geometric projections - unc charlotte · projection type (parallel or perspective) prp - point in...

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

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


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�


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.


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


Oblique 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.

⇒ Parallel edges converge at infinity, eg. AD,BC.ITCS 4120/5120 11 Geometric Projections

Perspective Projections:1-Point Perspective


Perspective Projections: 2, 3 point Perspective

VP�

2�

VP�

1

X�

Y�

Z�

VP�

2�

VP� 1

X�

Y�

Z�

VP�

3�


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


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



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


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.


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


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


View Volume

xv = ±hzv

d

yv = ±hzv

d

NoteClipping is more efficiently carried out in screen coordinates.


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


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


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



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.



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

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.


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


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)


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


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


Scale into symmetric (unit slope) view volume

Ms =

2d

xmax−xmin0 0 0

0 2dymax−ymin

0 00 0 1 00 0 0 1



Mpersp =

1 0 0 00 1 0 00 0 f/(f − n) −fn/(f − n)0 0 1 1


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.


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.


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


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

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??


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.


geometric projections - unc charlotte · projection type (parallel or perspective) prp - point in...

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