M4: Geometry (2016-2017)

2016-2017
Lecturer(s): 
Dr Richard Earl
Course Term: 
Michaelmas
Course Lecture Information: 

15 lectures

Course Overview: 

The course is an introduction to some elementary ideas in the geometry of euclidean space through vectors. One focus of the course is the use of co-ordinates and an appreciation of the invariance of geometry under an orthogonal change of variable. This leads into a deeper study of orthogonal matrices, of rotating frames, and into related co-ordinate systems.

Learning Outcomes: 

Students will learn how to encode a geometric scenario into vector equations and meet the vector algebra needed to manipulate such equations. Students will meet the benefits of choosing sensible co-ordinate systems and appreciate what geometry is invariant of such choices.

Course Syllabus: 
Course Synopsis: 

Euclidean geometry in two and three dimensions approached by vectors and coordinates. Vector addition and scalar multiplication. The scalar product, equations of planes, lines and circles. [3]

The vector product in three dimensions. Use of $\mathbf{a}, \mathbf{b}, \mathbf{a} \land \mathbf{b}$ as a basis. $\mathbf{r} \land \mathbf{a} = \mathbf{b}$ represents a line. Scalar triple products and vector triple products, vector algebra. [2]

Conics (normal form only), focus and directrix. Showing the locus $Ax^2 + Bxy + Cy^2 = 1$ can be put in normal form via a rotation matrix. Orthogonal matrices. $2\times 2$ orthogonal matrices and the maps they represent. Orthonormal
bases in $\mathbb{R}^3$. Orthogonal change of variable; $A\mathbf{u} \cdot A\mathbf{v} = \mathbf{u \cdot v}$ and $A(\mathbf{u} \land \mathbf{v}) = \pm A\mathbf{u} \land A \mathbf{v}$. Statement that a real symmetric matrix can be orthogonally diagonalized. Simple examples identifying conics not in normal form. [3]

$3 \times 3$ orthogonal matrices; $SO(3)$ and rotations; conditions for being a reflection. Isometries of $\mathbb{R}^3$. [2]

Rotating frames in $2$ and $3$ dimensions. Angular velocity. $\mathbf{v} = \omega \land \mathbf{r}$. [1.5]

Parametrised surfaces, including spheres, cones. Examples of coordinate systems including parabolic, spherical and cylindrical polars. Calculating normal as $\mathbf{r}_u \land \mathbf{r}_v$. Surface area. [3.5]

Reading List: 

1) J. Roe, Elementary Geometry (Oxford Science Publications, 1992), Chapters 1, 2.2, 3.4, 4, 5.3, 7.1--7.3, 8.1--8.3, 12.1.
2) M. Lunn, A First Course In Mechanics (Oxford Science Publications, 1991) Chapter 4