Geometric Projections - UNC Charlotte · Projection type (parallel or perspective) PRP - point in...
Transcript of Geometric Projections - UNC Charlotte · Projection type (parallel or perspective) PRP - point in...
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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.
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Perspective Projection
◦ Projectors converge to a point, the center of projectionITCS 4120/5120 3 Geometric Projections
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Parallel Projection
◦ Projectors are parallel meeting at infinityITCS 4120/5120 4 Geometric Projections
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Orthographic Projection
◦ Projectors are orthogonal to the projection plane
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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�
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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.
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Isometric Projections
◦ View Plane Normal makes equal angles with all principal axes.ITCS 4120/5120 8 Geometric Projections
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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
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Oblique Projections
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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.
⇒ Parallel edges converge at infinity, eg. AD,BC.ITCS 4120/5120 11 Geometric Projections
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Perspective Projections:1-Point Perspective
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Perspective Projections:1-Point Perspective
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Perspective Projections:1-Point Perspective
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Perspective Projections:1-Point Perspective
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Perspective Projections: 2, 3 point Perspective
VP�
2�
VP�
1
X�
Y�
Z�
VP�
2�
VP� 1
X�
Y�
Z�
VP�
3�
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Perspective Projections:2-Point Perspective
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Geometry of Perspective Projection
Eye( eu, ev, en)
P(pu, pv, pn)
P’
Projection Plane
Problem:
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
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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
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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
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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
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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
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Perspective Projection
Mp =
1 0 0 00 1 0 00 0 1 00 0 −1/en 1
which is the perspective projection transform.
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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
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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
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View Volume
xv = ±hzv
d
yv = ±hzv
d
NoteClipping is more efficiently carried out in screen coordinates.
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3D Screen Space
Perspective ProjectionP(xv, yv, zv)
(xs,ys)
d
xs = dxv/zv, ys = dyv/zv
In homogeneous coordinates,
X = xv, Y = yv, Z = zv, w = zv/d
Mpersp =
1 0 0 00 1 0 00 0 1 00 0 1/d 0
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Perspective Transformation
zv
yv
xv
zv
yv
xv
xs =dxv
hzv
ys =dyv
hzv
zs =f (1− d/zv)
f − dITCS 4120/5120 34 Geometric Projections
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Perspective Transformation
In homogeneous coordinates,
X = (d/h)xv, Y = (d/h)yv, Z =fzv
f − d− df
f − d, w = zv
and
Mpersp =
d/h 0 0 00 d/h 0 00 0 f/(f − d) −df/(f − d)0 0 1 0
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Perspective Transformation
Mpersp =
1 0 0 00 1 0 00 0 f/(f − d) −df/(f − d)0 0 1 0
d/h 0 0 0
0 d/h 0 00 0 1 00 0 0 1
= Mpersp2Mpersp1
Note:
⇒ Mpersp1 converts a truncated pyramid to a regular pyramid (unitslopes for the side planes)
⇒ Mpersp2 maps the regular pyramid into a box.
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Perspective Transformation
Consider the following points:
⇒ (0, h, d, 1) −→ (0, d, d, 1) −→ (0, d, 0, d)
⇒ (0,−h, d, 1) −→ ??
⇒ (0, fh/d, f, 1)??
⇒ (0,−fh/d, f, 1) −→ ??ITCS 4120/5120 37 Geometric Projections
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PHIGS Viewing System
� A View Reference Coordinate (VRC) system.
� View Reference Point (VRP) - origin of VRC
� Projection Reference Point (PRP) - center of projection
� Allows oblique projections (center of proj. to center of view plane isnot parallel to View Plane Normal (VPN)
� Near, Far clipping planes and projection plane defined.
� Projection window of any aspect ratio and located anywhere in theview plane.
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View Orientation
� VRP - point in World coords.
� VPN - vector in World coords.
� View Up (VUV) - vector in world coords.
~U = ~V UV × ~V PN~V = ~V PN × ~U
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View Mapping
� Projection type (parallel or perspective)
� PRP - point in VRC space
� View plane distance - in VRC space
� Front and Back plane distances
� View plane window size (4 limits)
� Projection viewport limits in normalized projection space (4 limits)
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Normalizing Transformation
� Translate PRP to the origin.
After translation,
� View plane distance = d
� Near plane distance = n
� Far plane distance = f
� Window parameters,
⇒ umax, umin become xmax, xmin
⇒ vmax, vmin become ymax, ymin
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Shear Projection Window Center
Shear the center of the projection window such that the center line be-comes the zv axis.
Msh =
1 0 xmax+xmin
2d0
0 1 ymax+ymin
2d0
0 0 1 00 0 0 1
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Scale into symmetric (unit slope) view volume
Ms =
2d
xmax−xmin0 0 0
0 2dymax−ymin
0 00 0 1 00 0 0 1
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Perspective Transformation
Mpersp =
1 0 0 00 1 0 00 0 f/(f − n) −fn/(f − n)0 0 1 1
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Normalizing Transformation - Properties
� Planes are preserved.
� Planes parallel to the view plane are shifted.
� The 4 sidewalls of the view volume become 4 faces of a paral-lelepiped, view point is at infinity.
� Also known as pre-warping.
� Useful for visible surface calculations.
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Clipping in 3D
N
U or V
F
B
U=N o
r V =
N
U=−N or V=−N
(1, 1)
N
View Space� Clipping Space
nmin
U or V
View�
Plane
Clip boundaries are as follows:
−gn ≤ gu ≤ gn
−gn ≤ gv ≤ gn
nmin ≤ gn ≤ 1
Use the Cohen-Sutherland Algorithm with these boundaries definingthe clip volume.
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Cohen-Sutherland Clipping Algorithm
� Extension of the 2D algorithm to 3D.
� Note that the scaling to unit slope planes facilitates efficient clippingin 3D.
⇒ gu > −gn - Point to right of volume
⇒ gu < gn - Point to left of volume
⇒ gv > −gn - Point above volume
⇒ gv < gn - Point below volume
⇒ gn < nmin - Point in front of volume
⇒ gn > 1 - Point behind volume
ITCS 4120/5120 47 Geometric Projections
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3-D Clipping in Homogeneous Coordinates
X
Y
Z
U
V
N
(rx, ry, rz)(0, 0, 0)
P(x, y, z)Q(a,b,c)
World Coords Camera Coords
−1 ≤ gu
gw
≤ 1,−1 ≤ gv
gw
≤ 1, 0 ≤ gn
gw
≤ 1
OR
−gw ≤ gu ≤ gw,−gw ≤ gv ≤ gw, 0 ≤ gn ≤ gw
Equalities reverse if gw ≤ 0.0ITCS 4120/5120 48 Geometric Projections
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Clipping Points and Lines
U or V
W = 1�
W�
P1’
P2’
W= −U or W = −V� W= U or W = V
�
P1
Projection of P1,P1’
� If gw component is negative, multiply through by -1, then clip.
� Same procedure when both endpoints of a line have negative gw
� When endpoints of a line have opposite gw, what do we do??
ITCS 4120/5120 49 Geometric Projections
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Clipping Points and Lines
U or V
W = 1�
W�
P1
P1’
P2
P2’
W= −U or W = −V� W= U or W = V
�
� P1 and P2 are on opposite side of W = 0 plane.
� The projection of P1P2 has 2 parts; one segment extends to −∞,the other to +∞.
� Solution: Clip twice, once with each region.
� Better, Clip with top region, negate end points, and clip again withtop region - Need to have only 1 clip region.
ITCS 4120/5120 50 Geometric Projections