Transcript Classes of Polygons • Planar polygons • Non-planar polygons – Simple

Classes of Polygons
• Planar polygons
– Simple
• Convex
• Star-shaped
– Non-simple
• Non-planar polygons
Planar Polygons
• Definition:
All points of the polygon must lie within a plane.
Simple Polygons
• Definition
A polygon P is simple, if no two non-consecutive
edges intersect.
Simple
Not simple
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
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
Convex Polygons
• Alternative Definition
Going from point Pi to Pi+1 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.
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.
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.
Star-Shaped Polygons
• The kernel can easily be determined by
clipping the polygon to the half-planes
defined by each edge.
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.
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, pi+1)
intersects the boundary of P only at pi-1 and pi+1.
Not a
principal vertex
diagonal
Convex hull
More Polygonal Definitions
• Ear – A principal vertex pi of a simple polygon P is called
an ear if the diagonal (pi-1, pi+1) 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, pi+1) lies in the exterior
of P.
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.
Polygonal Tricks
• Area
• Test for concave/convex.
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.
Triangle Area
• The good old cross product:
||AB|| = area of parallel-piped
= 2*area of triangle.
B
For 2D triangles, the
magnitude is the
absolute value of
the z-component.
Area = ½AxBy-AyBx 
A
Polygonal Area
• Vectors: Choose the point to be the origin,
then the vectors are simply the points on the
polygon.
• Convex polygon
n 1
Area =
1
2
 x y
i
i 0
i 1
 xi 1 yi 
Polygonal Area
• Wow!!! This actually works for any simple
polygon.
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.
Efficient Scan-Conversion
• Convex polygons
• Triangles
Scan-Converting Convex
Polygons
• What is the intersection of a line with a
convex polygon?
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 scanlines where new edges
start.
Scan-Converting Triangles
• Special cases
One horizontal edge.
• For y=ymin to ymax
Draw from x-left to x-right
Update x-left and x-right
Scan-Converting Triangles
• General cases:
• Divide into two
special cases.
• One quick split at middle vertex and 2 very
tight for loops.
Triangulations
• To avoid problems with non-planar
polygons, as well as simplify the scanconversion 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.
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.
Triangulating Convex Polygons
• Pick any vertex i.
• Connect the vertices (i,k,k+1) for all
k i, i-1.
Triangulating Concave Polygons
While polygon P is not a triangle
Find an ear of P.
Remove the ear from P.