An affine change of coordinates in R^{n} given by
where R is an invertible n \times n matrix, is called a Euclidean change of coordinates if the matrix R is an orthogonal matrix.
Coordinate-based calculation of lengths and angles is not affected by a Euclidean change of coordinates.
The classification of isometries outlined in the discussion of February 22 is not affected by a Euclidean change of coordinates.
Every rotation is a composition of the reflections in two lines that intersect at the center of rotation. For a given rotation one of the two lines through its center may be chosen arbitrarily.
Every translation is a composition of the reflections in two lines that are both perpendicular to the direction of translation. For a given translation one of the two parallel lines perpendicular to the direction of translation may be chosen arbitarily.
This is the date for the midterm announced at the beginning of the semester.
Let rho be rotation about the origin by the angle 2 theta, and let sigma be reflection in the y-axis. Find the axes of two other reflections xi and eta such that one has
Write translation by the vector (3, -4) as the composition of a reflection with a reflection in a line through the origin.
Explain why every glide reflection may be written as the product of three reflections.
Explain why the composition of any four reflections may always be written as the composition of two reflections.
What type of isometry may result from composing a rotation and a reflection?
When is the composition of a rotation and a reflection simply a reflection?
When is the composition of three reflections simply a reflection?