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Prove that two successive Rotations transformations are additive?
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  • Any sequence of transformations can be represented as a composite transformation matrix by calculating the product of the individual transformation matrices. Forming products by transformation matrices is usually referred to as a concatenation, or composition, of matrices.

Translations

  • Two successive translations of an object can be carried out by first concatenating the translations matrices, then applying the composite matrix to the coordinate points. Specifying the two successive translation distances as (Tx1, Ty1) and (Tx2, Ty2), we calculate the composite matrix as

$= \begin{vmatrix} \ 1 & 0 & 0 \\ \ 0 & 1 & 0 \\ \ T_{x1} & T_{y2} & 1 \\ \end{vmatrix} \begin{vmatrix} \ 1 & 0 & 0 \\ \ 0 & 1 & 0 \\ \ T_{x2} & T_{y2} & 1 \\ \end{vmatrix} \begin{vmatrix} \ 1 & 0 & 0 \\ \ 0 & 1 & 0 \\ \ T_{x1} + T_{x2}& T_{y1} +T_{y2} & 1 \\ \end{vmatrix}$

Which demonstrates that two successive translations are additive.

Scalings

Concatenating transformation matrices for two successive scaling operations produces the following composite scaling matrix:

S(Sx1, Sy1) . S(Sx2, Sy2) = S(Sx1.Sx2, Sy1.Sy2)

The resulting matrix in this case indicates that successive scaling operations are multiplicative. That is, if we were to triple the size of an object twice in succession, the final size would be nine times that of the original.

Rotations

The composite matrix for two successive rotations is calculated as

R(θ1) . R(θ2) = R(θ1 + θ2)

As is the case with translations, successive rotations are additive.

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