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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

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 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)

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)

- 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

- 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 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

Changing a smooth curve to a corner and vice versa

convert-anchor-point tool

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
- Gradient: linear, radial
- 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
- Reflective about a line
- 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
- 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
- http://www.povray.org/, Free

Radiosity

- (a) Actual photo (b) Radiosity image
- More accurately based on physics of light than other shading algorithms

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

- 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

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