Projections
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Projections. Projection. Demo. Projections - Outline. 3D Viewing Coordinate System & Transform Process Generalized Projections Taxonomy of Projections Perspective Projections Parallel Projections. 3D Viewing. Inherently more complex than 2D case. Extra dimension to deal with

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Projections

Projections

Projection

Graphics


Projections

Demo

Graphics


Projections outline

Projections - Outline

  • 3D Viewing

  • Coordinate System & Transform Process

  • Generalized Projections

  • Taxonomy of Projections

  • Perspective Projections

  • Parallel Projections

Graphics


3d viewing

3D Viewing

  • Inherently more complex than 2D case.

    • Extra dimension to deal with

    • Most display devices are only 2D

  • Need to use a projectionto transform 3D object or scene to 2D display device.

  • Need to clip against a 3D view volume.

    • Six planes.

    • View volume probably truncated pyramid

Graphics


Coordinate systems transform process

Coordinate Systems & Transform Process

Object coordinate systems.

Transform

World coordinates.

Clip

View Volume

Project

Screen coordinates.

Rasterize

Raster

Graphics


Viewing coordinate system

Body System

yv

xv

P0

zv

Front-Wheel System

Viewing plane

Viewing Coordinate System

zw

world

yw

xw

Viewer System

View Plane

View Window

Graphics


Specifying the viewing coordinate system

Specifying the Viewing Coordinate System

  • Viewing Coordinates system, [xv, yv, zv], describes 3D objects with respect to a viewer.

  • A viewing plane (projection plane) is set up perpendicular to zv and aligned with (xv,yv).

  • In order to set a viewing plane we have to specify:

Graphics


Projections

yv

xv

P

zv

Viewing plane

zw

v

L

N

  • P=(Px,Py,Pz) is a point where a camera is located.

  • L - is a point to look-at.

  • V - is the view up vector, whose projection onto the view-plane is directed up.

yw

xw

Graphics


Viewing window

Viewing Window

  • After objects were projected onto the viewing plane, an image is taken from a Viewing Window.

  • A viewing window can be placed anywhere on the view plane.

  • In general the view window is aligned with the viewing coordinates and is defined by its extreme points: (l,b) and (r,t)

View plane

yv

(r,t)

xv

View window

zv

Graphics

(l,b)


Viewing volume

Viewing Volume

  • Given the specification of the viewing window, we can set up a Viewing Volume.

  • Only objects inside the viewing volume will appear in the display, the rest are clipped.

Graphics


Projections

  • In order to limit the infinite viewing volume we define two additional planes: Near Plane and Far Plane.

  • Only objects in the bounded viewing volume will appear.

  • The near and far planes are parallel to the viewing plane and specified by znear and zfar.

  • A limited viewing volume is defined:

    • For orthographic: a rectangular parallelpiped.

    • For oblique: an oblique parallelpiped.

    • For perspective: a frustum.

Far

Plane

zv

Near

Plane

zv

window

Far

Plane

window

Near

Plane

Graphics


Projections

window

  • The Viewing Plane can be placed anywhere along the Zv axis, as long it does not contain the center of projection.

Far

Plane

Near

Plane

zv

Graphics


Defining the viewing volume

top

right

yv

xv

zv

bottom

near

left

far

Defining the viewing Volume

top

right

yv

xv

zv

bottom

near

left

far

Graphics


Generalised projections

Generalised Projections.

  • Transforms points in a coordinate system of dimension n into points in one of less than n (ie 3D to 2D)

  • The projection is defined by straight lines called projectors.

  • Projectors emanate from a centre of projection, pass through every point in the object and intersect a projection surface to form the 2D projection.

Graphics


Projections1

Projections.

  • In graphics we are generally only interested in planar projections – where the projection surface is a plane.

    • Most cameras employ a planar film plane.

      We will only deal with geometric projections – the projectors are straight lines.

  • Whether rays coming form object converges to COP or not

Graphics


Projections2

Projections.

  • Henceforth refer to planar geometric projections as just: projections.

  • Two classes of projections :

    • Perspective.

    • Parallel.

Parallel

A

Parallel

A

A

A

Centre of

Projection.

B

B

B

Centre of

Projection

at infinity

B

Perspective

Graphics


Projections3

Projections

  • Viewing 3D objects on a 2D display requires a mapping from 3D to 2D.

  • Projectors are lines from the center of projection through each point in the object.

  • A projection is formed by the intersection of the projectors with a viewing plane.

Center of

Projection

Projectors

Graphics


Projections

  • Center of projection at infinity results with a parallel projection.( projection lines are parallel)

  • A center of projection at a finite distance results with a perspective projection. (projection lines converges to COP)

Graphics


Projections

  • A parallel projection preserves relative proportions of objects, but does not give realistic appearance (commonly used in engineering drawing).

  • A perspective projection produces realistic appearance, but does not preserve relative proportions.

Graphics


A taxonomy of projections

Planar geometric projections.

Parallel

Perspective

Orthographic

Oblique

1 point

Axonometric

Cabinet

Cavalier

2 point

Isometric

3 point

Multi View

A Taxonomy of Projections

Graphics


Projections

  • Parallel Projection

    Projection lines (projectors) are parallel not converges

    Converges at infinity, COP is infinity

    Preserve the shape not used for realistic images

    Parallel line intersect perpendicularly to projection plane-Orthographic Projection

    When parallel line intersect plane at some angle not perpendicular –Oblique

Graphics


Projections

  • When parallel line perpendicularly intersect and View plane is parallel to principal plane (perpendicular to axies ) of object space-Multi view projection (shows one face of object-top, bottom, left, right, front, rear) it includes 2 dimensions(Lxb, bxh, hXL)

  • When parallel line perpendicularly intersect and View plane is not parallel to principal Plane( not perpendicular to principal axies) of object space-Axonometric projection (isometric, diametric, trimetric) it includes 3 dimensions(Lxbxh) projectors makes equal angel with all three principal axies –isometric

  • Multi view and orthographic combination provide 3 faces can be seen TFRi, BLRe

Graphics


Projections

  • In oblique projection only the face of object is parallel to view plane are shown, their shape, size, length, angle are preserved for these faces only, phases not parallel discarded. In oblique projection Line which is perpendicular to plane is shorter in length of actual line(projection rays )change in length of projected line is –foreshortening factor f

  • When f=0 projection is orthographic (cot 90=0) angle between projector and plane is 90

  • When f=1 then oblique projection is Cavalier projection(cot45=1) angle between projector and plane is 45

  • When f=1/2 then oblique projection is cabinet projection or break front or cupboard (cot63.435=1/2) angle is 63.435

Graphics


Parallel projections

Parallel Projections

  • Specified by a direction to the centre of projection, rather than a point.

    • Centre of projection at infinity.

  • Orthographic

    • The normal to the projection plane is the same as the direction to the centre of projection.

  • Oblique

    • Directions are different.

A

Parallel

A

B

Centre of

Projection

at infinity

B

Graphics


Parallel projection

Parallel Projection

  • Projectors are all parallel.

  • Orthographic: Projectors are perpendicular to the projection plane.

  • Oblique: Projectors are not necessarily perpendicular to the projection plane.

Orthographic

Oblique

Graphics


Orthographic projection

Orthographic Projection

  • Since the viewing plane is aligned with (xv,yv), orthographic projection is performed by:

(x,y,z)

(x,y)

yv

xv

P0

zv

Graphics


Projections

  • Lengths and angles of faces parallel to the viewing planes are preserved (Plan View).

  • Problem: 3D nature of projected objects is difficult to deduce.

Top View

Side View

Front view

Graphics


Projections

  • Orthographic: Projector is perpendicular to view plane

  • Oblique: projector is not perpendicular to view plane

  • Multi view: View plan parallel to principal planes

  • Axonometric : View plane not parallel to principal planes

Graphics


Orthographic projections

Orthographic Projections

Most common orthographic

Projection :

Front-elevation,

Side-elevation,

Plan-elevation.

Angle of projection parallel to principal axis; projection plane is perpendicular to axis.

Commonly used in technical drawings

Graphics


Orthographic projection1

Orthographic Projection

Orthographic Projection onto a plane at z = 0.

xp = x , yp = y , z = 0.

Graphics


Axonometric orthographic projections

Axonometric Orthographic Projections

  • Projection plane not parallel to principal Plane (not perpendicular to principal axis) normal of plane makes various angle axies

  • Show several faces of the object at once

  • Foreshortening is uniform rather than being related to distance(shortening factor f )

  • Parallelism of lines is preserved

  • Distances can be measured along each principal axis ( with scale factors )

Graphics


Isometric projection

Isometric Projection

  • Most common axonometric projection

  • Projection plane normal makes equal angles with each axis.

  • i.e normal is (dx,dy,dz), |dx| = |dy|=|dz|

Graphics


Projections

y

y

120º

120º

120º

x

z

x

  • All 3 axes equally foreshortened

  • measurements can be made

  • Hence the name iso-metric

Projection

Plane

z

Isometric Projection

Normal

Graphics


Oblique projections

Oblique projections.

  • Projection plane normal differs from the direction of projection.

  • Usually the projection plane is parallel to a principal axis.

    • Other faces can measure distance, but not angles.

    • Parallel rays intersect view plane at angle β

Graphics


Projections

  • cavalier projection :

    • Preserves lengths of lines perpendicular to the viewing plane.

    • 3D nature can be captured but shape seems distorted.

    • Can display a combination of front, side, and top views.

      Cabinet projection:

    • lines perpendicular to the viewing plane project at 1/2 of their length.

    • A more realistic view than the cavalier projection.

    • Can display a combination of front, side, and top views.

Graphics


Oblique projection

Oblique projection

Normal

Parallel to x axis

y

x

Projection

Plane

z

Graphics


Geometry of oblique projections

y

L.sin

L

z

P=(0,0,1)

x

L.cos

Geometry of Oblique Projections

  • Projection plane is x,y plane

  • L=1/tan()

  • - angle between plane and projection

    direction

    - Determines the type of projection

  •  is choice of horizontal angle.

  • Given a desired L and ,

  • Direction of projection is

  • (L.cos, L.sin,-1)

Graphics


Geometry of oblique projections1

Geometry of Oblique Projections

  • Point P=(0,0,1) maps to:

    P’=(l.cosa, l.sina, 0) on xy plane,

    and P(x,y,z) onto P’(xp,yp,0)

and

Graphics


Oblique projection1

Oblique Projection

  • Projectors are not perpendicular to the viewing plane.

  • Angles and lengths are preserved for faces parallel to the plane of projection.

  • preserves 3D nature of an object.

yv

(xp,yp)

xv

(x,y,z)

(x,y)

Graphics


Projections

Several Oblique Projections

1

1

1

1

1

1

=30o

=45o

cavalier Projections (=45o) of a cube

for two values of angle 

0.5

0.5

1

1

1

1

=45o

=30o

Cabinet Projections (= 63.4o) of a cube

for two values of angle 

Graphics


Perspective projections

Perspective Projections.

  • Defined by projection plane and centre of projection.

  • Visual effect is termed perspective foreshortening.

    • The size of the projection of an object varies inversely with distance from the centre of projection.

    • Similar to a camera - Looks realistic !

  • Not useful for metric information

    • Angles only preserved on faces parallel to the projection plane.

    • Distances not preserved

Graphics


Projections

Perspective Projections

  • A set of lines not parallel to the projection plane converge at a vanishing point.

    • point at infinity.

    • Homogeneous coordinate is 0 (x,y,0)

Graphics


Perspective projections1

Perspective Projections

  • Lines parallel to a principal axis converge at an axis vanishing point.

    • Categorized according to the number of such points

    • Corresponds to the number of axes cut by the projection plane.

y

y

x

x

z

z

Projection plane

Graphics


Painless perspective

Painless Perspective

Graphics


Why do parallel lines seem to converge

Why do parallel lines seem to converge?

Graphics


The eye as a camera

The eye as a camera

Graphics


The eye as a camera1

The eye as a camera

Graphics


Equal distances appear smaller

Equal distances appear smaller

Graphics


Simplified camera

Simplified camera

Graphics


Simplified camera1

Simplified camera

View plane

Z-axis

Graphics


Perspective

Perspective

The first ever painting (Trinity with the Virgin, St. John and Donors) done in perspective by Masaccio, in 1427.

Graphics


Projections

Graphics


1 point projection

1-Point Projection

Projection plane cuts 1 axis only.

Graphics


1 point perspective

1-Point Perspective

A painting (The Piazza of St. Mark, Venice) done by Canaletto in 1735-45 in one-point perspective

Graphics


2 point perspective

y

z

x

Projection plane

2-Point Perspective

Graphics


2 point perspective1

2-Point Perspective

Painting in two point perspective by Edward Hopper

The Mansard Roof 1923 (240 Kb); Watercolor on paper, 13 3/4 x 19 inches;

The Brooklyn Museum, New York

Graphics


3 point perspective

y

z

x

Projection plane

3-Point Perspective

Generally held to add little beyond 2-point perspective.

A painting (City Night, 1926) by Georgia O'Keefe, that is approximately in three-point perspective.

Graphics


Vanishing points

Vanishing Points

  • There are infinitely many general vanishing points.

  • There can be up to three principal vanishing points (axis vanishing points).

  • Perspective projections are categorized by the number of principal vanishing points, equal to the number of principal axes intersected by the viewing plane.

  • Most commonly used: one-point and two-points perspective.

Graphics


Projections

y

x

z

One point (z axis)

perspective projection

x axis

vanishing point.

z axis

vanishing point.

Two points

perspective projection

Graphics


Projections

(xp,yp,0)

y

d

x

center of

projection

(x,y,z)

  • Using similar triangles it follows: consider xz plane

z

x

xp

Preserve the angle

-z

d

(x,y,z)

Graphics


Projections

  • Thus, a perspective projection matrix is defined:

Graphics


Finding vanishing points

Finding Vanishing Points

  • Recall : An axis vanishing point is the point where the axis intercepts the projection plane  point at infinity.

Graphics


Mathematics of viewing

Mathematics of Viewing

  • Need to generate the transformation matrices for perspective and parallel projections

  • Should be 4x4 matrices to allow general concatenation

Graphics


Perspective projection simplest case

Perspective Projection – Simplest Case

Centre of projection at the origin,

Projection plane at z=d.

Projection

Plane.

y

P(x,y,z)

x

Pp(xp,yp,d)

z

d

Graphics


Perspective projection simplest case1

x

P(x,y,z)

xp

z

d

y

d

P(x,y,z)

z

x

Pp(xp,yp,d)

yp

P(x,y,z)

z

d

y

Perspective Projection – Simplest Case

Graphics


Perspective projection

Perspective Projection

Graphics


Perspective projection1

Perspective Projection

Graphics


Perspective projection2

Perspective Projection

Trouble with this formulation :

Centre of projection fixed at the origin.

Graphics


Alternative formulation

d

x

z

P(x,y,z)

xp

yp

P(x,y,z)

z

y

d

Projection plane at z = 0

Centre of projection at

z = -d

Alternative Formulation

Graphics


Alternative formulation1

d

x

z

P(x,y,z)

xp

yp

P(x,y,z)

z

y

d

Projection plane at z = 0,

Centre of projection at

z = -d

Now we can allow d

Alternative Formulation

Graphics


Perspective projection3

Perspective Projection

  • In a perspective projection, the center of projection is at a finite distance from the viewing plane.

  • The size of a projected object is inversely proportional to its distance from the viewing plane.

  • Parallel lines that are not parallel to the viewing plane, converge to a vanishing point.

  • A vanishing point is the projection of a point at infinity.

Z-axis vanishing point

y

x

Graphics

z


Projections

Graphics


Observations

Observations

  • Mper is singular (|Mper|=0), thus Mper is a many to one mapping.

  • Points on the viewing plane (z=0) do not change.

  • The homogeneous coordinates of a point at infinity directed to (Ux,Uy,Uz) are (Ux,Uy,Uz,0). Thus, The vanishing point of parallel lines directed to (Ux,Uy,Uz) is at [dUx/Uz, dUy/Uz].

  • When d, Mper Mort

Graphics


Projections

What is the difference between moving the center of

projection and moving the projection plane?

Original

z

Projection

plane

Center of

Projection

Moving the Center of Projection

z

Center of

Projection

Projection

plane

Moving the Projection Plane

z

Graphics

Projection

plane

Center of

Projection


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