• The displayed objects have 3 dimensions: width (x), height (y), and depth (z).
• Classification
• Real-time versus pre-rendered animation
• Immediate versus retained mode
• 3D Effects:
• perspective, hidden surfaces
• color, shading, transparency, texture mapping
• The collection of geometrical objects defined in the program in usually called the scene.

• 2D - Cartesian coordinates Oxy.
• Viewport: region of the application window used for drawing.
• 3D Cartesian coordinates Oxyz.
• Vertex: a position in the 3D space - a vertex.
• Most 3D APIs use real 3D coordinates
• Going from the physical 3D coordinates to the 2D discrete screen coordinates - projection.

Def. P(x, y, z) -> P'(x', y', z') in the projection plane p.
Usually the projection plane is defined by z'=0.

2 types of projection: parallel, perspective.
Parallel projection: orthographic or oblique.

The parallel projection

• Defined by a projection line (direction) d.
• If the line is perpendicular to the projection plane - orthographic projection. Otherwise oblique.

Features of the parallel projection:

• preserves the ratio between parallel objects;
• preserves the parallel lines;
• the orthogonal projection preserves the width of segments that are parallel to p;
• gives poor impression of image depth;
• often used for technical drawing, architecture plans, etc.

The perspective projection: from a projection center C representing the viewer's eye.

• Given a projection center C representing the viewer's eye.
• Consider the line L connecting C and P. Then P'  = intersection of L and p.

Features of the perspective projection:

• does not preserve proportions;
• does not preserve parallel lines;
• object size depends on the closeness to the center: close objects seem big, distant objects seem small;
• gives a good impression of image depth;
• contains a vanishing point, where all the lines that are perpendicular to the projection plane converge; this point is the orthogonal projection of C on the plane p. It also represents the infinity point (the horizon).
• often used for painting, drawing (especially comics), and 3D software.

• 2D parallel orthographic:

gluOrtho2D(left, right, bottom, top);
• 3D parallel orthographic:

glOrtho(left, right, bottom, top, near, far);
• 3D perspective

glFrustum(xmin, xmax, ymin, ymax, near, far); // camera view
gluPerspective(angle, aspect, near, far);

Camera

• For both the parallel and perspective projections, defining the view area is not enough. We also need to position the camera in the scene.
• Camera location and orientation (page 252 in the textbook):
• gluLookAt(eyex, eyey, eyez, atx, aty, atz, upx, upy, upz);

Def. A function F : R³ -> R³ such that P_o(x_o, y_o, z_o) -> P(x, y, z).

Homogeneous coordinates: represent each point as a vector with 4 coordinates.
Every transformation can be represented as a 4x4 matrix.
 

P =

x y z 1

Translation of vector T_v=(t_x, t_y, t_z)
 

x y z 1

=

1 0 0 0

0 1 0 0

0 0 1 0

tx ty tz 1

xo yo zo 1

Scaling of factors S_v=(s_x, s_y, s_z)
 

x y z 1

=

sx 0 0 0

0 sy 0 0

0 0 sz 0

0 0 0 1

xo yo zo 1

Rotation: more complex than in 2D.

Around the 0x axis of angle q :

x y z 1

=

1 0 0 0

0 cos q sin q 0

0 - sin q cos q 0

0 0 0 1

xo yo zo 1

Around the 0y axis of angle q :

x y z 1

=

cos q 0 sin q 0

0 1 0 0

- sin q 0 cos q 0

0 0 0 1

xo yo zo 1

Around the 0z axis of angle q :

x y z 1

=

cos q sin q 0 0

- sin q cos q 0 0

0 0 1 0

0 0 0 1

xo yo zo 1

Any arbitrary rotation about the origin R can be expressed as a unique product (composition) of a rotation about two of the 3 axes. For example: R = R_x R_z.We can determine the rotation matrix for R by multiplying the rotation matrix for R_x with the rotation matrix for R_z.

Example: rotate around Oy, by an angle of 30^o:
glRotatef(30.0, 0.0, 1.0, 0.0);

Composing the transformations

• Rotations are not commutative:  R_x · R_z¹ R_z · R_x.
• Translations and rotations are not commutative: R · T ¹ T · R.
• Translations are commutative: T₁ · T₂ = T₂ · T₁.
• Rotation about an arbitrary point P: T_p · R · T_-p:

• Both the parallel and the perspective projections can be written as 4x4 matrixes.
• To apply the perspective projection, one can first transform the object, then apply the parallel projection.

Matrix Mode in OpenGL

• There are 3 matrices in OpenGL,
• for the objects in the scene (model view),
• for the camera (point of view or projection),
• for the texture.
• Only one of them is active at any time.
• All of the transformations are defined as matrices that are multiplied with the active matrix.

Matrix Operations

• Switching the matrix mode:
  glMatrixMode(GL_PROJECTION);
  glMatrixMode(GL_MODELVIEW); // default
  glMatrixMode(GL_TEXTURE);
  
  
• Setting a matrix to the identity:
  glLoadIdentity();
  
  
• Loading a particular matrix:
  glFloat m[] = {1.0, 0.0, ... }; // 16 elem.
  glLoadMatrixf(m); // f or d
  
  
• Multipling the active matrix by a given matrix
  glMultMatrixf(m); // f or d
  
  
• The 3 matrices are organized in a stack. One can push the active matrix in the stack, then pop it out of the stack again.
  glPushMatrix(); // save
  glPopMatrix(); // restore