10/4/2023 0 Comments Transformation math![]() The blue arrows, going from left to right, show different rotations which take a point P in and moves it to point P out.The diagram on the left tries to illustrate this. The second type of algebra defines how rotations can be combined, that is, we first do 'rotation 1' then we do 'rotation 2' this must be equivalent to some combined rotation, say: 'rotation 3'.The first type of algebra defines how a given point is transformed, that is, a given rotation must define where every point, before the rotation, ends up after the rotation.There are two types of algebra associated with transformations (such as rotation) and these algebras must interwork correctly together. ![]() 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. Its hard to categorise these things in a definitive way, for example, projective transforms can be considered linear if we use homogeneous coordinates. Here is an (incomplete) attempt to categorise these things: We can also categorise by the types of algebra that can produce the transform, for example, linear ( matrix) algebra.Īlso we can do transforms in different numbers of dimensions. We can combine these in various ways: combinations of translation and rotation, known as isometries, are important because they represent the ways that solid objects can move in the world without changing shape. ![]() There are lots of ways to categorise transforms, one way is to start with the simple transforms that we intuitively understand like: This page introduces the subject of transformations and links on to other pages which analyse the various types of transform in more detail.Ī transform maps every point in space to a (possibly) different point. ![]()
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