A transform maps every point in space to a (possibly) different point.
Types of transform includes:
- Rotation
- Scaling
- Reflection
- Translate
- Shear
and many more complicated transforms.
There are different algebras that can represent transforms and help us calculate the effect of different operations, such as combining two transforms, not all of these algebras can represent all transforms. Useful algebras include:
There are a number of quantities associated with each transform including the following:
Determinants
This tends to be associated, in peoples minds, with matrices. In fact it can have a geometric meaning which can be calculated from other types of algebras.
It is a scalar value and it is related to:
- The scaling factor of the transform
- The left or right handedness of the transform.
In other words does it make an object bigger or smaller and does it reflect it (an odd number of times).
The calculation of determinants, in the case of matrices, is shown on these pages.
Eigenvalues and Eigenvectors
This also tends to be associated, in peoples minds, with matrices. Again these quantities can have a geometric meaning which can be calculated from other types of algebras.
This tells us something about the symmetry of a transform.
An eigenvector is a vector whose direction is not changed by the transform, it may be streached, but it still points in the same direction.
Each eigenvector has a corresponding eigenvalue which gives the scaling factor by which the transform scales the eigenvector. So the eigenvector is a vector and the eigenvalue is a scaler.
A given transform may have more than one eigenvector and eigenvalue pair depending on how many dimensions we are working in. For instance:
- If we are working in 2 dimensions there are upto 2 eigenvector and eigenvalue pairs.
- If we are working in 3 dimensions there are upto 3 eigenvector and eigenvalue pairs.
and so on.
The calculation of eigenvectors and eigenvalues, in the case of matrices, is shown on these pages.
As an example, if we have a rotation transform in 3 dimensions, then the eigenvector would be the axis of rotation since this is not altered by the transform and the corresponding eigenvalue would be +1 since the axis is not scaled by the rotation. If we have a rotation in 2 dimensions then the eigenvectors would be ±i where i is √-1 since all vectors in the plane change direction.
Another geometrical application of eigenvectors and eigenvalues is to attempt to factor transforms into rotational and scaling parts, this is discussed on this page.
A more physical application is to the inertia tensor where the eigenvectors indicate the axies that the solid object will rotate around without wobble.
Next
An important class of transforms are rotations and so we can go on to look at these in more detail on these pages.
