CS 450: COMPUTER GRAPHICS
REVIEW: OVERVIEW OF
POLYGONS
SPRING 2015
DR. MICHAEL J. REALE
NOTE: COUNTERCLOCKWISE ORDER
• Assuming:
•
Right-handed system
•
Vertices in counterclockwise order (looking at front of polygon)
FILL AREAS
• Fill area (or filled area) = area to be filled with color or pattern (or both)
•
Usually surface of object
•
Usually polygons
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
http://www.guru3d.com/articlespages/radeon-hd-5870-review-test,7.html
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
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
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
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
CONVEX VS. CONCAVE: CORNY MEMORY HOOK
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
http://www.spudislunarresources.co
m/blog/wpcontent/uploads/2013/03/Airplane.j
pg
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)
 x
v 
 y
 x
v   y 
 z 
x
 y
v 
z
 
 w
LENGTH OF A VECTOR AND THE ZERO VECTOR
• Use Euclidean distance for length:
v  x y z
2
2
• A vector with a length of zero is called the zero vector:
0 
v 
0 
0 
v  0
0
0 
0 
v 
0 
 
0 
2
SCALING VECTORS
• Scalar times a vector = scalar times the components of the vector
•
Example: multiply 5 by a 3D vector A
 x A  5 * x A 




5  A  5 *  y A   5 * y A 
 z A   5 * z A 
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.
v

v


x


2
2
2
 x y z 


v
y


x2  y2  z 2  x2  y2  z 2 


z
 2

2
2
 x  y  z 
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
 x A   xB   ( x A  xB ) 
A  B   y A    yB   ( y A  yB )
 z A   z B   ( z A  z B ) 
 x A   xB   ( x A  xB ) 
A  B   y A    yB   ( y A  yB )
 z A   z B   ( z A  z B ) 
DOT PRODUCT
A  B  xA xB  y A yB  z A zB
• 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:
A  B  A B cos 
• …where θ = smallest angle between the two vectors
• If the vectors are normalized, then:
A 1
B 1
A  B  cos 
A A  x  y  z  A
2
2
2
2
Remember:
DOT PRODUCT: CHECKING ANGLES
A  B  A B cos 
cos(0 )  1
• 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) = 0  vectors are orthogonal (i.e., perpendicular to each other) (θ = 90°)
•
(A · B) = -1  vectors pointing in the exact opposite direction (θ = 180°)
• (We’re going to use this trick for lighting calculations later  )
cos(90 )  0
cos(180 )  1
u y v z  u z v y 
 wx 




W   wy   U  V   u z v x  u x v z 
u x v y  u y v x 
 wz 


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:
W  U V  U V sin 
• …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!
CROSS PRODUCT: ORDER MATTERS!
• WARNING! ORDER MATTERS with the cross product!
U V  V U
• Property of anti-commutativity
• REMEMBER THE RIGHT-HAND-RULE!!!
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
U V  ex (u y vz )  e y (u z vx )  ez (u x v y )
 ex (u z v y )  e y (u x vz )  ez (u y vx )
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
( E1  E 2 ) z  0
SPLITTING BY VECTOR METHOD
• Transform to XY plane (if necessary)
• Get edge vectors in counterclockwise order:
Ek  Vk 1  Vk
• 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!
( E2  E 3 ) z  0
( E3  E 4 ) z  0
( E4  E 5 ) z  0
( E5  E 1 ) z  0
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
SPLITTING A CONVEX POLYGON
• Every 3 consecutive vertices 
make triangle
• Remove middle vertex
• Keep going until down to last 3 vertices
PLANE EQUATION
• Need plane equation:
•
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:
Ax  By  Cz  D  0
•
(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
CALCULATING THE PLANE EQUATION
• Divide the formula by D
• Pick any 3 noncollinear points in the polygon
( A / D) xk  ( B / D) yk  (C / D) zk  1
k  1,2,3
• Solve set of 3 simultaneous linear plane equations to get A/D, B/D, and C/D  use Cramer’s Rule
1
y1
z1
x1 1 z1
x1
y1 1
x1
y1
z1
A  1 y2
1 y3
z2
z3
B  x2 1 z2
x3 1 z3
C  x2
x3
y2 1
y3 1
D   x2
x3
y2
y3
z2
z3
A  y1 ( z 2  z3 )  y2 ( z3  z1 )  y3 ( z1  z2 )
B  z1 ( x2  x3 )  z 2 ( x3  x1 )  z3 ( x1  x2 )
C  x1 ( y2  y3 )  x2 ( y3  y1 )  x3 ( y1  y2 )
D   x1 ( y2 z3  y3 z 2 )  x2 ( y3 z1  y1 z3 )  x3 ( y1 z 2  y2 z1 )
QUICK REVIEW: DETERMINANT OF A MATRIX
M
1 2
3 4
 1* 4  2 * 3
i j k
M  1 2 3  i (2 * 6  3 * 5)  j (3 * 4  1* 6)  k (1* 5  2 * 4)
4 5 6
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
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)
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:
N  (V2  V1 )  (V3  V2 )
• NOTE: Counterclockwise looking from outside the polygon towards inside
POINT-NORMAL PLANE EQUATION
• Given the normal N and any point on the plane, the following holds true:
N  P  D
• A related, alternative equation for a plane is the point-normal form:
N  (V  P)  0
• …where V is any 3D point
EXAMPLE