Credits: 10 | Convenor: Dr. J.F. Watters | Semester: 1 (weeks 1 to 6) |
Prerequisites: | essential: MC144, MC145, MC147 | |
Assessment: | Coursework: 20% | One and a half hour exam: 80% |
Lectures: | 18 | Classes: | none |
Tutorials: | 6 | Private Study: | 51 |
Labs: | none | Seminars: | none |
Project: | none | Other: | none |
Total: | 75 |
To construct transformations with specific properties.
To be familiar with some geometric invariants of groups of transformations of the plane.
To understand stereographic projection and its mapping properties.
To have an understanding of bilinear transformations of the extended complex plane.
To use transformations to solve geometric problems.
To be able to identify the affine nature of a conic.
Developing understanding of mappings and some group-theoretical ideas.
Transformation as a problem solving technique.
Written presentation of logical argument in geometric settings.
Define a transformation of the plane as a bijective mapping.
Define an isometry; distinguish direct and opposite.
is reflection in the real axis.
is a translation.
is the rotation about O through arg a.
Prove that
is an isometry
iff |a| = 1.
Isometries map lines to lines and preserve parallelism and angles.
is a reflection and find its axis.
Obtain
form for reflection in line
thorugh given point in given direction.
Find criteria under which the transformation
is a reflection.
Describe the properties of the transformation
when
.
Define a glide reflection.
Prove that every isometry is uniquely determined by its effect on three
non-collinear points.
Classify direct and opposite isometries.
Classify the various types of isometry by their complex representations.
Describe the product of two reflections.
Describe the product of three reflections, distinguishing reflection and
glide reflection cases.
Prove that every isometry is a product of at most three reflections.
Describe products of rotations and translations as a rotation
identifying centre and angle.
Apply transformations to solve geometric construction problems.
Identify positional relationships between points and lines via equations
involving the corresponding half-turns and reflections and vice-versa.
Define similarity transformations and central dilations.
Describe where a is non-zero complex number.
Show that central dilations are given by
where a is non-zero real number
.
Show that all similarities are given by either
or
where a is non-zero complex number,
.
Prove that every similarity is the product of a central dilation and an
isometry.
Similarities map lines to lines, preserve parallelism and angles.
Know that there is a (unique)
similarity mapping one triangle onto a given similar triangle.
Define dilative rotations and dilative reflections recognising centre
and axes.
Express dilative rotations and dilative reflections as products of
central dilations and rotations or reflections.
Compute similarities (isometries) given two or three points and their
images.
Define and recognise groups of similarities (isometries).
Define the operation of inversion in a given circle as a transformation of the extended complex (inversive) plane. Give inversion in the unit circle as a complex transformation and relate this to the reciprocal map. Define bilinear transformations. Express bilinear transformations as products of the reciprocal map and similarities. Compute bilinear transformations given two or three points and their images. Define stereographic projection and be able to relate points on the sphere to points in the plane by their coordinates. Show that stereographic projection is conformal. Show that the reciprocal map and all bilinear maps are conformal. Understand the relationship between various circles on the unit sphere and lines and circles (circlines) in the plane. Know that lines through the centre of inversion are fixed. Know that lines not through the centre of inversion are mapped to circles through the centre and conversely. Know that circles not through the centre of inversion are mapped to circles of the same type. Use inversion to change circular problems into linear problems. Know that bilinear transformations are uniquely determined by their effect on three distinct points. Define the cross-ratio of four numbers in the extended complex plane. Show that bilinear transformations preserve cross-ratio. Prove that four points are collinear or concyclic if and only if their cross-ratio is real.
Define affine transformations. Prove that every affine transformation is a collineation and know that converse is true. Know that affine transformations are determined by action on three non-collinear points. Show that (collineations) affine transformations form a group of transformations. Construct affine transformations to map one triangle to another. Shoe that segment division ratios are preserved by affine transformations. Know that collinearity, parallelism, and segment division ratios, in particular mid-points, are affine properties. Show that the standard conics in the Euclidean plane are affine equivalent to the unit circle (ellipse), rectangular hyperbola xy = 1 (hyperbola), and y2 = x (parabola). Use affine equivalent forms to describe properties of conics. Use completion of squares to reduce general equation of the second degree in x,y to a standard form.
H. S. M. Coxeter, Introduction to Geometry, 2nd edition, Wiley.
M. Jeger, Transformation Geometry, Allen and Unwin.
P. J. Ryan, Euclidean and Non-Euclidean Geometry - an Analytic Approach, Cambridge University Press.