The composition of two rotations from the same center, is a rotation whose degree of rotation equals the sum of the degree rotations of the two initial rotations. You need to include a reflection in your transformations. Example: A glide reflection is commutative.In a composition, one transformation produces an image upon which the other transformation is then performed. Partial credit can be given. Let's look at some special situations involving combinations: In certain cases, a combination of transformations may be renamed by a single transformation. The double reflections are equivalent to a rotation of the pre-image about point P of an angle of rotation which is twice the angle formed between the intersecting lines theta. To get from D to A, go down 1 unit and 3 left. We illustrate with two examples. B' is -1, 8 To get from D to C, go down 3 units and 3 right. Composition of transformations is not commutative. This quadrupling of the area is reflected by a determinant with magnitude 4.
This process must be done from right to left!! We will obtain similar conclusions for higher-dimensional linear transformations in terms of the determinant of the associated matrix. May also be over any even number of parallel lines.B' is -1, 8 To get from D to C, go down 3 units and 3 right. It is not possible to rename all compositions of transformations with one transformation, however: Any translation or rotation can be expressed as the composition of two reflections. Each image is twice as far away from D as the original point was. Composition of transformations is not commutative. We colored the quarters of the square in different colors to help visualize how points within the square were mapped. You can change the linear transformation by typing in different numbers and change either quadrilateral by moving the points at its corners. This quadrupling of the area is reflected by a determinant with magnitude 4. You may also apply this rule to negative angles clockwise. There is no way to stretch and move the original unit square into the parallelogram without taking it out of the plane and flipping it or somehow moving the region through itself. The double reflections are equivalent to a rotation of the pre-image about point P of an angle of rotation which is twice the angle formed between the intersecting lines theta. Reversing the direction of the composition will not affect the outcome. A simple translation will not be good enough. To get from D to A, go down 1 unit and 3 left.
Remember that, by convention, the angles are read in a counterclockwise direction.