1. CS148: Introduction to Computer Graphics and Imaging
Triangles
CS148 Lecture 3

2. How to draw a triangle on the screen
Rasterization
CS148 Lecture 3

3. Window edges at integers
Pixel Coordinates
Window edges at integers y
(4,3)
Pixels
inside
window
(0,1)
(1,1)
(0,0)
(1,0) x

4. Pixel Coordinates
OpenGL: Pixel centers correspond to half-integer coordinates y
(.5, 1.5)
(1.5, 1.5)
(1.5, .5)
(.5, .5)
Note: Other graphics packages may use a different convention x

5. Triangle Rasterization Rule
Output fragment if pixel center is inside the triangle
CS148 Lecture 3

6. Triangle Rasterization
rasterize( vert v[3] ) {
for( int y=0; y<YRES; y++ )
for( int x=0; x<XRES; x++ )
if( inside3(v,x,y) )
fragment(x,y); } v0 l2 l1
Get rid of bounds l0 v2 v1
CS148 Lecture 3

7. Normal to the Line CS148 Lecture 3 Break into three slides.
First, just p0, p1, and t
Second, define Perp
Third,
Explain the function Perp(t)
Perp((x,y)) = (-y,x)
Line orientation is backwards; inside is normally to the left
CS148 Lecture 3

8. Line Equation This equation must be true for all point p on the line
Convention that n points to the left as we move from p0 to p1
Half-plane equation
Show this derivation
This equation must be true for all point p on the line
CS148 Lecture 3

9. Line Divides Plane into 2 Half-Spaces
Normal n points to the right of the line;
Inside (negative values) to the left of the line.
CS148 Lecture 3

10. Make a Line from Two Vertices
makeline( vert& v0, vert& v1, line& l ) {
l.a = v1.y - v0.y;
l.b = v0.x - v1.x;
l.c = -(l.a * v0.x + l.b * v0.y); }
V1.y * v0.x– v1.x*v0.y
If inside all three lines
int
mkedgev( e, x1, y1, x2, y2 )
struct edge *e;
value x1, y1;
value x2, y2; {
e->a = -(y1 - y2);
e->b = (x1 - x2);
e->c = -(e->a * x1 + e->b * y1);
e->v = 0.; } v0 v1 l
CS148 Lecture 3

11. Triangle Orientation v1 v2 v0 v1 v0 v2 CW (Back Facing) CCW
(Front Facing)
Convention: Inside a CCW polygon is equivalent to
inside every lines of the polygon (on their left)
CS148 Lecture 3

12. Point Inside Triangle Test
rasterize( vert v[3] ) {
line l0, l1, l2;
makeline(v[0],v[1],l2);
makeline(v[1],v[2],l0);
makeline(v[2],v[0],l1);
for( y=0; y<YRES; y++ ) {
for( x=0; x<XRES; x++ ) {
e0 = l0.a * x + l0.b * y + l0.c;
e1 = l1.a * x + l1.b * y + l1.c;
e2 = l2.a * x + l2.b * y + l2.c;
if( e0<=0 && e1<=0 && e2<=0 )
fragment(x,y); } v0 l2 l1 l0 v2 v1
CS148 Lecture 3

13. Singularities Singularities: Edges that touch pixels (e == 0)
Causes two fragments to be generated
Wasted effort drawing duplicated fragments
Problems with transparency (later lecture)
Not including singularities (e < 0) causes gaps
CS148 Lecture 3

14. Handling Singularities
Create shadowed edges (thick lines)
Don’t draw pixels on shadowed edges
Solid drawn; hollow not drawn
int shadow( line l ) {
return (l.a>0) || (l.a == 0 && l.b > 0); }
int inside( value e, line l ) {
return (e == 0) ? !shadow(l) : (e < 0);
CS148 Lecture 3

15. Better Point Inside Triangle Test
rasterize( vert v[3] ) {
line l0, l1, l2;
makeline(v[0],v[1],l2);
makeline(v[1],v[2],l0);
makeline(v[2],v[0],l1);
for( y=0; y<YRES; y++ ) {
for( x=0; x<XRES; x++ ) {
e0 = l0.a * x + l0.b * y + l0.c;
e1 = l1.a * x + l1.b * y + l1.c;
e2 = l2.a * x + l2.b * y + l2.c;
if( inside(e0,l0)&&inside(e1,l1)&&inside(e2,l2) )
fragment(x,y); } v0 l2 l1 l0 v2 v1
CS148 Lecture 3

16. Compute Bounding Rectangle (BBox)
bound3( vert v[3], bbox& b ) {
b.xmin = ceil(min(v[0].x, v[1].x, v[2].x));
b.xmax = ceil(max(v[0].x, v[1].x, v[2].x));
b.ymin = ceil(min(v[0].y, v[1].y, v[2].y));
b.ymax = ceil(max(v[0].y, v[1].y, v[2].y)); }
Calculate tight bound around the triangle
Round coordinates upward (ceil) to the nearest integer
CS148 Lecture 3

17. Compute Bounding Rectangle (BBox)
bound3( vert v[3], bbox& b ) {
b.xmin = ceil(min(v[0].x, v[1].x, v[2].x));
b.xmax = ceil(max(v[0].x, v[1].x, v[2].x));
b.ymin = ceil(min(v[0].y, v[1].y, v[2].y));
b.ymax = ceil(max(v[0].y, v[1].y, v[2].y)); }
Tested points indicated by filled circles
Don’t need to test hollow circles
CS148 Lecture 3

18. More Efficient Bounding Box Version
rasterize( vert v[3] ) {
bbox b; bound3(v, b);
line l0, l1, l2;
makeline(v[0],v[1],l2);
makeline(v[1],v[2],l0);
makeline(v[2],v[0],l1);
for( y=b.ymin; y<b.ymax, y++ ) {
for( x=b.xmin; x<b.xmax, x++ ) {
e0 = l0.A * x + l0.B * y + l0.C;
e1 = l1.A * x + l1.B * y + l1.C;
e2 = l2.A * x + l2.B * y + l2.C;
if( inside(e0,l0)&&inside(e1,l1)&&inside(e2,l2) )
fragment(x,y); }
CS148 Lecture 3

19. Interpolation
Convert discrete values to a continuous function by filling in “in between” values
CS148 Lecture 3

20. Linear Interpolation 0 1 CS148 Lecture 3
let’s say we have points y1 and y2 and we want to move smoothly between the two.
one way to do this is via the line connecting the two points
CS148 Lecture 3

21. Barycentric Interpolation
Edge
talk about balance point of the triangle
negative mass?
Triangle
CS148 Lecture 3

22. Barycentric Interpolation on Triangles
Can be used to interpolate colors
r = a0 r0 + a1 r1 + a2 r2
g = a0 g0 + a1 g1 + a2 g2
b = a0 b0 + a1 b1 + a2 b2
Can be used to interpolate texture coordinates
u = a0 u0 + a1 u1 + a2 u2
v = a0 v0 + a1 v1 + a2 v2
Can be used to interpolate z or depth
z = a0 z0 + a1 z1 + a2 z2
Can be used to interpolate normals…
CS148 Lecture 3

23. Finding Barycentric Coordinates
CS148 Lecture 3

24. How to Compute Triangle AreaB A
CS148 Lecture 3

25. Barycentric Coordinates: Linear System
Determinant of matrix is equal to zero if the points are co-linear
CS148 Lecture 3

26. Barycentric Coordinates: Basis Vectors
CS148 Lecture 3

27. Barycentric Coordinates vs. Basis Vector Coordinates
A point p inside triangle p0p1p2 can be expressed using barycentric coordinates as
Point p can also be expressed in the basis vector coordinates (of last slide as)
Therefore, we get the relations
CS148 Lecture 3