1. CS 450: Computer Graphics REVIEW: OVERVIEW OF POLYGONS
Spring 2015
Dr. Michael J. Reale

2. NOTE: COUNTERCLOCKWISE ORDER
Assuming:
Right-handed system
Vertices in counterclockwise order (looking at front of polygon)

3. FILL AREAS
Fill area (or filled area) = area to be filled with color or pattern (or both)
Usually surface of object
Usually polygons

4. WHY POLYGONS?
Why?
Boundaries = linear equations (efficient fill algorithms)
Can approximate most curved surfaces with polygons
Shading easier (especially with triangles  single plane per triangle)
Surface tessellation = approximating a curved surface with polygons
More polygons  better detail, but more vertices/polygons to process
Also called fitting the surface with a polygon mesh

5. POLYGON DEFINITIONS
Polygon = a figure with 3 or more vertices connected in sequence by straight-line segments (edges or sides of the polygon)
Most loose definition  any closed-polyline boundary
More finicky definitions  contained in single plane, edges have no common points other than their endpoints, no three successive points collinear
Standard polygon or simple polygon = closed-polyline with no crossing edges

6. DOES A POLYGON LIE IN A SINGLE PLANE?
In computer graphics, polygon not always in same plane:
Round-off error
E.g., after transformations
Fitting to surface makes non-planar polygons
E.g., quad “bent” in half
Thus, use triangles  single plane per triangle

7. DEGENERATE POLYGONS
Degenerate polygons = often used to describe polygon with:
3 or more collinear vertices  generates a line segment
E.g., in extreme case, triangle with no area
Repeated vertex positions  some edges have length 0

8. CONVEX VS. CONCAVE
Interior angle = angle inside polygon boundary formed by two adjacent edges
By interior angle:
Convex = all interior angles less than 180°
Concave = one or more interior angles greater than or equal to 180°
By looking at vertices compared to edge lines:
Convex = for all edge lines, all other vertices are on one side
Concave = for one or more edge lines, some of the vertices are on one side and some are on the other
Also, one or more edge lines will intersect another edge

9. CONVEX VS. CONCAVE: CORNY MEMORY HOOK

10. DEALING WITH DEGENERATES AND CONCAVE POLYGONS
OpenGL cannot deal with degenerate polygons or concave polygons  programmer must detect/preprocess them
Degenerate polygons  detect and remove
Concave polygons  detect and split into convex polygons

11. What’s your vector, victor?
What’s your vector, victor?
A vector = (N x 1) or (1 x N) matrix
(# of rows X # of columns)
In computer graphics:
N  usually 2, 3, or 4 (homogeneous coordinates)
Use column vectors (N x 1)
Components of the vector:
2D  x and y values
3D  x, y, and z values
4D  x, y, z, and w values
Vector interpreted as:
Location (w = 1)
Direction (w = 0)
Scalar = single value (or 1x1 vector)

12. LENGTH OF A VECTOR and THE ZERO VECTOR
Use Euclidean distance for length:
A vector with a length of zero is called the zero vector:

13. SCALING VECTORS
Scalar times a vector = scalar times the components of the vector
Example: multiply 5 by a 3D vector A

14. NORMALIZED VECTORS
Normalize a vector = divide vector by its length ║v║  makes length equal to 1
Equivalent to multiplying vector by 1/║v║
Resulting vector is called a normalized vector
WARNING: This is NOT the same as a NORMAL vector!
Although normal vectors are often normalized.

15. ADDING/SUBTRACTING VECTORS
Add/subtract vectors  add/subtract components
Geometric interpretation:
Adding  putting head of one vector on tail of the other
Subtracting  gives direction from one endpoint to the other

16. DOT PRODUCT
Result is a single number (i.e., scalar)  another name for the dot product is the scalar product
Dot product of vector with itself = square of length of vector:
Equivalent to:
…where θ = smallest angle between the two vectors
If the vectors are normalized, then:

17. DOT PRODUCT: CHECKING ANGLES
Remember:
DOT PRODUCT: CHECKING ANGLES
Look at sign of dot product to check angle:
(A · B) > 0  vectors pointing in similar directions (0 <= θ < 90°)
(A · B) = 0  vectors are orthogonal (i.e., perpendicular to each other) (θ = 90°)
(A · B) < 0  vectors pointing away from each other (90° < θ <= 180°)
For normalized vectors, dot product ranges from [-1, 1]:
(A · B) = 1  vectors pointing in the exact same direction (θ = 0°)
(A · B) = -1  vectors pointing in the exact opposite direction (θ = 180°)
(We’re going to use this trick for lighting calculations later  )

18. CROSS PRODUCT Also called vector product (results is a vector)
Given two vectors U and V  gives vector W that is orthogonal (perpendicular) to both U and V
U, V, and W form right-handed system!
I.e., can use right-hand-rule on U and V (IN THAT ORDER) to get W
Example: (X x Y) = Z axis!
The length of W (= U X V) is equivalent to:
…where again θ = smallest angle between U and V
If U and V are parallel  θ = 0°  sin θ = 0  get zero vector for W!

19. CROSS PRODUCT: ORDER MATTERS!
WARNING! ORDER MATTERS with the cross product!
Property of anti-commutativity
REMEMBER THE RIGHT-HAND-RULE!!!

20. COMPUTING THE CROSS PRODUCT: SARRUS’S SCHEME
Follow diagonal arrows  for each arrow  multiply elements along arrow times sign at top
ex = x axis, ey = y axis, ez = z axis

21. DETECTING AND SPLITTING CONCAVE POLYGONS
There are 2 ways we can do this:
Vector method
Check interior angles using cross product
Rotational method
Rotate each edge in line with X axis  check if vertex below X axis

22. SPLITTING BY VECTOR METHOD
Transform to XY plane (if necessary)
Get edge vectors in counterclockwise order:
For each pair of consecutive edge vectors, get cross product
If concave  negative Z component  split polygon along first vector in cross product pair
Have to intersect this line with other edges
Repeat process with two new polygons
NOTE: 3 successive collinear points anywhere  cross product becomes zero vector!

23. SPLITTING BY ROTATIONAL METHOD
Transform to XY plane (if necessary)
For each vertex Vk:
Move polygon so that Vk is at the origin
Rotate polygon so that Vk+1 is on the X axis
If Vk+2 is below X axis  polygon is concave  split polygon along x axis
Repeat concave test for each of the two new polygons
Stop when we’ve checked all vertices

24. SPLITTING A CONVEX POLYGON
Every 3 consecutive vertices  make triangle
Remove middle vertex
Keep going until down to last 3 vertices

25. PLANE EQUATION Need plane equation: Need normal of polygon:
Collision detection, raytracing/raycasting, etc.
Need normal of polygon:
Lighting/shading
Backface culling  don’t draw polygon facing away from camera
General equation of a plane:
(x,y,z) = any point on the plane
A,B,C,D = plane parameters
ON the plane  Ax + By + Cz + D = 0
BEHIND the plane  Ax + By + Cz + D < 0
IN FRONT OF the plane  Ax + By + Cz + D > 0

26. CALCULATING THE PLANE EQUATION
Divide the formula by D
Pick any 3 noncollinear points in the polygon
Solve set of 3 simultaneous linear plane equations to get A/D, B/D, and C/D  use Cramer’s Rule

27. QUICK REVIEW: DETERMINANT OF A MATRIX

28. If THE POLYGON IS NOT CONTAINED IN A PLANE
Either:
Divide into triangles
OR:
Find approximating plane
Divide vertices into subsets of 3
Get plane parameters for each subset
Get average plane parameters

29. NORMAL VECTOR Normal vector = gives us orientation of plane/polygon
Points towards OUTSIDE of plane
From back face to front face
Perpendicular to surface of plane
Normal N = (A,B,C)  parameters from plane equation!
Although it doesn’t have to be, the normal vector is often normalized (i.e., length = 1)

30. GETTING NORMAL AND PLANE EQUATION
Solve for plane equation  normal = (A,B,C) OR
Get normal from edges using cross-product  solve for D in plane equation
V1, V2, and V3 = consecutive vertices in counterclockwise order:
NOTE: Counterclockwise looking from outside the polygon towards inside

31. POINT-nORMAL PLANE EQUATION
Given the normal N and any point on the plane, the following holds true:
A related, alternative equation for a plane is the point-normal form:
…where V is any 3D point

32. EXAMPLE