • Same control points but in different orders

Bezier Curves











bezier cubic curves
Bezier Cubic Curves
  • x(t) = ax t3 +bxt2 + cxt + x1y(t) = ay t3 +byt2 + cyt + y1
  • p1= (x1, y1)
  • p2=(x1 + cx/3, y1 + cy/3)
  • p3=(x2 + (cx +bx )/3, y2 + (cy +by )/3)
  • p4=(x1 + cx +bx +ax, y1 + cy +by +ay)
smooth joins between curves
Smooth Joins between Curves
  • Fig. 20
    • Length of direction linesis the same on each side
  • Smoothness of joins when control points line up and direction lines are the same length
  • Corner point
    • Direction lines of adjacent segments ate not lines up, Fig. 4.21
  • Joined curves and lines
  • Open path
  • Closed path
Each line or curve is called a segment of the path
  • Anchor points: where segments join
  • Pencil tool: freehand
    • Bezier curve segments and straight lines are being created to approximate the path your cursors follows
    • A higher tolerance leads to a more efficient path with fewer anchor points which may smooth out of the smaller movements you made with pencil tool
stroke and fill
Stroke and Fill
  • Apply stroke to path
    • Drawing program have characteristics such as weight and color, which determine their appearance.
    • Weight= width of stroke
    • Dashed effects
      • Length of dashes
      • Gaps between them
line caps joins
Line Caps & Joins
  • Line cap
    • Butt cap
    • Round cap
    • Projecting cap
  • Line Joins
    • Miter
    • Rounded
    • Bevel
  • Most drawing programs also allow to fill an open path
    • Close the path with straight line between its endpoints
    • Flat color, pattern or gradients
    • Gradient: linear, radial
    • Texture
  • Pattern
    • Tiles: a small piece of artwork
    • Use pattern to stroke paths, a textured outline
      • Arrange perpendicular to path, not horizontally
      • Include special corner tiles
  • If you want to fill a path, you need to know which areas are inside it. (Fig. 4.27)
    • Non-zero winding number rule
      • Draw a line from the point in any direction
      • Every time the path crosses it from left to right, add one to winding number; every time the path crosses from right to left, subtract one from winding number
      • If winding number is zero, the point is outside the path, otherwise it is inside.
      • Depends on the path’s having a direction
transformations and filters
Transformations and Filters
  • Transformations
    • Translations: linear movement
    • Scaling, rotation about a point
    • Reflective about a line
    • Shearing: a distortion of angles of axes of an objects
  • Free manipulation of control points
  • Roughening
    • moves anchor points in a jagged array from the original object, creating a rough edge on the object
Scribbling filter
    • randomly distorts objects by moving anchor points away from the original object
    • converts the corner points of an object to smooth curves
    • Filter > Stylize > Round Corners
  • Only relatively few points need to be re-computed
3 d graphics
3-D Graphics
  • Axes in 3D: Fig. 4.35
  • Rotations in 3D: Fig. 4.36
3 d graphics1
3-D Graphics
  • Right-handed coordinate system, Fig. 4.37
  • 2D: define shapes by paths3D: define objects by surfaces
  • Hierarchical modeling
    • A bicycle consists of a frame, two wheels, …
  • In 3D, we have a mathematical model of objects in space, but we need a flat picture.
    • Viewpoint
    • Position of camera
    • Scaling with distance
    • Lighting: position, intensity, type
      • Interaction of light: underwater, smoke-filled room
    • Texture
    • Physical impossibilities: negative spotlights, absorbing unwanted light
3d models
3D Models
  • Constructive solid geometry
    • A few primitive geometric solids such as cube, cylinder, sphere and pyramid as elements from which to construct more complex objects
    • Operators: union, intersection, and difference, Figs. 38-40
    • Physical impossibility
free form modeling
Free Form Modeling
  • Mesh of polygons
  • A certain regular structure or symmetry from 2D shapes
    • Treat a 2D shape as cross section and define a volume by sweeping the cross section along a path
      • Extrusion
    • To produce more elaborate objects
      • Curved path can be used
      • Size of cross section can be altered
      • Lathing
procedural modeling
Procedural Modeling
  • Models described by equations
  • Fractals, Fig. 4.42
    • Coastlines
    • Mountains
    • Edges of cloudy
    • Fig. 4.43, snowflake
    • Fractal mountainside, Fig. 4.44
procedural modeling1
Procedural Modeling
  • 3D Fractals
    • Fig. 4.45
    • Fractal terrain, Fig. 4.46
procedural modeling2
Procedural Modeling
  • Metaballs
    • Model soft objects
    • Fig. 4.47
    • Complex objects can be built by sticking metaballs together
  • Particle systems
    • Features made out of many particles
    • Rains, fountains, fireworks, grass
  • Metaballs are also included and make a great addition to helping create base models for further editing.A Metaball can be used in either a positive or negative way in trueSpace 5.
  • Particle explosion
  • Particle Storm 2.0
  • Wire frame, Fig. 4.48
  • Hidden surface removal
  • Surface properties
    • Color and reflectivity
  • Lights
    • Shading
      • A color for each polygon
      • Interpolate color
        • Gouraud shading
        • Phong shading: specular reflection
Ray tracing
    • Tracing the path of a ray of light back from each pixel , Plate 8
    • Photo-realistic graphics
    • High-performance workstations
  • Radiosity
    • Interactions between objects
    • Model complex reflections that occur between surfaces that are close together
pov ray
  • Persistence of Vision Ray Tracer
  •, Free
  • (a) Actual photo (b) Radiosity image
  • More accurately based on physics of light than other shading algorithms
texture mapping
Texture Mapping
  • Adding surface details
    • Mathematically wrapped over surface of object
    • Produce appearance of object
  • Bump mapping
    • Apply bumpiness or roughness and transparency mapping and reflection mapping, which modify the corresponding optical characteristics on the basis of a 2D map
Planar Mapping
  • Mapping To A Cube
Cylindrical Mapping
  • Spherical Mapping
bump mapping
Bump Mapping
  • St.Mattew head model is first simplified, reducing its geometry from 4 millions of faces (left) to just 5 hundreds (middle).The detail lost is then reproduced with an ad-hoc bumpmap (right).The resulting model is dynamically shaded, very similar to the original, but rendered incomparably faster.
specialized types of 3d graphics
Specialized Types of 3D Graphics
  • Build a body out of arms and legs
  • Rendering engine can use algorithms that are optimized for the characteristic models produced within limits set by the modeller
    • MetaCreations Bryce for landscapes andPoser for human and animal figures
  • Corel
  • Constructed from a grayscale image whose brightness represents height of 3D terrain model
  • Terrains can be based on an imported image, or one painted by hand; they can be generated using fractals or built from satellite data.
  • Sky, atmosphere, clouds, fog, haze, sun, moon, stars, rainbows
  • Curious Labs
  • Manikins (人體模型)
  • Physically realizable: hand on a figure of a person cannot turn 360