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# GR2 Advanced Computer Graphics AGR - PowerPoint PPT Presentation

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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Presentation Transcript

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)
• selecting the view plane (or camera film plane)
• selecting the type of projection
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
• 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
• 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

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)

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

yv

Q

xv

yV

camera

zv

zV

view plane

Perspective Projection Calculation

zQ

zVP

zC

looking along x-axis

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
• 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

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
• 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
• 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
• 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

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

Edward

Hopper

Lighthouse

at Two Lights

-see

www.postershop.com

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

yv

Q

xv

yV

zv

zV

view plane

Parallel Projection Calculation

zQ

zVP

looking along x-axis

Q

yV

zV

view plane

Parallel Projection Calculation

P

yP = yQ

and similarly xP = xQ

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

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
• 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

back

plane

view frustum

view window

camera

front

plane

zV

View Volume - Parallel Projection

back

plane

view volume

view window

front

plane

zV

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