Lect24 Projections 1

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159.235 Graphics & Graphical

Programming

Lecture 24 - Projections - Part 1


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Projections - Outline

3D Viewing

Coordinate System & Transform Process

Generalised Projections

Taxonomy of Projections

Perspective Projections


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3D Viewing

Inherently more complex than 2D case.Extra dimension to deal with

Most display devices are only 2D

Need to use aprojection to transform 3Dobject or scene to 2D display device.

Need to clip against a 3D view volume.

Six planes.View volume probably truncated pyramid


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Coordinate Systems & Transform

ProcessObject coordinate systems.

World coordinates.

View Volume

Screen coordinates.

Raster

Transform

Project

Clip

Rasterize


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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 calledprojectors.

Projectors emanate from a centre of projection,pass

through every point in the object and intersect a

projection surface to form the 2D projection.


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Projections. In graphics we are generally only interested in planar

projectionswhere the projection surface is a plane.

Most cameras employ a planarfilm plane.

But the retina is not a plane - future devices such asdirect retina devices may need more complex projections

We will only deal with geometric projectionsthe

projectors are straight lines.

Many projections used in cartography are either non-

geometric or non-planar.

ExceptionImage-based rendering - advanced topic


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Projections.

Henceforth refer to planar geometric projections as just:

projections.

Two classes of projections :

Perspective. Parallel.

A

B

A

B

A

B

A

B

Centre ofProjection.

Centre of

Projection

at infinity

Parallel

Perspective

Parallel


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A Taxonomy of Projections

Planar geometric projections.

Parallel Perspective

Orthographic Oblique 1 point

2 point

3 point

Axonometric

Isometric

CavalierCabinet

Elevations


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Perspective Projections. Defined by projection plane and centre of projection.

Visual effect is termedperspective foreshortening.

The size of the projection of an object varies inversely withdistance from the centre of projection.

Similar to a camera - Looks realistic !

Not useful for metric information

Parallel lines do not in general project as parallel.

Angles only preserved on faces parallel to the projectionplane.

Distances not preserved


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Perspective

The first ever painting(Trini ty w ith the Virgin ,

St. John and Donors)

done in perspective by

Masaccio, in 1427.


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A set of lines not parallel to

the projection plane

converge at a vanishing

point.

Can be thought of in 3D as theprojection of a point at

infinity.

Homogeneous coordinate is 0

(x,y,0)


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z

x

y

Projection plane

xz

y

Lines parallel to a principal axis converge at an axisvanishing point. Categorized according to the number of such points

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


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1-Point Projection

Projection plane cuts 1

axis only.


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1-Point Perspective

A painting (The

Piazza of St. Mark,

Venice) done byCanaletto in 1735-

45 in one-point

perspective


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2-Point Perspective

y

z x

Projection plane


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2-Point Perspective

Painting in two point

perspective by

Edward Hopper

The Mansard Roo f

1923 (240 Kb);

Watercolor on paper,13 3/4 x 19 inches;

The Brooklyn

Museum, New York


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3-Point PerspectiveGenerally held to add little beyond 2-point perspective.

y

z x

Projection plane

A painting (City

Night, 1926) by

Georgia O'Keefe, that

is approximately in

three-point

perspective.


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Intro to Projections -Summary 3D Viewing

Coordinate System & Transform Process

Generalised Projections

Taxonomy of Projections

Perspective Projections Clipping can be done in image

space if more efficientapplication dependent.

Parallel Projections next Acknowledgement - Thanks to Eric McKenzie, Edinburgh, from whose Graphics

Course some of these slides were adapted.


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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 thedirection to the centre of projection.

Oblique Directions are different.


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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


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Axonometric Orthographic Projections

Projection plane not normal to principal axis

Show several faces of the object at once

Foreshortening is uniform rather than being

related to distance Parallelism of lines is preserved

Angles are not

Distances can be measured along each principalaxis ( with scale factors )


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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|

Only 8 directions that satisfy this

condition.


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Isometric Projection

Normal

x

z

y

Projection

Plane

y

z x

120

120

120

All 3 axes equally foreshortened

- measurements can be made

- Hence the name iso-metric


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Oblique projections.

Projection plane normal differs from the direction

of projection.

Usually the projection plane is normal to aprincipal axis.

Projection of a face parallel to this plane allows

measurement of angles and distance.

Other faces can measure distance, but not angles.

Frequently used in textbooks : easy to draw !


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Oblique projection

x

z

y

Projection

Plane

Normal

Parallel to x axis


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Geometry of Oblique Projections

Projection plane is x,y plane

L=1/tan()

- angle between normal and projectiondirection

- Determines the type of projection

is choice of horizontal angle.

Given a desired L and ,

Direction of projection is

(L.cos, L.sin,-1)

z

y

x

P

L

P=(0,0,1)

L.sin

L.cos


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Geometry of Oblique Projections

Point P=(0,0,1) maps to:

P=(l.cos, l.sin, 0) on xy plane,

and P(x,y,z) onto P(xp,yp,0)

)sin(

)cos(

lzyy

lzxx

p

p

1000

0000

0sin10

0cos01

l

l

Moband


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Mathematics of Viewing

Need to generate the transformation

matrices for perspective and parallel

projections. Should be 4x4 matrices to allow general

concatenation.

And theres still 3D clipping and moreviewing stuff to look at.


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Projections - Summary

Orthographic matrix - replace (z) axis withpoint.

Perspective matrixmultiply w by z.Clip in homogeneous coordinates.

Preserve z for hidden surface calculations.

Can find number of vanishing points.

Acknowledgments - thanks to Eric McKenzie, Edinburgh, from whoseGraphics Course some of these slides were adapted.

Lect24 Projections 1

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