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Computer Graphics: PROJECTIONS

PROJECTIONS

A Projection is defined as a process which transforms points in a coordinate system of dimension n into points in a

coordinate system of dimension less than n.

The mapping of 3D objects onto the 2D screen is done by straight projection rays (called projectors) emanating from

a center of projection, passing through each point of the object, and intersecting a projection plane to form the

projection.

Projections can be divided into two basic classes:

Perspective or Vanishing Point Method (VPM)1.

Parallel Projection2.

The distinction is in the relation of center of projection and projection plane. If the distance between the

center of projection and projection plane is finite then the projection is perspective and if the distance is

infinite, the projection is parallel.

Projective Projection

Projection Plane

Perspective Projections:

Perspective projections are planar geometric projections where all the projectors intersect in one point, center of

projection.

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Kumar Satyam's Blog: Computer Graphics: PROJECTIONS http://www.asksatyam.com/2011/01/projections.html

1 of 8 4/16/2014 10:04 PM

A property of perspective projections is the existence of one or more vanishing points.

A vanishing point is located on an axis of the object's coordinate system. Lines that

are parallel to the axis intersect in that point. There can be only one vanishing point

for each axis, and the axis needs to intersect the projection plane. Depending on the

number of vanishing points a perspective projection is also called one-point, two-point,

or three-point perspective.

One-point perspective

Two-point perspective

Three-point perspective

Only one principal axis (Z1) Z2 and X2 axises pierce the pierces the projection plane, projection plane, but Y2 axis

does not.

Parallel projections are planar geometric projections where all the projectors are parallel to each other. The direction

of projection or the angle between the projectors and the projection plane leads to further subclassing. All parallel

projections have the following properties in common: angles between lines that are located on a plane that is parallel

to the projection plane remain in the projected image; the projection of any pair of parallel lines is also parallel;

parallel lines not parallel to the projection plane are equally foreshortened; as long as no scaling is applied to the

projection, the distance between two points on a plane parallel to the projection plane is the same as the distance

between the projection of those points.

1.) Orthographic Parallel Projections: The orthographic projection is a parallel projection with the projectors

perpendicular to the projection plane.

The most common types of orthographic projections are:

1.a) Multiple View Projection

1.b) Axonometric Projection

Parallel Projections:

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2 of 8 4/16/2014 10:04 PM

1.a) Multiple View Projection : It is further classified into

a) Front Elevation Projection

b) Top Elevation or Plan-elevation Projection

c) Side Elevation Projection

1.b) Axonometric orthogonal projections -Axonometric orthogonal projections use projection planes that are

not normal to a principal axis and therefore show several faces of an object at once. They resemble the

perspective projection in this way, but differ in that the foreshortening is uniform rather than being related to the

distance from the center of projection. It is further classified into

i) Isometric Projection

ii ) Dimetric Projection

iii) Trimetric Projection

Isometric Projection -

Direction of viewing is such that the three axes of space appear equally foreshortened, of which

the displayed angles among them and also the scale of foreshortening are universally known.

Dimetric projections- Directions of viewing are such that two of the three axes of

space appear equally foreshortened, of which the attendant scale and angles of presentation are

determined according to the angle of viewing; the scale of the third direction (vertical) is

determined separately.

Trimetric projection-Direction of viewing is such that all of the three axes of space appear

unequally foreshortened. The scale along each of the three axes and the angles among them

are determined separately as dictated by the angle of viewing.

2.) Oblique Parallel Projection :

In this projection plane normal and the direction of projection differ. It combines properties of the front, top and side

orthogonal projections with those of the axonometric projections, the projection plane is normal to a principal axis, so

the projections of the face of the object parallel to this plane allows measurement of angles and distances.

Oblique Projection

Two frequently used oblique projections are:

2.1) Cavalier Projection –Direction of projection makes a 45 o angle with the projection plane.

2.2) Cabinet Projection - Direction of projection that makes an angle of 63.4 o with the projection plane.


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PERSPECTIVE PROJECTION

Perspective projection is a projection in which Center of projection is at a finite distance from a projection plane.

The technique of perspective projection is used in preparing perspective drawings of three dimensional objects and

scenes. Perspective Projection is characterized by perspective foreshorting and vanishing points.

Perspective foreshortening is the illusion that objects and length appear smaller as their distance from the center of

projection increases. The illusion that certain sets of parallel lines appear to meet at a point is another feature of

perspective drawings. These points are Vanishing Points.

Let P1P2 be an object and we take a projection of this object. Straight Projectors called projectors emanating from a

center of projection, passing through each point of object and intersecting a projection plane form a perspective

image P1'P2'of the object P1P2.

P2' is the perspective image of P2 and P1' is the perspective image of P1'.

Let the distance between the center of projection and plane of projection is d (focal length).

In Perspective projection, image size is not true size; it depends upon center of projection, plane of projection and

size of the object. The size of the image will be bigger when distance between center of projection and plane of

projection decreases (i.e. d decreases).

There are two standards in implementation.

Mathematical Description of Perspective Projection


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Standard 1: Center of projection is located at the origin.

From center of projection, a ray will be projected. Plane of projection is kept parallel to x-y plane at a distance of d.

P '(x',y',z') is the perspective positions of the point

P (x, y, z) and Coordinates of P and d is known and we want to calculate x', y 'and z'.

For calculation of y', we consider elevation.

y' / d = y/z

x ' / d = x/z

x' = xd/z

x' and y' are used to convert a given point to a perspective point.

Mathematically,

This matrix includes floating point calculation and we want to use integer point calculation as far as possible. Here x'

is not final x', it is xh . As we are using Homogenous Equation, we have to use xh/h .

So we convert the above matrix into –

Consider similar triangles AP'B & APC,

For calculation of x, we consider top view


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Standard 2 :

Centre of projection is on the negative side of z at a distance of d.

Calculation of y -

Consider similar Triangles,

y'/d = y/z+d

y'= yd/z+d

For calculating x', we consider top view


6 of 8 4/16/2014 10:04 PM

Posted by rockysatyam

Labels: Computer Graphics

Consider similar triangles ,

x' /d = x /d+z

x' = xd/ z+d

Perspective Matrix

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1 comment:

Patra Ishita Monday, March 31, 2014 at 6:35:00 PM GMT+5:30

The description is good for us. Thanx.

Reply


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Kumar Satyam's Blog_ Computer Graphics_ PROJECTIONS