Vector Graphics Digital Multimedia Chap.4

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Vector GraphicsDigital Multimedia Chap.4

• 2006. 4. 4
• ???
• icebyung_at_hufs.ac.kr

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Vector Graphics

• Vector Graphics
• Compact
• Scaleable
• Resolution-independent
• Easy to edit
• Attractive for networked multimedia

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Coordinates and Vectors

• Coordinates
• Image is stored as a rectangular array of pixels
• Rectangular array of pixels, each pixel is
  identified by row and column numbers (r,c)
  ?coordinates
• Coordinates of pixels in an image must be integer
  values between zero and the horizontal or
  vertical
• Coordinate transformation
• Coordinates in one system are transformed into a
  different one
• Vector
• Specify a displacement by movements in x and y
  directions
• Displacement is also a pair of numbers
• P2 - P1 (x2-x1, y2-y1), displacement (movement)
  from P1 to P2
• P2 - P1 ? P1 - P2

lt Vector gt
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Rendering

• Represent lines, curves, shapes,... by parameters
  of their defining equations
• e.g. equation of straight line y mx c,
  store m and c (or endpoints)
• Compute pixels which must be set in order to draw
  the object
• Pixels coordinates are always integers
• Cant set the value of just part of a pixel
• The pixel image can only ever approximate the
  ideal mathematical object
• Result exhibit jaggies or staircasing

Figure 4.4 Approximating a straight line
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Anti-aliasing

• Rendering is sampling and reconstruction
• Abstract continuous line c must be approximated
  by discrete pixels of finite size
• Jaggies are form of aliasing caused by
  undersampling
• Anti-aliasing
• Anti-aliasing is used to reduce aliasing impact
• Mitigate effect by coloring pixels in shades of
  grey (for a black line), brightness varies
  according to extent of intersection with an
  idealized 1 pixel wide line

Figure 4.6 Anti-aliased line
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Shapes

• Drawing programs and vector graphics languages
  provide basic repertoire of shapes that can
  easily be represented mathematically
• Rectangles and squares
• Ellipses and circles
• Straight lines, polylines and polygons
• Smooth (Bézier) curves
• Shapes built up out of these elements can be
  filled with color, patterns or gradients

polyline
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Carves

• Bézier Curves
• A class of curve
• Bézier curves (smooth curves) completely
  specified by four points
• 2 endpoints (P1, P2)
• 2 direction points (P3, P4)
• Sweep of the curve is determined by length and
  direction of lines from endpoints to direction
  points
• Constructing a Bézier curves
• Figure 4.10

figure 4.8 Bézier Curves (p.92)
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Paths

• Bézier curves can be combined to make elaborate
  curves and irregular shapes
• Joining two Bézier curves
• Path
• Collection of lines and curves
• Closed path joins up on itself, open path doesnt

Open path
Closed path
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Stroke and Fill

• Path is an abstract mathematical entity
• Path is infinitesimally thin
• Apply stroke to make path visible (like tracing
  with ink)
• Specify width and color etc
• Line cap subtle feature of stroke (shape of
  end)
• Butt cap, round cap, projecting cap
• Apply fill to closed path or shape (like coloring
  it in)
• Specify color or a gradient or pattern
• Gradients may be linear or radial

figure 4.20 Gradient fills
figure 4.22 pattern fills
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Transformations

• The objects that make up a vector image are
  stored in the form of a few values
• Manipulate vector objects in certain ways by
  changing stored values that describe them
• Translation (liner movement)
• Scaling
• Reflection
• Rotation
• Shearing (skewing)

Original shape
scaling
reflection
rotation
shearing
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3-D Graphics

• 3-D Vector Graphics
• Conceptually simple extension of 2-D
• Add z-axis at right angles to x- and y-axes
• Point is defined by (x, y, z) coordinates, vector
  is triple of values
• 3-D Rendering
• Start with mathematical model of objects in 3-D
  space, need a 2-D picture, usually in perspective
• Need to consider viewpoint ( or camera position)
• Need to consider lighting (position of light
  source)
• Surfaces appearance (texture, diffuse) can be
  determine by position and lighting condition

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3D-Models

• Three general approaches to modeling 3-D object
• Constructive solid geometry
• Free form modeling
• Procedural modeling
• Constructive solid geometry
• Simplest approach
• Build 3-D models out of a few primitive objects
  such as cube, cylinder, sphere and pyramid
• Apply distortions such as squashing or stretching
• Combine objects using spatial equivalents of set
  operations
• Union, intersection, difference

difference
union
intersection
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3D-Models

• Free form modeling
• Use representation of an objects boundary
  surface to define it
• Build surfaces out of flat polygons or curved
  patches
• Construct surface as mesh of flat polygons
• Can generalize Bézier curves to 3-D surfaces
• Generate objects by sweeping 2-D cross section
  along a path
• Extrusion and lathing
• Procedural modeling
• Define an object by giving algorithm for
  constructing it
• Fractals
• Exhibit same structure at all level of detail
• Model landscape features, clouds
• Metaballs

Figure 4.38 constructing a well-known fractal
curve
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3D-Rendering

• 3D models only exist inside computer
• When working on a model in 3D application, a
  rendering operation is needed
• Rendering high quality images is time consuming
• Necessary to compromise on quality
• Wire frame
• Used as preview images in modeling program
• Absence of any surfaces
• Hidden surface removal
• Necessary to determine which surfaces of objects
  are visible

Figure 4.45 wire frame
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3D-Rendering

• Shading algorithms
• Interpolate color across each polygon to disguise
  discontinuities
• gouraud shading and phong shading
• Paong shading takes account of specular
  reflection
• Only deal with each object in isolation
• Ray tracing
• Works by tracing the path of a ray of light
• Radiosity
• Interactions between objects
• Attempts to model the complex reflections that
  occur between surfaces
• Texture mapping
• Adding surface details to 3D models