Geometric Projections PLANAR
PARALLEL
PERSPECTIVE
1−pt OBLIQUE
2−pt
3−pt
ORTHOGRAPHIC
MULTI−VIEW
AXONOMETRIC
ISOMETRIC
DIMETRIC
TRIMETRIC
Parallel Projection: projectors are parallel to each other. Perspective Projection: projectors converge to a point.
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Geometric Projections
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◦ Projectors converge to a point, the center of projection 3
Geometric Projections
Parallel Projection
Perspective Projection
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◦ Projectors are parallel meeting at infinity Geometric Projections
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Geometric Projections
Orthographic Projection
Multi-View Orthographic Projection ◦ projectors are also aligned with principal axes, common in architectural drawings (Top, Side and Front Views) TOP VIEW
FRONT VIEW
SIDE VIEW
◦ Projectors are orthogonal to the projection plane ITCS 4120/5120
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Geometric Projections
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Geometric Projections
Isometric Projections Axonometric Projections View plane Normal not aligned with principal axes. Useful in revealing 3D nature of objects, but does not preserve shape.
Classified into Isometric, Dimetric, Trimetric projections Isometric projections are the most common.
◦ View Plane Normal makes equal angles with all principal axes. ITCS 4120/5120
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Geometric Projections
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Geometric Projections
Oblique Projections Oblique Projections Combines the advantages of orthographic and axonometric projections.
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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Geometric Projections
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Perspective Projections
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Geometric Projections
Perspective Projections: 1-Point Perspective
Classifed as 1-Point, 2-Point and 3-Point Perspective
VP
H
G C
D
F
E
A
B
Vanishing Point ⇒ Lines and faces recede from a view and converge to a point, known as a vanishing point.
⇒ Parallel edges converge at infinity, eg. AD, BC . ITCS 4120/5120
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Geometric Projections
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Geometric Projections
Perspective Projections: 1-Point Perspective
Perspective Projections: 1-Point Perspective
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Geometric Projections
Perspective Projections: 1-Point Perspective
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Geometric Projections
Perspective Projections: 2, 3 point Perspective
Y
VP1
VP 2
Y
VP 1
X
VP 2
Z
X
VP3
Z
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Geometric Projections
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Geometric Projections
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Geometric Projections
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Geometric Projections
Geometric Projections
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Geometric Projections
Perspective Projections: 2-Point Perspective
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Geometric Projections
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Geometric Projections
Geometry of Perspective Projection
P’ Eye( eu, ev, en) P(pu, pv, pn) Projection Plane
Problem: ~ 0, the projection of the P~ (pu, pv , pn), To determine P a point on the object, given the view point E(eu , ev , en ), ~ , the view plane normal (VPN). and N Simplification: ⇒ The View Plane contains the origin. ITCS 4120/5120
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Geometric Projections
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Geometric Projections
Geometry of Perspective Projection
Geometry of Perspective Projection
P(pu, pv, pn) t=1 P’
~ is on the N ~ axis. Simplification: E
t=0 Eye(eu, ev, en)
N
n=0 Projection Plane
~ = (eu, ev , en) = (0, 0, en) E So en pv pu , P 0 = eenn−p n en −pn
pn(t) = en + t(pn − en) pn(t0) = en + t0(pn − en) = 0 Hence t0 =
=
en en −pn
Projection Point: Pu0 (t0) = eu + t0(pu − eu) n (pu − eu) = = eu + ene−p n Similarly, −ev pn Pv0 (t0) = enepnv −p n ITCS 4120/5120
pu , pv 1−pn /en 1−pn /en
Thus P 0 projects to the 2D point en pu −eu pn en −pn
(p0u, p0v ) =
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Geometric Projections
pu , pv 1−pn /en 1−pn /en
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Perspective Projection Destroys Depth Information!
Geometric Projections
Pseudo Depth
P2(x2, y2, z2) P2(x1, y1, z1)
“Need to retain depth information after projection for use in shading and visible surface calculations.”
P’
Define pn 0 =
Eye (eu, ev, en) Projection Plane
~0 All points on the projector project to the same point P Assume ~a to be any point on the projector.
pu (eu = ev 1−pn /en eu +t(au −eu ) 1−(en +t(an −en ))/en au 1−an /en
= = Similarly,
Pv0 (t) =
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av 1−an /en
as a measure of P ’s depth.
As pn increases, pn 0 increases. Hence
P (t) = ~e + t(~a − ~e) Pu0 (t) =
pn , 1−pn /en
(pu0, pv 0, pn0) =
= 0)
pu , pv , pn 1−pn /en 1−pn /en 1−pn /en
In Homogeneous coordinates,
P 0 = (pu, pv , pn, 1 − pn/en) ~ p pu pv pn 1 T =M
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Geometric Projections
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Geometric Projections
Perspective Projection: General Case Perspective Projection 1 0 Mp = 0 0
0 0 1 0 0 1 0 −1/en
~ is off the N ~ axis If E P 0 = MpMsP~
0 0 0 1
and,
1 0 Ms = 0 0
which is the perspective projection transform.
0 −eu/en 1 −ev /en 0 1 0 0
0 0 0 1
~ off the N ~ axis introduces a shear, given by Ms ⇒ Moving E ITCS 4120/5120
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Geometric Projections
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Geometric Projections
View Volume View Space Operations Back Face Elimination Compare the orientation of the polygon with the view point (or center of projection). Define
~ p = polygon normal N ~ = vector to view point N
hzv d hzv yv = ± d
xv = ±
~p • N ~ >0 visibility = N Note
Clipping is more efficiently carried out in screen coordinates. ITCS 4120/5120
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Geometric Projections
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Geometric Projections
Perspective Transformation
3D Screen Space Perspective Projection
yv
P(xv, yv, zv) zv
(xs,ys) xv
yv
d
xs = dxv /zv , ys = dyv /zv
zv xv
In homogeneous coordinates,
dxv hzv dyv ys = hzv f (1 − d/zv ) zs = f −d
X = xv , Y = yv , Z = zv , w = zv /d
Mpersp
1 0 = 0 0
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0 0 1 0 0 1 0 1/d
xs =
0 0 0 0
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Geometric Projections
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In homogeneous coordinates,
and
Mpersp
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f zv df − , w = zv f −d f −d
d/h 0 0 0 0 0 0 d/h = 0 0 f /(f − d) −df /(f − d) 0 0 1 0
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Geometric Projections
Perspective Transformation
Perspective Transformation
X = (d/h)xv , Y = (d/h)yv , Z =
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Mpersp
1 0 0 0 d/h 0 0 0 0 0 0 d/h 0 0 0 1 = 0 1 0 0 0 f /(f − d) −df /(f − d) 0 0 0 0 1 0 0 1 0 = Mpersp2Mpersp1
Note: ⇒ Mpersp1 converts a truncated pyramid to a regular pyramid (unit slopes for the side planes)
⇒ Mpersp2 maps the regular pyramid into a box.
Geometric Projections
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Geometric Projections
Perspective Transformation 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 is not parallel to View Plane Normal (VPN)
Consider the following points:
Near, Far clipping planes and projection plane defined.
⇒ (0, h, d, 1) −→ (0, d, d, 1) −→ (0, d, 0, d)
Projection window of any aspect ratio and located anywhere in the view plane.
⇒ (0, −h, d, 1) −→ ?? ⇒ (0, f h/d, f, 1)?? ⇒ (0, −f h/d, f, 1) −→ ?? ITCS 4120/5120
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Geometric Projections
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View Orientation
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View Mapping
VRP - point in World coords.
Projection type (parallel or perspective)
VPN - vector in World coords.
PRP - point in VRC space
View Up (VUV) - vector in world coords.
View plane distance - in VRC space Front and Back plane distances
~ = VU ~V ×VP ~N U ~N × U ~ V~ = V P
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Geometric Projections
View plane window size (4 limits) Projection viewport limits in normalized projection space (4 limits)
Geometric Projections
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Geometric Projections
Normalizing Transformation Shear Projection Window Center
Translate PRP to the origin. After translation,
Shear the center of the projection window such that the center line becomes the zv axis.
View plane distance = d
1 0 Msh = 0 0
Near plane distance = n Far plane distance = f Window parameters, ⇒ umax, umin become xmax, xmin ⇒ vmax, vmin become ymax, ymin
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Geometric Projections
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Scale into symmetric (unit slope) view volume Ms =
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2d xmax −xmin
0 0 0
0
2d ymax −ymin
0 0
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0 0 1 0
xmax +xmin 2d ymax +ymin 2d
1 0
0 0 0 1
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Geometric Projections
Perspective Transformation
0 0 0 1
1 0 Mpersp = 0 0
Geometric Projections
0 1 0 0
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0 0 0 1 0 0 0 f /(f − n) −f n/(f − n) 0 1 1
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Geometric Projections
(1, 1)
=
N
Clipping in 3D or N U=
Normalizing Transformation - Properties
V
U or V
U or V
N
N
Planes are preserved.
n min U=
F
Planes parallel to the view plane are shifted. The 4 sidewalls of the view volume become 4 faces of a paral-
−N
B
View Plane
or
V=
−N
Clipping Space
View Space
Clip boundaries are as follows:
lelepiped, view point is at infinity.
Also known as pre-warping.
−gn ≤ gu ≤ gn −gn ≤ gv ≤ gn nmin ≤ gn ≤ 1
Useful for visible surface calculations.
Use the Cohen-Sutherland Algorithm with these boundaries defining the clip volume. ITCS 4120/5120
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Geometric Projections
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Geometric Projections
3-D Clipping in Homogeneous Coordinates Cohen-Sutherland Clipping Algorithm
Y
P(x, y, z)
U
Extension of the 2D algorithm to 3D.
Q(a,b,c)
Note that the scaling to unit slope planes facilitates efficient clipping
Z
in 3D.
(rx, ry, rz) V
⇒ gu > −gn - Point to right of volume
Camera Coords
World Coords
⇒ gu < gn - Point to left of volume
X
⇒ gv > −gn - Point above volume
−1 ≤
⇒ gv < gn - Point below volume
gu gv gn ≤ 1, −1 ≤ ≤ 1, 0 ≤ ≤1 gw gw gw OR
⇒ gn < nmin - Point in front of volume ⇒ gn > 1 - Point behind volume
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N (0, 0, 0)
−gw ≤ gu ≤ gw , −gw ≤ gv ≤ gw , 0 ≤ gn ≤ gw 47
Geometric Projections
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Equalities reverse if gw ≤ 0.0 48
Geometric Projections
Clipping Points and Lines
Clipping Points and Lines
W P1
W= U or W = V
W= −U or W = −V W W= −U or W = −V
P1’
W= U or W = V P1
W=1 Projection of P1,P1’
P2’
W=1
P2’
U or V
U or V
P2
P 1 and P 2 are on opposite side of W = 0 plane.
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
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Geometric Projections
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 with top region - Need to have only 1 clip region.
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Geometric Projections
