### M4: Geometry (2022-23)

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 coordinates and an appreciation of the invariance of geometry under an orthogonal change of variables. This leads to a deeper study of orthogonal matrices, of rotating frames, and into related coordinate systems. Another focus is on the construction and geometric properties of lines, planes, and surfaces.
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. Students will gain introductory insight into the geometry of surfaces and lines.
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. Scalar triple products and vector triple products, vector algebra. [2]

Conics (normal form), focus and directrix. Degree two equations in two variables. [2]

Orthogonal matrices, and the maps they represent in $$\mathbb{R}^2$$. Orthonormal bases in $$\mathbb{R}^3$$. Orthogonal change of variable. Statement of Spectral Theorem and use in simple examples. Identifying conics not in normal form. [2]

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

Parametrised surfaces, including spheres, cones; surfaces of revolution. Examples of finding shortest path on a surface. Surface isometries. Arc length and surface area; isometries and area. [4]