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Methods of Projections Parallel and central projection

Learning Objectives • Introduce the classical views

• Compare and contrast image formation by computer with how

images have been formed by architects, artists, and engineers

• Learn the benefits and drawbacks of each type of view

• Introduce the mathematics of projection

Linear mapping

1. Image of a point is point

2. Image of a line is line (or point)

3. Image of a plane is plane (or line)

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View

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We call the pictures for observing objects with different

ways are views.

View results are related with the shape and size of

scene objects, the position and direction of viewpoint.

Planar Geometric Projections

• Standard projections project onto a plane

• Projectors are lines that either – converge at a center of projection

– are parallel

• Such projections preserve lines – but not necessarily angles

• Nonplanar projections are needed for applications such as map construction

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Classical Projections

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Perspective vs Parallel

• Computer graphics treats all projections the same and implements them with a matrix

• Classical viewing developed different techniques for drawing each type of projection

• Fundamental distinction is between parallel and perspective viewing even though mathematically parallel viewing is the limit of perspective viewing

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Multiview Orthographic Projection

• Projection plane parallel to principal face

• Usually form front, top, side views

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isometric (not multiview

orthographic view)

front

side top

in CAD and architecture,

we often display three

multiviews plus isometric

Advantages and Disadvantages

+ Preserves both distances and angles

Shapes preserved

Can be used for measurements

• Building plans

• Manuals

- Cannot see what object really looks like because many surfaces hidden from view

Often we add the isometric

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Types of Axonometric Projections

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Isometric projection

Isometric projection- a drafting

style in which a two-dimensional

object is drawn to look three-

dimensional. The horizontal edges

of the object are drawn at a 30° or

150° angle, and all vertical lines at

90° (remain vertical).

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Advantages and Disadvantages

• Lines are scaled (foreshortened) scaling factors are known only for a axis (principal directions)

Projection of a circle in a plane not parallel to the projection plane is an ellipse

• Does not look real for large objects because far objects are scaled the same as near objects

• Used in CAD applications (with perspective) 13

Axonometry

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M. C. Escher, Waterfall, 1961

Taska: Tangent plane of the cone Regular hexagon Helix Hyperboloid of revolution

Oblique Projection

Arbitrary relationship between projectors and projection plane

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Advantages and Disadvantages

• Can pick the angles to emphasize a particular face

– Architecture: plan oblique, elevation oblique

• Angles in faces parallel to projection plane are preserved while we can still see “around” side

• In physical world, cannot create with simple camera;

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Orthographic

and Oblique drawings

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Pohlke’s Theorem (1860) The principal theorem of oblique axonometry

Three straight line segments of arbitrary length in a plane, drawn

from a point under arbitrary angles form a parallel projection of

three equal segments drawn from the origin on three

perpendicular coordinate axes.

However, only one of the segments or one of the angles

may vanish.

Any complete plane quadrangle can serve

as the parallel projection of a tetrahedron

similar to a given one.

https://www.geogebra.org/m/CsamWsfy

Sketching

Method of pictorial sketching of an object

1. Sketch an enclosing box of the given object. If the object

decomposes from different parts we sketch the boxes

circumscribed to these parts.

2. Sketch views in the principal faces (thin lines).

3. Sketch auxiliary parallel lines

4. Thick lines to highlight visible edges.

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Sketching-isometric grid

The horizontal edges of the object are drawn at a 30° or

150° angle, and all vertical lines at 90° (remain vertical).

Perspective Projection

Projectors converge at center of projection

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Naturally we see things in perspective o Objects appear smaller the farther away they are; o Rays from view point are not parallel.

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Vanishing Points • Parallel lines (not parallel to the projection plan) on the object

converge at a single point in the projection (the vanishing point)

• Drawing simple perspectives by hand uses these vanishing point(s)

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Types of Perspective

Three-Point Two-Point One-Point

• Number of principal face parallel to projection plane

• Number of vanishing points for cube

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Advantages and Disadvantages

• Objects further from viewer are projected smaller than the same sized objects closer to the viewer (diminution)

– Looks realistic

• Equal distances along a line are not projected into equal distances (nonuniform foreshortening)

• Angles preserved only in planes parallel to the projection plane

• More difficult to construct by hand than parallel projections (but not more difficult by computer)

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3D space

2D image

Projection

Z

Xp

Yp

Window coordinate

Screen coordinate system World coordinate system

Viewport

Window

Far plane

Near plane

Viewpoint

Image

plane

Viewing Volume

Matrix Reprezentation of Projection

Orthogonal projection to coordinate plane p = (xy)

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3

0

( , , ) ( , , )

1 0 0

; 0 1 0

0 0 0

p

p

p

p p p

x x

y y

z

X x y z E X x y z

X PX P

p

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Simple Perspective

Consider a simple perspective with the COP at the origin, projection (picture) plane p parallel to coord. plane (x,y).

p

p

p

x d

x z

y d

y z

z d

( , , )X x y z

( , , )p p pX x y z

Matrix reprezentation only in the projective space!

Perspective Matrices

Simple projection matrix in homogeneous coordinates

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0 0 0

0 0 0' ;

0 0 0

0 0 1 0

d

dX PX P

d

p

p

p

dx x

z

dy y

z

z d

, , , , ,1 , , ,d d d d

X x y d x y d xd yd zd zz z z z