3 D Transformation

8/13/2019 3 D Transformation

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3-D Transformation

Shubhangi Shinde


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Projections

Display device (a screen) is 2D How do we map 3D objects to 2D space?

2D to 2D is straight forward 2D window to world.. and a viewport on the 2D surface.

Clip what won't be shown in the 2D window, and map theremainder to the viewport.

3D to 2D is more complicated Solution : Transform 3D objects on to a 2D plane usingprojections


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Projections

Projections: key terms

Projectionfrom 3D to 2D is defined by straightprojection rays

(projectors) emanating from the 'center of projection', passing

through each point of the object, and intersecting the

'projectionplane' to form a projection.


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Types of projections

2 types of projections

perspectiveandparallel.

Key factor is the center of projection.

if distance to center of projection is finite : perspective

if infinite : parallel


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Perspective v Parallel Perspective:

visual effect is similar to human visual system...

has 'perspective foreshortening'

size of object varies inversely with distance from the center ofprojection.

angles only remain intact for faces parallel to projection plane. Lines of projecton are not parallel. Instead, they all converge at

a single point called the center of projection or projectionreference point.

Parallel:

less realistic view because of no foreshortening

however, parallel lines remain parallel.

angles only remain intact for faces parallel to projection plane.

Z coordinate is discarded and parallel lines from each vertex on

the object are extended until they intersect the view plane. The point of intersection is the projection of the vertex.


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Perspective Projections

Any parallel lines notparallel to the projection plane,

converge at a vanishing point.

There are an infinite number of these, 1 for each of the

infinite amount of directions line can be oriented.

If a set of lines are parallel to one of the three

principle axes, the vanishing point is called an axis

vanishing point.

There are at most 3 such points, corresponding to the

number of axes cut by the projection plane.


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Example:

if z projection plane cuts the z axis: normal to it, so only zhas a principle vanishing point, as x and y are parallel andhave none.

Can categorise perspective projections by thenumber of principle vanishing points, and thenumber of axes the projection plane cuts.


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2 different examples of a one-point perspective

projection of a cube.

(note: x and y parallel lines do not converge)


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Two-point perspective projection:

This is often used in architectural, engineering and

industrial design drawings.

Three-point is used less frequently as it adds little

extra realism to that offered by two-point

perspective projection.


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Two-point perspective projection:


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Orthographic Projection

Orthographic projection shows complex objects by

doing a 2D drawing of each side to show the main

features.

Orthographic drawings usually consist of a front view,a side view and a top view, but more views may be

shown for complex objects with lots of detail.

Projection plane is perpendicular to the principle

axis.

These projections are used in engineerring drawing

to depict machine parts,assemblies building and so

on.


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Here are three orthographic views of an object.


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Axonometric Projection

Orthographic projection can display more then oneface of an object called axmonometric projection

Projection planes are not normal to a principle axis.

the ability to show the inclined position of an object

with respect to the plane of projection.

Parallelism of lines are preserved but angles are not.

Kinds: IsometricIso (one or equal) and Metrus (measures);

equal measures.

Dimetrican axonometric drawing into two angle.

Trimetricutilizes three different angles.


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Isometric Projection

Align view plane so that it intersects each coordinateaxis in which the object is defined at the same distance

from the origin.

Obtained by aligning the projection vector with the

cube diagnol.In isometric projection the angles between the

projection of the axes are equal i.e. 120.

It is important to appreciate that it is the angles

between the projection of the axes that are being

discussed and not the true angles between the axes

themselves which is always 90.


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Dimetric Projection

The angles between the projection of the axes in

dimetric projection renders two of the three to be

equal.

To draw the outline of an object in dimetricprojection, two scales are required.

The scales are generated the same as for isometric.
http://www.ul.ie/~rynnet/keanea/isometri.htmhttp://www.ul.ie/~rynnet/keanea/isometri.htm


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Dimetric Projection


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Dimetric Projection


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Trimetric Projection

In trimetric projection the projection of the three

angles between the axes are unequal.

Thus, three separate scales are needed to generate atrimetric projection of an object.

The scales are constructed using the same method

described in isometric and dimetric projection.


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Parallel Projections

2 principle types:

orthographicand oblique.

Orthographic : direction of projection = normal to the projection plane.

Oblique :

direction of projection != normal to the projection plane.


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Parallel Projection

Angel Figure 5.4

Center of projection is at infinity

Direction of projection (DOP) same for all points

DOP

View

Plane


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Orthographic (or orthogonal) projections:

front elevation, top-elevation and side-elevation.

all have projection plane perpendicular to a principle axes.

Useful because angle and distance measurements can bemade...

However, As only one face of an object is shown, it can be

hard to create a mental image of the object, even whenseveral view are available.


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Orthogonal projections:


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Oblique parallel projections

Obtained by projecting points along parallel linesthat are not perpendicular to the projection plane.

View plane and direction of projection are not same. Objects can be visualised better then with orthographic

projections

Can measure distances, but not angles*

* Can only measure angles for faces of objects parallel to the plane

2 common oblique parallel projections:

Cavalierand Cabinet


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Cavalier: The direction of the projection makes a 45 degree angle with the

projection plane.

The projection of a line perpendicular to the view plane has thesame length as the line itself.

Because there is no foreshortening, this causes an exaggeration ofthe z axes.


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Cabinet: The direction of the projection makes a 63.4 degree angle with the

projection plane.

Lined perpendicular to the viewing surface, are projected at one halftheir actual length.

This results in foreshortening of the z axis, and provides a more

realistic view.


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Oblique Parallel Projections

b= 45 Cavalier projection

b= 63.4 Cabinet projection

b= 90 Orthogonal projection


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x

y

la

(xs,ys)(0,0,1)

P

Consider the point P:

P can be represented in 3D

space - (0,0,1)

P can be represented in 2D

(screen coords) - (xs,ys)


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At (0,0,1)

xs= lcos a

ys= lsin a Generally

multiply by z and allow for (non-zero) x and y

xs= x + z.l.cos ays= y + z.l.sin a


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1

.

1000

0000

0sin100cos01

1

0 z

y

x

y

x

s

s

alal


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3 D Transformation

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