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GR2 Advanced Computer Graphics AGR. Lecture 3 Viewing - Projections. Viewing. Graphics display devices are 2D rectangular screens Hence we need to understand how to transform our 3D world to a 2D surface This involves: selecting the observer position (or camera position)

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gr2 advanced computer graphics agr

GR2Advanced Computer GraphicsAGR

Lecture 3

Viewing - Projections

viewing
Viewing
  • Graphics display devices are 2D rectangular screens
  • Hence we need to understand how to transform our 3D world to a 2D surface
  • This involves:
    • selecting the observer position (or camera position)
    • selecting the view plane (or camera film plane)
    • selecting the type of projection
perspective projections
Perspective Projections
  • There are two types of projection: perspective and parallel
  • In a perspective projection, object positions are projected onto the view plane along lines which converge at the observer

P1

P1’

camera

P2

P2’

view plane

parallel projection
Parallel Projection
  • In a parallel projection, the observer position is at an infinite distance, so the projection lines are parallel

P1

P2

view plane

perspective and parallel projection
Perspective and Parallel Projection
  • Parallel projection preserves the relative proportions of objects, but does not give a realistic view
  • Perspective projection gives realistic views, but does not preserve proportions
    • Projections of distant objects are smaller than projections of objects of the same size which are closer to the view plane
viewing coordinate system

yV

xV

zV

Viewing Coordinate System
  • Viewing is easier if we work in a viewing co-ordinate system, where the observer or camera position is on the z-axis, looking along the negative z-direction

Camera is positioned at:

(0 , 0, zC)

view plane

yv

xv

zv

View Plane
  • We assume the view plane is perpendicular to the viewing direction

The view plane

is positioned at:

(0, 0, zVP)

Let d = zC - zVP be the

distance between the

camera and the plane

perspective projection calculation

yv

Q

xv

yV

camera

zv

zV

view plane

Perspective Projection Calculation

zQ

zVP

zC

looking along x-axis

perspective projection calculation1

Q

yV

camera

zV

view plane

Perspective Projection Calculation

P

zQ

zVP

zC

By similar triangles,

yP / yQ = (zC - zVP) / (zC - zQ)

and so

yP = yQ * (zC - zVP) / (zC - zQ)

or

yP = yQ * d / (zC - zQ)

xP likewise

using matrices and homogeneous coordinates
Using Matrices and Homogeneous Coordinates
  • We can express the perspective transformation in matrix form
  • Point Q in homogeneous coordinates is (xQ, yQ, zQ, 1)
  • We shall generate a point H in homogeneous coordinates (xH, yH, zH, wH), where wH is not 1
  • But the point (xH/wH, yH/wH, zH/wH, 1) is the same as H in homogeneous space
  • This gives us the point P in 3D space, ie xP = xH/wH, sim’ly for yP
transformation matrix for perspective
Transformation Matrix for Perspective

xQ

yQ

zQ

1

xH

yH

zH

wH

=

1 0 0 0

0 1 0 0

0 0 -zVP/d zVPzC/d

0 0 -1/d zC/d

Then xP = xH / wH

ie

xP = xH / ( (zC - zQ) / d )

ie

xP = xQ / ( (zC - zQ) / d )

yP likewise

exercises
Exercises
  • There are two special cases which you can now derive:
    • camera at the origin (zC = 0)
    • view plane at the origin (zVP = 0)
  • Follow through the operations just described for these two cases, and write down the transformation matrices
note for later
Note for Later
  • The original z co-ordinate of points is retained
    • we need relative depth in the scene in order to sort out which faces are visible to the camera
vanishing points
Vanishing Points
  • When a 3D object is projected onto a view plane using perspective, parallel lines in object NOT parallel to the view plane converge to a vanishing point

vanishing point

one-point

perspective

projection

of cube

view plane

one point perspective
One-point Perspective

This is:

Trinity with the Virgin,

St John and Donors,

by Mastaccio in 1427

Said to be the first

painting in perspective

two point perspective
Two-point Perspective

Edward

Hopper

Lighthouse

at Two Lights

-see

www.postershop.com

parallel projection two types
Orthographic parallel projection has view plane perpendicular to direction of projection

Oblique parallel projection has view plane at an oblique angle to direction of projection

Parallel Projection - Two types

P1

P1

P2

P2

view plane

view plane

We shall only consider orthographic projection

slide22

yv

Q

xv

yV

zv

zV

view plane

Parallel Projection Calculation

zQ

zVP

looking along x-axis

parallel projection calculation

Q

yV

zV

view plane

Parallel Projection Calculation

P

yP = yQ

and similarly xP = xQ

parallel projection calculation1
Parallel Projection Calculation
  • So this is much easier than perspective!
    • xP = xQ
    • yP = yQ
    • zP = zVP
  • The transformation matrix is simply

1 0 0 0

0 1 0 0

0 0 zVP/zQ 0

0 0 0 1

view volumes view window

yv

xv

zv

View Volumes - View Window
  • Type of lens in a camera is one factor which determines how much of the view is captured
    • wide angle lens captures more than regular lens
  • Analogy in computer graphics is the view window, a rectangle in the view plane

view window

view volume front and back planes
View Volume - Front and Back Planes
  • We will also typically want to limit the view in the zV direction
  • We define two planes, each parallel to the view plane, to achieve this
    • front plane (or near plane)
    • back plane (or far plane)

zV

back plane

front plane

view frustum perspective projection
View Frustum - Perspective Projection

back

plane

view frustum

view window

camera

front

plane

zV

view volume parallel projection
View Volume - Parallel Projection

back

plane

view volume

view window

front

plane

zV

view volume
View Volume
  • The front and back planes act as important clipping planes
  • Can be used to select part of a scene we want to view
  • Front plane important in perspective to remove near objects which will swamp picture