Classes of Polygons

1
Classes of Polygons

• Planar polygons
• Simple
• Convex
• Star-shaped
• Non-simple
• Non-planar polygons

2
Planar Polygons

• Definition
• All points of the polygon must lie within a
  plane.

3
Simple Polygons

• Definition
• A polygon P is simple, if no two non-consecutive
  edges intersect.

Simple
Not simple
4
Simple Polygons

• The polygon interior is well-defined. If we
  traverse the polygon in a counter-clockwise
  direction, the interior is always to our left.

Simple
Not simple
5
Convex Polygons

• Definition
• A polygon P is convex, if and only if, for any
  pair of points x,y, in P the line segment between
  x and y lies entirely in P.
• Every point can see every other point.

Convex
Concave
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Convex Polygons

• Alternative Definition
• Going from point Pi to Pi1 involves making only
  left had turns.
• Implies that the polygon vertices are ordered in
  a counter-clockwise fashion.
• Implies cross-products of adjacent edges are
  positive.
• Convex polygons are of course, simple.

7
Triangles

• Any three non-colinear points determine a
  triangle.
• Planar
• Convex
• Cross products can determine the normal (A,B,C)T
  to the triangle (plane).
• Plane Ax By Cz D 0, where D can be
  solved using any of the three points.

8
Star-Shaped Polygons

• Definition
• A star-shaped polygon, P, is a simple polygon
  which contains a non-empty set, K of points, such
  that every point in K is visible from any other
  point in the polygon. The set K is called the
  kernel of P.
• A convex polygon is a star-shaped polygon whose
  kernel is equal to the interior of P.

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Star-Shaped Polygons

• The kernel can easily be determined by clipping
  the polygon to the half-planes defined by each
  edge.

10
Non-Planar Polygons

• In 3D computer graphics, we often have to deal
  with non-planar polygons.
• Longitude/latitude grid on a sphere.
• Polygonal approximations to smooth surfaces.

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More Polygonal Definitions

• Convex Hull The convex hull of a polygon P, is
  the smallest convex polygon that contains P.
• Diagonal A line segment lying entirely inside
  polygon P and joining two non-consecutive
  vertices pi and pk
• Principal vertex A vertex pi of simple polygon
  P is called a principal vertex if the diagonal
  (pi-1, pi1) intersects the boundary of P only at
  pi-1 and pi1.

Not a principal vertex
diagonal
Convex hull
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More Polygonal Definitions

• Ear A principal vertex pi of a simple polygon P
  is called an ear if the diagonal (pi-1, pi1)
  lies in the interior of P.
• Mouth A principal vertex pi of a simple polygon
  P is called a mouth if the diagonal (pi-1, pi1)
  lies in the exterior of P.

13
Some Theorems

• One-Mouth Theorem Except for convex polygons,
  every simple polygon has at least one mouth.
• Two-Ears Theorem Except for triangles, every
  simple polygon has at least two non-overlapping
  ears.

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Polygonal Tricks

• Area
• Test for concave/convex.

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Polygonal Area

• Convex polygons
• Take any point inside the polygon and compute the
  area of all triangles formed with one edge and
  the point.

Actually, we can take any point and do this, if
we give the area of the triangles a signed value.
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Triangle Area

• The good old cross product
• A?B area of parallel-piped
• 2area of triangle.
• For 2D triangles, the magnitude is the absolute
  value ofthe z-component.

B
A
Area ½?AxBy-AyBx ?
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Polygonal Area

• Vectors Choose the point to be the origin, then
  the vectors are simply the points on the polygon.
• Convex polygon
• Area

18
Polygonal Area

• Wow!!! This actually works for any simple polygon.

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Convexity Test

• We can easily determine whether a polygon is
  convex or concave, and whether it is order
  clockwise or counter-clockwise.
• Examine the cross products between adjacent
  edges. If they are all the same sign, then the
  polygon is convex, otherwise it is concave.
• If the area computed as before is positive, then
  the ordering is in a counter-clockwise fashion.

20
Efficient Scan-Conversion

• Convex polygons
• Triangles

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Scan-Converting Convex Polygons

• What is the intersection of a line with a convex
  polygon?

22
Scan-Converting Convex Polygons

• Any line will intersect a convex polygon in at
  most 2 locations.
• This greatly simplifies our active edge table.
• Simply keep a current x-left and an x-right
  intersection.
• Update the slopes on scan-lines where new
  edgesstart.

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Scan-Converting Triangles

• Special cases
• One horizontal edge.
• For yymin to ymax
• Draw from x-left to x-right
• Update x-left and x-right

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Scan-Converting Triangles

• General cases
• Divide into two special cases.
• One quick split at middle vertex and 2 very tight
  for loops.

25
Triangulations

• To avoid problems with non-planar polygons, as
  well as simplify the scan-conversion process, we
  can triangulate a polygon before we rasterize it.
• A triangulation of a simple polygon P consists of
  n-3 diagonals, or n-2 triangles.

26
Triangulation Algorithms

• Convex polygons are trivial.
• Star-shaped polygons can also be dealt with
  rather easily.
• Can also use splitting algorithms as described in
  the book. These add additional vertices, but may
  be simpler.

27
Triangulating Convex Polygons

• Pick any vertex i.
• Connect the vertices (i,k,k1) for all k? i, i-1.

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Triangulating Concave Polygons

• While polygon P is not a triangle
• Find an ear of P.
• Remove the ear from P.