159.235 Graphics & Graphical Programming

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159.235 Graphics Graphical Programming

• Lecture 20 - 3D Shapes Polygons

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Polygons -Outline

• What are polygons?
• Types
• Convexity/concavity
• Triangles
• Storage of vertices and edges
• Rendering wire-frames as lines and points

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Polygonal Surfaces

• Polygon surfaces are a simple form of
  representation used in most applications
• Used in all real-time displays as fast to process
• Other descriptions such as Splines might be used
  in some applications but are generally reduced
  to polygons for processing.
• Polygon surfaces readily integrated with
  scan-line algorithms.

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Scan Conversion of Polygons

• Fill the polygon with colour.
• Incorporate in a scan conversion algorithm.
• We may not be drawing actual pixels due to
  anti-aliasing - jargon is that we tile the
  polygon with pixel fragments.

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Common Types of Polygon

• Concave
• Convex
• Triangles
• Trapezoids
• Quadrilaterals
• Self-intersecting
• Multiple loops
• Holes

Concave
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Convexity/Concavity Definition

• A polygon is convex if for all edges, all other
  vertices lie on the same side of the edge
• Otherwise it is concave.
• Concave polygons often difficult to process
  (algorithms get stuck or miss bits)

Concave
Convex
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Triangles Always Convex

• Mathematically, uses simple linear equations.
• Three points guaranteed co-planar by definition!
• Any polygon can be decomposed into triangles
• Triangles can approximate arbitrary shapes
• For any orientation on screen, a scan line will
  intersect only a single segment (scan).

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Polygon decomposition into Triangles
Convex polygons easy to decompose but if not
convex may have to add extra vertices to avoid
overlaps or intersection ambiguities
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Arbitrary shapes with triangles
Any 2D shape, or 3D surface, can be approximated
with locally linear polygons. To improve just
increase the number of edges
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Quadrilaterals are simple too and often mixed
with triangles
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Java Swing and Java 3D

• Swing has various 2D shape classes that allow
  us to implement various geometries
• Point3d and Vector3d objects are a convenience

• Java 3D is a Sun Microsystems (currently free)
  graphics library that builds on the 2D stuff
  from Swing
• Supports some very sophisticated 3D rendering
• Start by building our own 3D simple renderer

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Polygon Representation

• Could store all vertices explicitly - for each
  polygon
• Makes it hard to edit the vertices (multiple
  copies)
• Easier to store a table of vertices

• Index into the vertex table for each polygon
• Need to search for adjacent polygons
• Do end up drawing edges twice
• Tradeoffs!

• Store all polygon vertices explicitly.
• Inefficient
• Cannot manipulate vertex positions.

Polygonal Geometry.
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Common Polygonal Data Structures
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Drawing a Wire-frame 3d Polygon

• Break down into vertices and edges
• For each edge determine the two vertices
• Project the vertices from world space or model
  coordinates to screen coordinates

• Draw the projected line (edge)
• Repeat for each polygon under given projection

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Filling, or tiling a triangle.

• Calculate bounding box for triangle.
• Loop through pixels in box
• Test for points lying inside the triangle
• Draw pixel fragment if inside box
• Lets us colour in the shape (Java 2D or Swing
  will do this part for us)

Bounding box
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Polygons - Summary

• Various sorts of Polygons
• Convexity often an important property
• Triangles often useful simplest polygonal form
• Can store polygons (or triangles) in various
  ways - tradeoffs depending upon algorithm
• Render wire-frame by decomposing into
  points/lines - need transform to screen
  coordinates