## Chapter 11Projections

• 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 (, )
LOST FIGURE ????

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

• Cabinet projection: tan = 2 ( = 63.4o; 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.

### Parallel

xp=x - (z/tan) * cos yp=y - (z/tan) * sin zp=0

### Perspective

xp __ d = x __ z+d yp d = y __ z+d

=

• d = : Orthogonal parallel projection.

• Does not reduce to oblique parallel projection

### Unified

• (ux , uy , uz ) a unit vector

• Parametric equations for projection point on projector
xp =ux L + (x - ux L)t yp =uy L + (y - uy L)t 0=uz L + (z - uz L)t

### 11.4ProjectiveDepth

• For hidden features removal

• Project to (xp , yp , zp ) instead of (xp , yp , 0)

### Depth z

xp __ d = x_ z+d yp __ d = y_ z+d zp =z

• Depth preserved
• Straight lines don’t transform into straight lines
 Ax + Bz + C = 0 y + D = 0 A‘xp zp + B‘zp + C‘ = 0 yp zp + D‘ = 0

### Normalized Depth

xp__ d = x_ z+dy p__ d = y_ z+d zp= 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.