Question

Classify the following isometries. State, with reason, if the isometry is a translation, a rotation, a reflection, or a glide reflection. If the isometry is a rotation, give the angle and the centre of the rotation. Express the centre as a complex number in standard form, a + ib. If the isometry is a reflection, give the line of reflection.

(a) f(z)=iz+1+2i
(b) h(z)=z_ + 1 + i
(c) g(z)=-z_+3
(d) p(z)=iz_

Answer 1

The isometries include a mixture of glide reflections and rotations. The transformations involve combinations of rotation, reflection, and translation in the complex plane.

Step-by-step explanation:

In this question, we are asked to classify different isometries of complex numbers into translations, rotations, reflections, or glide reflections. Isometries are transformations that preserve distances between points.

(a) f(z)=iz+1+2i is a rotation. Multiplication by i corresponds to a rotation of 90 degrees counterclockwise about the origin in the complex plane. Adding 1+2i to the result translates the image which means this is a rotation followed by a translation, commonly known as a glide reflection.

(b) h(z)=z_ + 1 + i appears to have a typo. Assuming the underscore is a placeholder for conjugation, 'h(z)' is a reflection in the real axis followed by a translation by 1+i. This type of transformation is not purely a reflection or a translation; it's a glide reflection as well.

(c) g(z)=-z_+3 is a reflection. The conjugate 'z_' reflects the point across the real axis, and the negation reflects the point across the imaginary axis, and these combined result in a 180-degree rotation. Adding 3 to the result translates the image, making it a glide reflection.

(d) p(z)=iz_ is a rotation. The conjugation reflects the point across the real axis, and multiplication by i rotates points 90 degrees counterclockwise. This is simply a 90-degree rotation without any translation, rather than a glide reflection.