Parallel and Perspective Projection

In three-dimensional computer graphics, objects are created and represented in a 3D coordinate system using three coordinates: P(x, y, z)

where:

• x represents the position of the point along the X-axis.
• y represents the position of the point along the Y-axis.
• z represents the position of the point along the Z-axis.

A computer display screen is a two-dimensional surface. It can only display objects using two coordinates: P'(x', y')

Therefore, before displaying a 3D object on a 2D screen, the 3D coordinates must be converted into 2D coordinates.

This conversion process is called Projection.

Projection is an important step in the 3D graphics pipeline that transforms a three-dimensional scene into a two-dimensional image.

Need for Projection in 3D Graphics

A real-world object exists in three dimensions:

• Width
• Height
• Depth

However, output devices such as monitors, projectors, and mobile screens are two-dimensional.

For example, a cube in the real world has:

• Length
• Width
• Height

But when displayed on a screen, it appears as a 2D image containing only:

• X-coordinate
• Y-coordinate

Therefore, projection is required to represent a 3D object on a 2D display.

Projection Concept in 3D Transformation

Projection can be defined as the process of mapping points from a three-dimensional coordinate system onto a two-dimensional projection plane.

The basic idea of projection is:

1. A 3D object is placed in a coordinate system.
2. A viewer or camera observes the object.
3. Projection lines are drawn from the object to the projection plane.
4. The intersection points form the final 2D image.

The transformation can be represented as: $(x, y, z) \rightarrow (x', y')$

where:

• x, y, z are original 3D coordinates.
• x′, y′ are projected 2D coordinates.

Projection Terminology

1. Projection Plane

The projection plane is the surface on which the image of a 3D object is generated.

It is also called:

• View plane
• Image plane

Examples:

• Computer monitor
• Camera sensor
• Display screen

The final projected image appears on this plane.

2. Projectors

Projectors are imaginary lines that connect points on a 3D object to their corresponding points on the projection plane.

The direction of these projector lines determines the type of projection.

There are two possibilities:

1. Parallel projector lines
2. Converging projector lines

3. Center of Projection

The center of projection is the point from where projection lines originate.

It represents the position of the observer or camera.

In perspective projection, all projection lines meet at the center of projection.

Types of Projection in 3D Transformation

Projection methods are mainly classified into two categories:

$$ Projection = \begin{cases} Parallel\ Projection\\ Perspective\ Projection \end{cases} $$

1. Parallel Projection

Parallel projection is a projection technique in which all projection lines remain parallel to each other.

In this method, the center of projection is considered to be located at an infinite distance from the object.

Because all projection lines are parallel, the size of the object does not change with distance.

Principle of Parallel Projection

In parallel projection:

• The object is placed in a 3D coordinate system.
• Projection lines are drawn parallel to each other.
• These lines intersect the projection plane.
• The resulting points form the 2D image.

The mathematical representation is: P' = M_p P

where:

• P is the original 3D point.
• M_p​ is the projection matrix.
• P′ is the projected point.

Characteristics of Parallel Projection

1. Parallel Projectors

All projection lines are parallel.

2. No Perspective Effect

The size of an object remains constant regardless of its distance.

3. Accurate Measurements

Objects maintain their actual dimensions.

4. Less Realistic

It does not represent human vision because depth is not visually reduced.

Types of Parallel Projection

Parallel projection is further divided into:

1. Orthographic Projection
2. Oblique Projection

2. Perspective Projection

Perspective projection is a projection technique that produces images similar to human vision.

In this method:

• Objects closer to the viewer appear larger.
• Objects farther away appear smaller.

This effect is called the perspective effect.

Principle of Perspective Projection

In perspective projection:

• All projection lines originate from one point.
• The point is called the center of projection.
• Projection lines are not parallel.

The camera model is based on perspective projection.

Perspective Projection Geometry

Consider a point: P(x, y, z)

projected onto a plane located at distance $d$d.

The projected coordinates are: 

$x'=\frac{dx}{z}$

$y'=\frac{dy}{z}$

From these equations, we can observe:

• When z increases, x′ and y′ decrease.
• Objects farther away appear smaller.

Difference Between Parallel and Perspective Projection

Applications of Projection in 3D Transformation

Computer Games

Perspective projection creates realistic 3D environments.

Animation

3D movies use perspective projection to generate realistic scenes.

CAD Systems

Orthographic projection is used for accurate engineering drawings.

Virtual Reality

Projection techniques create immersive environments.

Simulation Systems

Flight simulators and training systems use projection methods.