1. Angel: Interactive Computer Graphics5E © Addison-Wesley 2009
Frames & Geometry
Angel
Angel: Interactive Computer Graphics5E © Addison-Wesley 2009

2. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Objectives
Discuss frames
Introduce simple data structures for building polygonal models
Vertex lists
Edge lists
OpenGL vertex arrays
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

3. Homogeneous Coordinates
The homogeneous coordinates form for a three dimensional point [x y z] is given as
p =[x’ y’ z’ w] T =[wx wy wz w] T
We return to a three dimensional point (for w0) by
xx’/w
yy’/w
zz’/w
If w=0, the representation is that of a vector
Note that homogeneous coordinates replaces points in three dimensions by lines through the origin in four dimensions
For w=1, the representation of a point is [x y z 1]
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

4. Homogeneous Coordinates and Computer Graphics
Homogeneous coordinates are key to all computer graphics systems
All standard transformations (rotation, translation, scaling) can be implemented with matrix multiplications using 4 x 4 matrices
Hardware pipeline works with 4 dimensional representations
For orthographic viewing, we can maintain w=0 for vectors and w=1 for points
For perspective we need a perspective division
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

5. Change of Coordinate Systems
Consider two representations of a the same vector with respect to two different bases. The representations are
a=[a1 a2 a3 ]
b=[b1 b2 b3]
where
v=a1v1+ a2v2 +a3v3 = [a1 a2 a3] [v1 v2 v3] T
=b1u1+ b2u2 +b3u3 = [b1 b2 b3] [u1 u2 u3] T
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

6. Representing second basis in terms of first
Each of the basis vectors, u1,u2, u3, are vectors that can be represented in terms of the first basis v
u1 = g11v1+g12v2+g13v3
u2 = g21v1+g22v2+g23v3
u3 = g31v1+g32v2+g33v3
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

7. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Matrix Form
The coefficients define a 3 x 3 matrix
and the bases can be related by
see text for numerical examples
M =
a=MTb
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

8. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Change of Frames
We can apply a similar process in homogeneous coordinates to the representations of both points and vectors
Any point or vector can be represented in either frame
We can represent Q0, u1, u2, u3 in terms of P0, v1, v2, v3 u2 u1 v2
Consider two frames:
(P0, v1, v2, v3)
(Q0, u1, u2, u3) Q0 P0 v1 u3 v3
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

9. Representing One Frame in Terms of the Other
Extending what we did with change of bases
u1 = g11v1 + g12v2 + g13v3
u2 = g21v1 + g22v2 + g23v3
u3 = g31v1 + g32v2 + g33v3
Q0 = g41v1 + g42v2 + g43v3 + g44P0
defining a 4 x 4 matrix
M =
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

10. Working with Representations
Within the two frames any point or vector has a representation of the same form
a=[a1 a2 a3 a4 ] in the first frame
b=[b1 b2 b3 b4 ] in the second frame
where a4 = b4 = 1 for points and a4 = b4 = 0 for vectors and
The matrix M is 4 x 4 and specifies an affine transformation in homogeneous coordinates
a=MTb
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

11. Affine Transformations
Every linear transformation is equivalent to a change in frames
Every affine transformation preserves lines
However, an affine transformation has only 12 degrees of freedom because 4 of the elements in the matrix are fixed and are a subset of all possible 4 x 4 linear transformations
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

12. The World and Camera Frames
When we work with representations, we work with n-tuples or arrays of scalars
Changes in frame are then defined by 4 x 4 matrices
In OpenGL, the base frame that we start with is the world frame
Eventually we represent entities in the camera frame by changing the world representation using the model-view matrix
Initially these frames are the same (M=I)
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

13. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Moving the Camera
If objects are on both sides of z=0, we must move camera frame
M =
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

14. Angel: Interactive Computer Graphics5E © Addison-Wesley 2009
Geometric Modeling
Angel: Interactive Computer Graphics5E © Addison-Wesley 2009

15. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Representing a Mesh
Consider a mesh
There are 8 nodes and 12 edges
5 interior polygons
6 interior (shared) edges
Each vertex has a location vi = (xi yi zi) e2 v5 v6 e3 e9 e8 v8 v4 e1
e11
e10 v7 e4 e7 v1
e12 v2 v3 e6 e5
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

16. Simple Representation
Define each polygon by the geometric locations of its vertices
Leads to OpenGL code such as
Inefficient and unstructured
Consider moving a vertex to a new location
Must search for all occurrences
glBegin(GL_POLYGON);
glVertex3f(x1, y1, z1);
glVertex3f(x6, y6, z6);
glVertex3f(x7, y7, z7);
glEnd();
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

17. Inward and Outward Facing Polygons
The order {v1, v6, v7} and {v6, v7, v1} are equivalent in that the same polygon will be rendered by OpenGL but the order {v1, v7, v6} is different
The first two describe outwardly
facing polygons
Use the right-hand rule =
counter-clockwise encirclement
of outward-pointing normal
OpenGL can treat inward and
outward facing polygons differently
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

18. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Geometry vs Topology
Generally it is a good idea to look for data structures that separate the geometry from the topology
Geometry: locations of the vertices
Topology: organization of the vertices and edges
Example: a polygon is an ordered list of vertices with an edge connecting successive pairs of vertices and the last to the first
Topology holds even if geometry changes
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

19. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Vertex Lists
Put the geometry in an array
Use pointers from the vertices into this array
Introduce a polygon list
x1 y1 z1
x2 y2 z2
x3 y3 z3
x4 y4 z4
x5 y5 z5.
x6 y6 z6
x7 y7 z7
x8 y8 z8 v1 v7 v6 P1 P2 P3 P4 P5 v8 v5 v6
topology
geometry
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

20. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Shared Edges
Vertex lists will draw filled polygons correctly but if we draw the polygon by its edges, shared edges are drawn twice
Can store mesh by edge list
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

21. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Edge List v1 v2 v7 v6 v8 v5 v3 e1 e8 e3 e2
e11 e6 e7
e10 e5 e4 e9
e12
x1 y1 z1
x2 y2 z2
x3 y3 z3
x4 y4 z4
x5 y5 z5.
x6 y6 z6
x7 y7 z7
x8 y8 z8 e1 e2 e3 e4 e5 e6 e7 e8 e9 v1 v6
Note polygons are
not represented
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

22. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Modeling a Cube
Model a color cube for rotating cube program
Define global arrays for vertices and colors
GLfloat vertices[][3] = {{-1.0,-1.0,-1.0},{1.0,-1.0,-1.0},
{1.0,1.0,-1.0}, {-1.0,1.0,-1.0}, {-1.0,-1.0,1.0},
{1.0,-1.0,1.0}, {1.0,1.0,1.0}, {-1.0,1.0,1.0}};
GLfloat colors[][3] = {{0.0,0.0,0.0},{1.0,0.0,0.0},
{1.0,1.0,0.0}, {0.0,1.0,0.0}, {0.0,0.0,1.0},
{1.0,0.0,1.0}, {1.0,1.0,1.0}, {0.0,1.0,1.0}};
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

23. Drawing a polygon from a list of indices
Draw a quadrilateral from a list of indices into the array vertices and use color corresponding to first index
void polygon(int a, int b, int c , int d) {
glBegin(GL_POLYGON);
glColor3fv(colors[a]);
glVertex3fv(vertices[a]);
glVertex3fv(vertices[b]);
glVertex3fv(vertices[c]);
glVertex3fv(vertices[d]);
glEnd(); }
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

24. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Draw cube from faces
void colorcube( ) {
polygon(0,3,2,1);
polygon(2,3,7,6);
polygon(0,4,7,3);
polygon(1,2,6,5);
polygon(4,5,6,7);
polygon(0,1,5,4); } 5 6 2 1 7 4 3
Note that vertices are ordered so that
we obtain correct outward facing normals
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

25. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Efficiency
The weakness of our approach is that we are building the model in the application and must do many function calls to draw the cube
Drawing a cube by its faces in the most straight forward way requires
6 glBegin, 6 glEnd
6 glColor
24 glVertex
More if we use texture and lighting
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

26. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Vertex Arrays
OpenGL provides a facility called vertex arrays that allows us to store array data in the implementation
Six types of arrays supported
Vertices
Colors
Color indices
Normals
Texture coordinates
Edge flags
We will need only colors and vertices
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

27. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Initialization
Using the same color and vertex data, first we enable
glEnableClientState(GL_COLOR_ARRAY);
glEnableClientState(GL_VERTEX_ARRAY);
Identify location of arrays
glVertexPointer(3, GL_FLOAT, 0, vertices);
glColorPointer(3, GL_FLOAT, 0, colors);
data array
data contiguous
3d arrays
stored as floats
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

28. Mapping indices to faces
Form an array of face indices
Each successive four indices describe a face of the cube
Draw through glDrawElements which replaces all glVertex and glColor calls in the display callback
GLubyte cubeIndices[24] = {0,3,2,1,2,3,7,6
0,4,7,3,1,2,6,5,4,5,6,7,0,1,5,4};
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

29. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Drawing the cube
Method 1:
Method 2:
number of indices
what to draw
for(i=0; i<6; i++) glDrawElements(GL_POLYGON, 4,
GL_UNSIGNED_BYTE, &cubeIndices[4*i]);
format of index data
start of index data
glDrawElements(GL_QUADS, 24,
GL_UNSIGNED_BYTE, cubeIndices);
Draws cube with 1 function call!!
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009