Chapter 11
Projections

• Determine a center of projection (COP)

• Project each point P to intersection of line (COP,P) with projective plane

• The line (P,COP) is called the projector of P.
• Lines project to lines.
• Generalizations to planar geometric projections: curved surfaces and curved projectors.

11.1 Parallel Projections

• Center of projection at infinite
  • Choose direction of this point (angles or vector)
  
• Invariant to distance
• Preserve length and orientation of lines that are parallel to projective plane.

• Dependent on orientation of objects
• Dependent on direction of projection

Orthogonal Projections

Projectors perpendicular to projective plane.

• Point (x,y,z) projected to (x,y,0).

• Elevation projection. Projective plane perpendicular to an axis in the object coordinate system.

• Axonometric projection--Projective plane not normal to any axis of the object coordinate system (normally shows more than one face of the object)

• Isometric projection. Axonometric projection in which the unit direction vector is (±1,±1,±1) within the object coordinates (same direction with respect to each axis).

Oblique

Projectors not perpendicular to plane of projection.

• Projectors at direction (, )

• Cavalier projection: = 45^o (lines perpendicular to projective plane preserve their length)

• Cabinet projection: tan = 2 ( = 63.4^o; lines perpendicular to the projective plane are halved)

11.2 Perspective Projections

• Center of projectionat finite distance
  • Choose location of this point
• perspective shortening: lengths vary inversely with distance

• Projections of parallel lines meet at a vanishing point, if they are not parallel to the view plane.

• Projections of lines parallel to a principal axis meet at an axis/principal vanishing point.

  Example: Cubes can have 1, 2, or 3 vanishing points.

11.3 Transformation Matrices

Parallel

x_p=x - (z/tan) * cos y_p=y - (z/tan) * sin z_p=0

Perspective

x_p __ d = x __ z+d y_p d = y __ z+d

=

• d = : Orthogonal parallel projection.

• Does not reduce to oblique parallel projection

Unified

• (u_x , u_y , u_z ) a unit vector

• Parametric equations for projection point on projector

11.4 Projective Depth

• For hidden features removal

• Project to (x_p , y_p , z_p ) instead of (x_p , y_p , 0)

Depth z

x_p __ d = x_ z+d y_p __ d = y_ z+d z_p =z

• Depth preserved
• Straight lines don’t transform into straight lines

Normalized Depth

x_p__ d = x_ z+dy _p__ d = y_ z+d z_p= z_ z+d

• Lines and planes are invariant under this transformation.
  Ax + By + Cz + D	= 0
  zp + D‘	= 0

• Some other entities change their shape.