Geometric Projections PLANAR PARALLEL PERSPECTIVE OBLIQUE ORTHOGRAPHIC AXONOMETRIC MULTI-VIEW 1-pt 2-pt 3-p ISOMETRIC DIMETRIC TRIMETRIC Parallel Projection: projectors are parallel to each other. Perspective Projection: projectors converge to a point. ITCS 4120/5120 1 Geometric Projections
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Geometric Projections - UNC Charlotte · Projection type (parallel or perspective) PRP - point in VRC space View plane distance - in VRC space Front and Back plane distances View
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Geometric Projections
PLANAR
PARALLEL PERSPECTIVE
OBLIQUE ORTHOGRAPHIC
AXONOMETRIC� MULTI−VIEW
1−pt 2−pt�
3−pt�
ISOMETRIC DIMETRIC TRIMETRIC�
� Parallel Projection: projectors are parallel to each other.
� Perspective Projection: projectors converge to a point.
ITCS 4120/5120 1 Geometric Projections
ITCS 4120/5120 2 Geometric Projections
Perspective Projection
◦ Projectors converge to a point, the center of projectionITCS 4120/5120 3 Geometric Projections
Parallel Projection
◦ Projectors are parallel meeting at infinityITCS 4120/5120 4 Geometric Projections
Orthographic Projection
◦ Projectors are orthogonal to the projection plane
ITCS 4120/5120 5 Geometric Projections
Multi-View Orthographic Projection
◦ projectors are also aligned with principal axes, common in architec-tural drawings (Top, Side and Front Views)
FRONT VIEW
SIDE VIEW
TOP VIEW�
ITCS 4120/5120 6 Geometric Projections
Axonometric Projections
� View plane Normal not aligned with principal axes.
� Useful in revealing 3D nature of objects, but does not preserveshape.
� Classified into Isometric, Dimetric, Trimetric projections
� Isometric projections are the most common.
ITCS 4120/5120 7 Geometric Projections
Isometric Projections
◦ View Plane Normal makes equal angles with all principal axes.ITCS 4120/5120 8 Geometric Projections
Oblique Projections
Combines the advantages of orthographic and axonometric projec-tions.
� Viewpoint is off the VPN axis.
� Line joining the viewpoint to the projection point is at an angle α.
� projectors are still orthogonal to the viewplane.
� Special Cases:
⇒ Cabinet: tanα = 1.0
⇒ Cavalier tanα = 0.5
ITCS 4120/5120 9 Geometric Projections
Oblique Projections
ITCS 4120/5120 10 Geometric Projections
Perspective Projections
� Classifed as 1-Point, 2-Point and 3-Point Perspective
A� B
CD
E F
GH
VP�
Vanishing Point
⇒ Lines and faces recede from a view and converge to a point, knownas a vanishing point.
To determine ~P ′, the projection of the ~P (pu, pv, pn),a point on the object, given the view point E(eu, ev, en),and ~N , the view plane normal (VPN).
Simplification:
⇒ The View Plane contains the origin.ITCS 4120/5120 24 Geometric Projections
Geometry of Perspective Projection
Eye(eu, ev, en) N
Projection Plane
t = 0
t = 1
P(pu, pv, pn)
P’
n = 0
pn(t) = en + t(pn − en)
pn(t′) = en + t′(pn − en) = 0
Hence t′ = en
en−pn
Projection Point:P ′u(t
′) = eu + t′(pu − eu)= eu + en
en−pn(pu − eu) = enpu−eupn
en−pn
Similarly,P ′v(t
′) = enpv−evpn
en−pn
ITCS 4120/5120 25 Geometric Projections
Geometry of Perspective Projection
Simplification: ~E is on the ~N axis.
~E = (eu, ev, en) = (0, 0, en)So
P ′ =(
enpu
en−pn, enpv
en−pn
)=(
pu
1−pn/en, pv
1−pn/en
)Thus P ′ projects to the 2D point
(p′u, p′v) =
(pu
1−pn/en, pv
1−pn/en
)ITCS 4120/5120 26 Geometric Projections
Perspective Projection Destroys DepthInformation!
Eye (eu, ev, en)
Projection Plane
P2(x1, y1, z1)
P2(x2, y2, z2)
P’
All points on the projector project to the same point ~P ′Assume ~a to be any point on the projector.
P (t) = ~e + t(~a− ~e)
P ′u(t) = pu
1−pn/en(eu = ev = 0)
= eu+t(au−eu)1−(en+t(an−en))/en
= au
1−an/en
Similarly,P ′v(t) = av
1−an/en
ITCS 4120/5120 27 Geometric Projections
Pseudo Depth
“Need to retain depth information after projection for use in shading and visiblesurface calculations.”
Define pn′ = pn
1−pn/en, as a measure of P ’s depth.
As pn increases, pn′ increases.
Hence
(pu′, pv
′, pn′) =
(pu
1−pn/en, pv
1−pn/en, pn
1−pn/en
)In Homogeneous coordinates,
P ′ = (pu, pv, pn, 1− pn/en)
= ~Mp
[pu pv pn 1
]TITCS 4120/5120 28 Geometric Projections
Perspective Projection
Mp =
1 0 0 00 1 0 00 0 1 00 0 −1/en 1
which is the perspective projection transform.
ITCS 4120/5120 29 Geometric Projections
Perspective Projection: General Case
If ~E is off the ~N axis
P ′ = MpMs~P
and,
Ms =
1 0 −eu/en 00 1 −ev/en 00 0 1 00 0 0 1
⇒ Moving ~E off the ~N axis introduces a shear, given by Ms
ITCS 4120/5120 30 Geometric Projections
View Space Operations
Back Face Elimination
Compare the orientation of the polygon with the view point (or center ofprojection). Define
~Np = polygon normal~N = vector to view point
visibility = ~Np • ~N > 0
ITCS 4120/5120 31 Geometric Projections
View Volume
xv = ±hzv
d
yv = ±hzv
d
NoteClipping is more efficiently carried out in screen coordinates.