Understanding 3D Projection Properties

3D projections are fundamental concepts in computer graphics used to represent a three-dimensional object on a two-dimensional plane. The two main types are orthographic projection and perspective projection, each having distinct properties. We need to analyze the given properties and classify them based on whether they apply to perspective projection only, orthographic projection only, both, or neither.

Orthographic Projection Explained

In orthographic projection, projection rays are parallel to each other and perpendicular to the projection plane. This type of projection is often used in engineering and architectural drawings where true dimensions are important. A key characteristic is that objects retain their relative sizes regardless of their distance from the viewer or the projection plane.

Perspective Projection Explained

In perspective projection, projection rays converge at a single point called the center of projection (or eye point). This simulates how human vision works. Objects appear smaller as they move farther away from the viewer, creating a sense of depth and realism. This projection involves foreshortening.

Analyzing Each Property

Property (A): Straight lines are mapped to straight lines.

This property is true for both orthographic and perspective projections. Both are types of geometric transformations, and specifically, projective transformations preserve the straightness of lines. If you project a straight line in 3D space onto a 2D plane using either orthographic or perspective projection, the result on the 2D plane will still be a straight line.

Classification: (iii) both orthographic and projective transformations.

Property (B): Distances and angles are (in general) preserved.

This property is generally not true for either orthographic or perspective projections when considering 3D space distances and angles mapped to 2D. Orthographic projection does not preserve distances or angles for features not parallel to the projection plane (e.g., the diagonal of a cube face not parallel to the view plane will appear shorter). Perspective projection dramatically distorts both distances and angles due to foreshortening. Only in very specific cases (like projecting a line segment parallel to the projection plane in orthographic projection) are lengths preserved. Angles are also generally not preserved. Therefore, this property is true for neither projection type in the general sense.

Classification: (iv) neither orthographic nor projective transformation.

Property (C): Far away objects appear the same size as closer ones

This is a defining characteristic of orthographic projection. Because the projection rays are parallel, the size of the projected image of an object depends only on the object's actual size and orientation relative to the projection plane, not its distance from the viewer or projection plane. In contrast, perspective projection causes distant objects to appear smaller.

Classification: (ii) an orthographic projection only.

Property (D): Requires homogeneous coordinates in order for it to be encoded into a linear transformation.

Orthographic projection can be represented as a linear transformation using standard 3D coordinates (matrix multiplication). However, perspective projection involves a division operation (typically by the z-coordinate or a similar term related to distance), which is not a linear operation in standard Cartesian coordinates. To represent perspective projection using matrix multiplication within standard graphics pipelines, homogeneous coordinates are used. This allows the non-linear division to be incorporated as part of a 4x4 matrix multiplication followed by a perspective divide step. Therefore, while homogeneous coordinates are often used for both in a pipeline for consistency, they are required specifically for perspective projection to be represented as a linear transformation in this manner.

Classification: (i) a perspective projection only.

Summary of Properties and Classifications

Ordering Based on Classification

The question asks for the order based on properties true of:

• (i) a perspective projection only
• (ii) an orthographic projection only
• (iii) both orthographic and projective transformations
• (iv) neither orthographic nor projective transformation

Based on our analysis:

• (i) is Property (D)
• (ii) is Property (C)
• (iii) is Property (A)
• (iv) is Property (B)

Therefore, the correct order of properties is D, C, A, B.

Revision Table: 3D Projection Comparison

Additional Information: Homogeneous Coordinates and Projections

Homogeneous coordinates are a system of coordinates used in projective geometry. For a 3D point \((x, y, z)\), the homogeneous coordinate representation is typically \((wx, wy, wz, w)\), where \(w\) is a non-zero scalar. A common representation is \((x, y, z, 1)\) for 3D points. The power of homogeneous coordinates in graphics is that translation, rotation, scaling, and even perspective projection can all be represented as matrix multiplications using 4x4 matrices.

For a perspective projection, the transformation matrix might produce a point in homogeneous coordinates \((x', y', z', w')\). The final 3D or 2D coordinates are then obtained by dividing by the \(w'\) component (the "perspective divide"): \((x'/w', y'/w', z'/w')\) or \((x'/w', y'/w')\) for the 2D projection plane coordinates. This division step is non-linear, but the matrix multiplication part in homogeneous space is linear, allowing the entire transformation pipeline to be handled efficiently using matrix operations.