1. Polygons

2. Polygons

3. Triangle Meshes

4. Polygon Tables Vertex Table Edge Table Surface Table v1: x1, y1, z1
E1: v1, v2
E2: v2, v3
E3: x3, v1
E4: v3, v4
E5: v4, v5
E5: v5, v1
Surface Table
S1: E1, E2, E3
S2: E3, E4, E5, E6

5. Front and Back Faces Any point (x, y, z) is: on the plane:
A.x + B.y + C.z + D = 0
behind the plane:
A.x + B.y + C.z + D < 0
in front of the plane:
A.x + B.y + C.z + D > 0 y
in front of (outside)
the plane x
front face
behind (inside)
the plane z

6. Front and Back Faces A.x + B.y + C.z + D = 0 N = (v2-v1) x (v3-v1)
N.P = -D y
Normal vector
N=(A, B, C) v3 v2 x v1 z

7. all interior angles are
Polygons
convex
all interior angles are
<1800
concave
at least one angle is
>1800

8. Identifying Concave Polygons
Method 1
if some vertices are on one side and some on the other side of an extension line, then concave v4 E3 E5 v3 E2 v1 v2 E1

9. Splitting Concave Polygons
Rotational Method
Shift polygon so that each vertex vk in turn is at the coordinate origin
Rotate polygon about the origin in clockwise direction so that next vertex vk+1 is on the x axis
If vk+1 is below the x-axis, polygon is concave
Split polygon along x-axis y v1 v2 v3 v4 v5 x

10. Splitting Concave Polygons
Rotational Method
Shift polygon so that each vertex vk in turn is at the coordinate origin
Rotate polygon about the origin in clockwise direction so that next vertex vk+1 is on the x axis
If vk+1 is below the x-axis, polygon is concave
Split polygon along x-axis v1 v2 v3 v4 v5 y x

11. Identifying Concave Polygons
Method 2
if cross product of adjacent vector pairs is <0, then concave v4 E3 E5 v3 E2 v1 v2 E1

12. Splitting Concave Polygons
Vector Method
Find the cross products of edge vectors (Ek x Ek+1)
If z component is negative then split polygon along the line of the first edge vector in the cross product pair
Then find the intersection of the line with other edges of the polygon E4 v5 v4 E3 E5 v3 E2 v1 v2 E1

13. Splitting Convex Polygons into Triangles

14. Polygon Filling: Inside-Outside Test
Odd-Even Test
Draw a line from any point P to a point outside the closed polyline
If the number of line-segment crossings is:
odd => P is an interior point
even => P is an exterior point

15. Polygon Filling: Scan-Line Algorithm
Find intersection of polygon boundaries with screen scan lines (solving the pair of simultaneous linear equations where scanline is y, which is a constant)
Apply fill-color to scan line sections that lie within the fill region (use odd-even rule)

16. Polygon Filling: Scan-Line Algorithm
Monotonically increasing/decreasing

17. Polygon Filling: Scan-Line Algorithm
Incremental calculation of intersection points
yk+1 – yk =1
m = 1/ (xk+1 – xk )
xk+1 = xk + 1/m
(xk+1, yk+1)
yk+1 yk
(xk, yk)
xk xk+1

18. Polygon Filling: Boundary-Fill Algorithm
4-connected
8-connected

19. Polygon Filling: Boundary-Fill Algorithm
void boundaryFill4 (int x, int y, int fillColor, int borderColor) {
int interiorColor;
/* Set current color to fillColor, then perform following oprations. */
getPixel (x, y, interiorColor);
if ((interiorColor != borderColor) &&
(interiorColor != fillColor))
setPixel (x, y); // Set pixel color to fillColor
boundaryFill4 (x + 1, y , fillColor, borderColor);
boundaryFill4 (x - 1, y , fillColor, borderColor);
boundaryFill4 (x , y + 1, fillColor, borderColor);
boundaryFill4 (x , y - 1, fillColor, borderColor) }

20. Polygon Filling: Boundary-Fill Algorithm
void boundaryFill4 (int x, int y, int fillColor, int borderColor) {
int interiorColor;
/* Set current color to fillColor, then perform following oprations. */
getPixel (x, y, interiorColor);
if ((interiorColor != borderColor) &&
(interiorColor != fillColor))
setPixel (x, y); // Set pixel color to fillColor
boundaryFill4 (x + 1, y , fillColor, borderColor);
boundaryFill4 (x - 1, y , fillColor, borderColor);
boundaryFill4 (x , y + 1, fillColor, borderColor);
boundaryFill4 (x , y - 1, fillColor, borderColor) }

21. Polygon Filling: Boundary-Fill Algorithm
void boundaryFill4 (int x, int y, int fillColor, int borderColor) {
int interiorColor;
/* Set current color to fillColor, then perform following oprations. */
getPixel (x, y, interiorColor);
if ((interiorColor != borderColor) &&
(interiorColor != fillColor))
setPixel (x, y); // Set pixel color to fillColor
boundaryFill4 (x + 1, y , fillColor, borderColor);
boundaryFill4 (x - 1, y , fillColor, borderColor);
boundaryFill4 (x , y + 1, fillColor, borderColor);
boundaryFill4 (x , y - 1, fillColor, borderColor) }

22. Polygon Filling: Boundary-Fill Algorithm2 2 1 1

23. Polygon Filling: Boundary-Fill Algorithm3 3 1 1

24. Polygon Filling: Boundary-Fill Algorithm5 6 6 5 4 1 4 1

25. Polygon Filling: Boundary-Fill Algorithm7 7 5 4 1 5 4 1

26. Polygon Filling: Boundary-Fill Algorithm5 5 4 1 4 1

27. Polygon Filling: Boundary-Fill Algorithm8 8 4 1 4 1

28. Polygon Filling: Boundary-Fill Algorithm4 1 4 1

29. Polygon Filling: Boundary-Fill Algorithm9 1 9 1

30. Polygon Filling: Boundary-Fill Algorithm10 1 10 1

31. Polygon Filling: Boundary-Fill Algorithm11 1 1 11

32. Polygon Filling: Boundary-Fill Algorithm12 1 1 12

33. Polygon Filling: Boundary-Fill Algorithm1 1

34. Polygon Filling: Flood-Fill Algorithm
void floodFill4 (int x, int y, int fillColor, int interiorColor) {
int color;
/* Set current color to fillColor, then perform following operations. */
getPixel (x, y, color);
if (color = interiorColor) {
setPixel (x, y); // Set pixel color to fillColor
floodFill4 (x + 1, y, fillColor, interiorColor);
floodFill4 (x - 1, y, fillColor, interiorColor);
floodFill4 (x, y + 1, fillColor, interiorColor);
floodFill4 (x, y - 1, fillColor, interiorColor) }

35. OpenGL Polygon Functions glRect*(x1, y1, x2, y2); Ex:
i (integer)
s (short)
f (float)
d (double)
v (vector)
Ex:
glRecti(50, 250, 200, 100);
int v1[ ]={50, 250};
int v2[ ]={200, 100};
glRectiv(v1, v2);
(50,250)
(200,100)

36. OpenGL Ex: GL_POLYGON GL_TRIANGLES GL_TRIANGLE_STRIP GL_TRIANGLE_FAN
glBegin (GL_POLYGON);
glVertex2iv(p1);
glVertex2iv(p2);
glVertex2iv(p3);
glVertex2iv(p4);
glVertex2iv(p5);
glVertex2iv(p6);
glEnd(); p6 p5 p1 p4 p2 p3
GL_POLYGON
GL_TRIANGLES
GL_TRIANGLE_STRIP
GL_TRIANGLE_FAN

37. OpenGL Ex: GL_QUADS GL_QUAD_STRIP glBegin (GL_QUADS); glVertex2iv(p1);
glEnd(); p8 p1 p4 p5 p7 p6 p3 p2
GL_QUADS
GL_QUAD_STRIP