Transformations and Viewing

Question 1

Homogeneous coordinates

Answer 1

four-dimensional representation for both points and vectors in three dimensions
fourth component of a vector is set to 0
fourth component of a point is set to 1

Question 2

Advantages of Homogeneous coordinates

Answer 2

• All affine (line-preserving) transformations can be represented as matrix multiplications
• We can carry out operations on points and vectors using their homogeneous-coordinate representations and ordinary matrix algebra
• The uniform representation of all affine transformations makes carrying out successive transformations (concatenation) far easier than in three-dimensional space
• less arithmetic work is involved
• Modern hardware implements homogeneous-coordinate operations directly, using parallelism to achieve high-speed calculations

Question 3

Non-uniform foreshortening

Answer 3

The images of objects farther from the centre of projection are reduced in size(diminution) compared to the images of objects closer to the COP

Question 4

Projections as part of parallel projection

Answer 4

• Orthographic
• Axonometric
• Oblique

Question 5

Orthographic projection

Answer 5

In all orthographic (or orthogonal) views, the projectors are perpendicular to the projection plane
It is well suited for working drawings

Question 6

Axonometric projections

Answer 6

the projectors are still orthogonal to the projection plane, but the projection plane can have any orientation with respect to the object.

Question 7

Views part of Axonometric projection

Answer 7

• Isometric
• Diametric
• Trimetric

Question 8

Isometric view

Answer 8

The projection plane is placed symmetrically with respect to the three principal faces that meet at a corner of a rectangular object

Question 9

Diametric view

Answer 9

The projection place is placed symmetrically with respect to two of the principal faces of a rectangular object

Question 10

Trimetric view

Answer 10

The projection plane can have any orientation with respect to the object (the general case).

Question 11

Oblique projections

Answer 11

It is the most general parallel view. We obtain an oblique projection by allowing the projectors to make an arbitrary angle with the projection plane

Question 12

Perspective projections

Answer 12

• All perspective views are characterized by diminution of size: the farther an object is moved from the viewer, the smaller its image becomes
• perspective views are used by applications where it is important to achieve natural-looking images

Question 13

Projection normalization

Answer 13

• a technique that converts all projections into simple orthogonal projections by distorting the objects such that the orthogonal projection of the distorted objects is the same as the desired projection of the original objects
• done by applying a matrix called the normalization matrix, also known as the projection matrix

Question 14

Planar geometric projections

Answer 14

The class of projections produced by parallel and perspective systems because the projection surface is a plane and the projectors are lines

Question 15

Synthetic-camera model

Answer 15

A paradigm in which we look at the creation of a computer-generated image as being similar to forming an image using an optical system, such as a camera

Question 16

Functions in the API for Synthetic-camera model

Answer 16

• Objects: The geometry of an object is usually defined by sets of vertices.
• A viewer: We can define a viewer or camera by specifying a number of parameters, including: position, orientation, focal length, and the size of the projection plane.
• Light sources: Light sources are defined by their location, strength, colour, and directionality.
• Material properties: Material properties are characteristics, or attributes, of the objects, and such properties are specified through a series of function calls at the time that each object is defined.

Question 17

define the camera frame, we require three parameters

Answer 17

• View-Reference Point (VRP): Specifies the location of the COP, given in world coordinates.
• View-Plane Normal (VPN): Also known as 𝑛, specifies the normal to the projection plane.
• View-up vector (VUP): Specifies what direction is up from the camera’s perspective. This vector need not be perpendicular to 𝑛.

Question 18

Affine transformation

Answer 18

A transformation is a function that takes a point (or vector) and maps it into another point (or vector). When we work with homogeneous coordinates, any affine transformation can be represented by a 4×4 matrix that can be applied to a point or vector by pre-multiplication

𝐪=𝐓𝐩.

All affine transformations preserve lines. Common affine transformations include rotation, translation, scaling, shearing, or any combination of these.

Question 19

Frustum

Answer 19

The shape of the view volume for a perspective projection

Question 20

Translation

Answer 20

an operation that displaces points by a fixed distance in a given direction

Question 21

Rotation

Answer 21

an operation that rotates points by a fixed angle about a point or line

Question 22

Scaling

Answer 22

an affine non-rigid-body transformation by which we can make an object bigger or smaller

Question 23

Uniform scaling

Answer 23

The scaling factor in all directions is identical. The shape of the scaled object is preserved.

Question 24

Non-uniform scaling

Answer 24

The scaling factor of each direction need not be identical. The shape of the scaled object is distorted.