Chapter 4 Vector Graphics

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# Chapter 4 Vector Graphics - PowerPoint PPT Presentation

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

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.
• 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 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.
Introduction
• Vector Graphics
• Compact
• Scaleable
• Resolution independent
• Easy to edit
• Attractive for networked multimedia
Introduction
• 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.
Introduction
• 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
• 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 Vectors
• Points

A(3,7)

B(7,3)

O(0,0)

Anti-aliasing
• Approximating a straight line
• Using intermediate grey values
• Brightness is proportional to area of intersection
• At expense of fuzziness
Shapes
• 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
Polylines
• 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)

T1

T0

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

b1

b2

b3

b0

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 Curves
• Constructing a Bezier curve
• Fig. 4.10-13
• Finding mid-points of lines
Bezier Curves
• Figs. 4.14-18
• Same control points but in different orders

Bezier Curves

b11

b1

b2

b12

b02

b03

b01

b21

b0

b3

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
• 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
Paths
• 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
• 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 cap
• Butt cap
• Round cap
• Projecting cap
• Line Joins
• Miter
• Rounded
• Bevel
Fill
• Most drawing programs also allow to fill an open path
• Close the path with straight line between its endpoints
• Flat color, pattern or gradients
• Texture
Fill
• 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
• Translations: linear movement
• Scaling, rotation about a point
• Shearing: a distortion of angles of axes of an objects
Filters
• 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
Rounding
• 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
• Axes in 3D: Fig. 4.35
• Rotations in 3D: Fig. 4.36
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, …
Rendering
• 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
• 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
• 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
• Models described by equations
• Fractals, Fig. 4.42
• Coastlines
• Mountains
• Edges of cloudy
• Fig. 4.43, snowflake
• Fractal mountainside, Fig. 4.44
Procedural Modeling
• 3D Fractals
• Fig. 4.45
• Fractal terrain, Fig. 4.46
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
• 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.
Lightwave
• Particle explosion
• Particle Storm 2.0
Rendering
• Wire frame, Fig. 4.48
• Hidden surface removal
• Surface properties
• Color and reflectivity
• Lights
• A color for each polygon
• Interpolate color
Ray tracing
• Tracing the path of a ray of light back from each pixel , Plate 8
• Photo-realistic graphics
• High-performance workstations
• Interactions between objects
• Model complex reflections that occur between surfaces that are close together
POV-Ray
• Persistence of Vision Ray Tracer
• http://www.povray.org/, Free
• (a) Actual photo (b) Radiosity image
• More accurately based on physics of light than other shading algorithms
Texture Mapping
• 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
• 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
• 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
Bryce
• 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
Poser
• Curious Labs
• Manikins (人體模型)
• Physically realizable: hand on a figure of a person cannot turn 360