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.

- 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

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).

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)

- 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.

x_{p}=x - (z/tan) * cos
y_{p}=y - (z/tan) * sin
z_{p}=0

x_{p }__
d = x __
z+d y_{p}
d = y __
z+d

=

- d = : Orthogonal parallel projection.
- Does not reduce to oblique parallel projection

- (u
_{x }, u_{y }, u_{z }) a unit vector - Parametric equations for projection point on projector

- For hidden features removal
- Project to
(x
_{p }, y_{p }, z_{p }) instead of (x_{p }, y_{p }, 0)

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

Ax + Bz + C | = 0 |

y + D | = 0 |

A‘x_{p } z_{p } + B‘z_{p } + C‘ | = 0 |

y_{p } z_{p } + D‘ | = 0 |

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 z _{p }+ D‘= 0 - Some other entities change their shape.