Modeling, Transformations, Viewing
Modeling
Representation of geometry
- vertices, lines, polygons -- simple primitives. Widely supported.
- triangle, triangle strips -- efficient for rendering.
- curves, surfaces, volumes -- richer representation.
Modeling is an often cumbersome process requiring manual measurement of lengths
breadths, position etc. of real objects. Digitizers are now commercially available
that can be used to sample a real object at a large number of points. Such
representation is typically followed by a conversion to a more continuous representation,
e.g. polygons or surfaces. Similarly, CT scans, MRI and ultrasound help acquire images
of various parts of human body. Construction of models from images is a useful techniques.
It has been extensively used for medical models and is a hot research topic.
It is not always that you want to model an existing object. In fact, one vaunted power of
computer graphics is its ability to show non-existing or conceptual models. A number of
modeling tools exist that provide ``intuitive'' interfaces to generate models.
For example, Bézier surfaces (see definition in the text) can be represented just
be its control point and just manipulating its control points. More sophisticated
techniques exist.
Hierarchical modeling
Modeling and representing each piece of geometry in an environment separately
can be a difficult and time consuming process. Organizing models into a hierarchy
often alleviates these problems. Instead of modeling each chair in a classroom
separately, we can model one chair and then transform and place it different positions
Or to construct a robot, one could design a left hand -- place it at the left shoulder
and then reflect it and place it at the right shoulder to model the right hand. The hand
itself consist of an a palm that is attached to a fore-arm, which itself is attached
to the upper-arm. The palm would consist of a number of fingers appropriately attached
to it. Thus the model looks like a tree. A transformation is associated with each
node of the tree which places the part of the model represented by its subtree in a
bigger model, a bigger environment. Once we perform all the transformation up till
the root, you have the position of each piece in the ``world'':
we go from modeling to world coordinates.
Note that a number of objects at the leaf nodes look exactly like some others. For
example in the class-room all chairs look alike. In fact each chair may itself consist
of parts that look alike. Instead of storing a detailed representation of each object
at the leaf, we can just store one copy and maintain pointers to that copy at the leaf
node. This helps reduce memory requirements. Another advantage of hierarchical modeling
is its easy modification. If we want throw away the old chairs in the class-room and
buy new ones, all we need to do is replace the master copy of the chair and all other
copies -- all instances automatically get updated.
Transformations
Common operations are translation and rotation -- these are rigid body transformations.
Other transformations are scaling, reflection, shearing etc. Each operation can be
written in terms of matrix multiplication, representing the transformation as a 4x4
homogeneous matrix and the vertices as a 4x1 homogeneous matrix:
[Transformed point] = [Transformation Matrix] X [Original point]
The most standard homogeneous representation of a 3D points [x, y, z] is
[x, y, z, 1]. For vectors we use [x, y, z, 0] to ensure that
multiplying them with a translation matrix leaves them unchanged. In general,
we can get the 3D point by dividing each coordinate by the fourth coordinate
(also called the homogeneous coordinate). When the homogeneous coordinate equals
zero, interesting thing happen -- but we will not talk about those in this course.
The derivation of these matrices is not
repeated here but the matrix themselves are reproduced. We use the right-handed
coordinate system, i.e. if you curl your right hand fingers from positive X
towards positive Y, your thumb points towards positive Z.
Matrices
| Translate | Rotate (by ø about X axis) |
Scale |
Reflect (about YZ plane) |
Shear (perpendicular to Z axis) |
|
|
 |
| 1 | 0 | 0 | 0 |
| 0 | cos(ø) | -sin(ø) | 0 |
| 0 | sin(ø) | cos(ø) | 0 |
| 0 | 0 | 0 | 1 |
|
 |
|
|
|
|
The translation matrix above corresponds to a translation by the vector
[tx, ty, tz, 0].
The rotation matrix rotates a point about the X axis. All (pure)
rotation matrices are orthonormal -- the magnitude of each row (or column) is
1 and the dot product of any pair of rows (or columns) is 0. Thus a rotation
matrix can also represent an orthonormal basis. (The top left 3x3 submatrix is
a basis for the three dimensional Euclidean space)
The scaling matrix scales a point [x, y, z] to
[sxx, syy, szz].
Operations can be composed. Thus a rotation followed by a translation of point p
can be written as the matrix expression: R.T.p. Since matrix operations
are associative, we can first multiply R.T (or any sequence of matrices)
and use the result M as a composite matrix to multiply each desired point by.
Note that transformations need not be commutative in general, since matrix multiplication
is not commutative. Thus the order in which we carry out the transformations matters.
However, check that the rigid body motions: translation and rotation are additive.
Scaling is multiplicative, Reflection is a special type of scaling.
Shearing is not commutative.
General transformations
While most transformations we will need in this course can be described in terms
of rotation, translation etc., we may need more general axis of rotation, or more
general direction of scale. In the transformation matrices listed above the major
restriction comes from the fact that the origin and the orthonormal basis vectors
at the origin are first class citizens. We can change this by creating an appropriately
positioned and oriented set of orthonormal basis vectors and transform about that.
Another way of looking at the same thing, sometimes more intuitive, is to first
move the object to the origin (and sometimes, re-orient it), perform the transformation
and finally move it back (after un-orienting it). Moving a point to the origin is
simple it involves translating by the vector from the point to the origin. A general
re-orientation can be done as follows.
Take a row of a generic rotation matrix R and transform it by R.
What happens? Multiplying R to the first row results in [1, 0, 0, 0].
Similarly the other two rows generate [0, 1, 0, 0] and [0, 0, 0, 1] respectively.
This means that if we rotate the three vectors (rows), we align them with our
basis vectors -- the primary axes. In other words we convert from the original
basis coordinates to a new set of coordinates which corresponds exactly to the
basis given by the rows. That is to say any coordinate in the original
basis can be rotated by R to obtain its representation in the basis given
by the three rows. Confusing enough? Try a few examples and it will straighten itself
out. To go back from the new basis to the original basis we can use
R-1, which is the same as RT.
Viewing
Parameters commonly used to specify viewing:
- VRP: View reference point -- this is where we place the origin of the view
coordinate system (or the camera coordinate system).
- VPN: View plane normal (this determines the orientation of the view plane -- and gives
one of the basis vectors (or camera axes) in our view coordinate system (VCS).
- UP: This determines orientation of the world on the view-plane i.e. it gives
the second vector in VCS. Actually, we do not restrict UP to be of unit length (we know
how to normalize) or perpendicular to VPN (we can later derive a new UP' that is
indeed perpendicular to VPN).
UP must not be parallel to VPN or we cannot build a VCS.
A vector parallel to VPN does not mean much either as its projection on the view-plane
is null and thus the orientation of the world on the plane cannot be fixed. Using
VPN and UP, we can derive the third vector of the basis -- V = PU×VPN. (Remember, we
must also normalize each vector, since we need an orthonormal basis.) UP' (or U) can
now be obtained by VPN×V. We denote the camera axes by <X, Y, Z>
in the following discussion; not to be confused with world axes.
- PRP: Projection reference point (Now we fix the position of the eye -- important
for perspective projection. For parallel projection we only need the direction of
projection.
To transform from the world coordinates to viewing coordinates (or camera coordinates)
we use the general transformation stuff we learned about in the previous section.
The VCS provides the rotation matrix and VRP provides the translation required to
achieve our transformation. This transformation simplifies the projection equation.
(For example, we know that the projection is along the Z axis and
the view plane is perpendicular to it etc. Note that more general projections are
possible, for which projection plane is not perpendicular to Z axis, but the
basic math remains the same.) Let us project a point (x,y,z) on to
the view plane Z=zv.
Can we write these in terms of matrix multiplication? It is equivalent to multiplying
with the following matrix.
