Homogeneous coordinates
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
Advantages of Homogeneous coordinates
- 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
Non-uniform foreshortening
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
Projections as part of parallel projection
- Orthographic
- Axonometric
- Oblique
Orthographic projection
In all orthographic (or orthogonal) views, the projectors are perpendicular to the projection plane
It is well suited for working drawings
Axonometric projections
the projectors are still orthogonal to the projection plane, but the projection plane can have any orientation with respect to the object.
Views part of Axonometric projection
- Isometric
- Diametric
- Trimetric
Isometric view
The projection plane is placed symmetrically with respect to the three principal faces that meet at a corner of a rectangular object
Diametric view
The projection place is placed symmetrically with respect to two of the principal faces of a rectangular object
Trimetric view
The projection plane can have any orientation with respect to the object (the general case).
Oblique projections
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
Perspective projections
- 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
Projection normalization
- 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
Planar geometric projections
The class of projections produced by parallel and perspective systems because the projection surface is a plane and the projectors are lines
Synthetic-camera model
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
Functions in the API for Synthetic-camera model
- 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.
define the camera frame, we require three parameters
- 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.
Affine transformation
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.
Frustum
The shape of the view volume for a perspective projection
Translation
an operation that displaces points by a fixed distance in a given direction
Rotation
an operation that rotates points by a fixed angle about a point or line
Scaling
an affine non-rigid-body transformation by which we can make an object bigger or smaller
Uniform scaling
The scaling factor in all directions is identical. The shape of the scaled object is preserved.
Non-uniform scaling
The scaling factor of each direction need not be identical. The shape of the scaled object is distorted.
