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

2. INTRODUCTION TO POLYGONS

3. INTRODUCTION
So far, we’ve talked mostly about points, lines, and curves
Fill area (or filled area) = some area that needs to be filled in with a solid color or pattern
Usually used to describe the surface of an object (but not always)
Usually polygons (but not always)

4. WHY POLYGONS?
Most graphics libraries force you to use polygons for filled areas
Why?
Polygon boundaries can be described with linear equations
Makes fill algorithms more efficient
Can approximate most curved surfaces with polygons
Similar to approximating curve with line segments
Makes shading easier (especially if polygons are triangles  single plane per triangle)
Can still create the illusion of a curved/complicated surface with tricks like phong shading and normal mapping
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. WHAT IS A POLYGON, EXACTLY?
The official definition varies, but at bare minimum:
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
I.e., points originally in single plane, but, after transformation, the points may be slightly off
Fitting to surface makes non-planar polygons
E.g., if using quads, approximation may “bend” the quad in half
Thus, we usually use triangles to avoid this problem

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  generates shape with:
Extraneous lines
Overlapping edges
Edges with length 0

8. CONVEX VS. CONCAVE: BY INTERIOR ANGLE
Interior angle = angle inside polygon boundary formed by two adjacent edges
Convex = all interior angles less than 180°
Concave = one or more interior angles greater than or equal to 180°

9. CONVEX VS. CONCAVE: BY PICKING SIDES
Another way to define convex vs. concave is to look at each line formed by each edge and see if all other vertices on one side or not
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

10. CONVEX VS. CONCAVE: CORNY MEMORY HOOK

11. DEALING WITH DEGENERATES AND CONCAVE POLYGONS
Most graphics software packages (i.e., OpenGL) can’t really deal with the degenerate polygons
To avoid extra processing, however, they leave it as the programmer’s problem
Concave polygons are more complicated to deal with when it comes to filling algorithms
Solution: split concave polygon into convex polygons (e.g., triangles)
Again, often have to do this yourself
OpenGL “requires” that all polygons be convex (or at least, there’s no guarantee the filling will work if not convex)

12. BRIEF VECTOR MATH REVIEW

13. INTRODUCTION
Before we go any further, we’ll need to quickly review some concepts from linear algebra
We will revisit these concepts in a future, more thorough review
For now, let’s talk about:
Vectors
Vector length
Scaling vectors
Normalized vectors
Adding/subtracting vectors
Dot product
Cross product

14. What’s your vector, victor?
A vector is basically a (N x 1) or (1 x N) matrix
(# of rows X # of columns)
In computer graphics:
N is usually 2, 3, or 4 (homogeneous coordinates)
Usually 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
Scalar = single value (or 1x1 vector)

15. INTERPRETING VECTORS A vector can also be interpreted as: …in space
Location (w = 1)
Direction (w = 0)
…in space

16. LENGTH OF A VECTOR
We usually use Euclidean distance to get the length of a vector:
The square of the distance (not surprisingly) is given by:

17. ZERO VECTOR
A vector with a length of zero is called the zero vector:

18. SCALING VECTORS
To multiply a scalar by a vector, you multiple that scalar by the components of the vector
Example: multiply 5 by a 3D vector A

19. 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.

20. ADDING/SUBTRACTING VECTORS
To add/subtract vectors, you add/subtract their respective components:

21. GEOMETRIC INTERPRETATION OF ADDING/SUBTRACTING (DIRECTION) VECTORS
Adding  putting head of one vector on tail of the other
Subtracting  gives direction from one endpoint to the other

22. DOT PRODUCT
Given two 3D vectors A and B, the dot product of A and B is given by:
Basically:
Multiple corresponding components
Add them together
NOTE: Result is a single number (i.e., scalar)  another name for the dot product is the scalar product

23. DOT PRODUCT WITH YOURSELF
The dot product of a vector A with itself is equal to the square of its length:

24. DOT PRODUCT: WHAT DOES IT MEAN?
The dot product of two vectors A and B is in fact equivalent to:
…where θ = smallest angle between the two vectors
(Basically it has to do with projecting one vector on another; we’ll discuss this later)
If the vectors are normalized, then:

25. DOT PRODUCT: CHECKING ANGLES
Remember:
DOT PRODUCT: CHECKING ANGLES
What this means:
(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  )

26. CROSS PRODUCT 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!

27. LENGTH OF CROSS PRODUCT
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!

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

29. COMPUTING THE CROSS PRODUCT
To compute the cross product:

30. COMPUTING THE CROSS PRODUCT: SARRUS’S SCHEME
An easier way to remember this is with a method called 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

31. IDENTIFYING AND SPLITTING CONCAVE POLYGONS

32. DETECTING CONCAVE POLYGONS
We can do this one of two ways:
Check interior angles for one greater than 180°
For each edge line, check if vertices are all on the same side  if not, concave

33. CHECKING INTERIOR ANGLES
Treat each edge as a vector (go counterclockwise):
For each pair of consecutive edge vectors, get cross product
If convex  signs of all cross products will be the same
In 2D  cross product will always be +Z or –Z
In 3D  transform polygon to XY plane (as best we can)
If concave  one or more will be different
NOTE: This breaks down if we have 3 successive collinear points anywhere  cross product becomes zero vector!

34. SPLITTING A CONCAVE POLYGON
There are 2 ways we can do this:
Vector method
Rotational method

35. 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 a cross product has a negative z component  split polygon along first vector in cross product pair
Have to intersect this line with other edges

36. 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

37. SPLITTING A CONVEX POLYGON
Once we have a convex polygon(s), it is usually beneficial to split it into triangles:
Every 3 consecutive vertices  make triangle
Remove middle vertex
Keep going until down to last 3 vertices

38. POLYGONS, PLANES, and NORMALS

39. INTRODUCTION
We’re going to assume at this point that each polygon we have fits into a single plane (triangle or not)
One very important piece of information we need to know is what is the front (and back) of our polygon
Need this for:
Lighting/shading
Backface culling (i.e., not drawing polygons that face away from the camera)
More generally, we also would like to know the equation of the plane the polygon rests in
Useful for collision detection and/or raytracing

40. PLANE EQUATION
Each polygon in our 3D scene is contained within a plane of infinite extent
The general equation of a plane is as follows:
(x,y,z) = any point on the plane
A,B,C,D = plane parameters

41. CALCULATING THE PLANE EQUATION
To get the plane parameters:
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:

42. USING CRAMER’S RULE
It turns out you can use something called Cramer’s Rule to solve this
Basically, the solutions for A, B, C, and D are given by the following determinants:
More information on Cramer’s Rule:

43. QUICK REVIEW: DETERMINANT OF A MATRIX

44. GETTING THE DETERMINANTS
Expanding the determinants gives us the answers for A,B,C, and D
This will actually work even when D = 0 (technically you can’t use Cramer’s Rule under those circumstances, but the workaround actually gives you the same answer)

45. 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

46. FRONT AND BACK Each polygon has a front face and a back face
We can use the plane equation to determine if a given point (x,y,z) is:
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
NOTE: This is valid for a right-handed coordinate system, so we should also make sure our vertices are ordered counterclockwise!

47. 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)

48. GETTING THE NORMAL VECTOR FROM EDGES
You can also compute the normal vector directly from the edges using the cross-product
Assuming V1, V2, and V3 are consecutive vertices of a convex polygon in counterclockwise order:
Can also use this approach to get A, B, and C, and then solve for D.
NOTE: Counterclockwise looking from outside the polygon towards inside

49. 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
Basically, think of a vector going from P (a point on the plane) to some other point V. If that vector is on the plane, it should be orthogonal to the normal (hence, the dot product with be 0).

50. EXAMPLE