General description
This module introduces some of the fundamental notions of geometry - points, lines, curves, planes and surfaces - analytically, in the language of coordinate geometry. Conics are classified in terms of their equations and geometric properties. The concepts of tangent and normal are developed. There are brief introductions to protective geometry and to vector methods.

Aims
To develop geometric intuition and the ability to view geometric problems analytically and vice versa.

Learning outcomes
On completion of this module, a student should be able to:

• identify the type of a conic from its equation;
• obtain any centre, focus or axis of a conic;
• determine the equations of the tangents and normals to a general conic;
• use vectors to solve elementary problems in geometry;
• express the coordinates of a general point of certain curves parametrically;
• determine the equations of lines and planes in real 3-space;
• explain the concept of a tangent plane and determine its equation in special cases.

Syllabus
1. COORDINATE GEOMETRY IN THE REAL PLANE: The straight line. Conics - particular forms and the general form. Identification of centres, foci and major and minor axes. Cases of degeneracy. Changes of axes. The general equation of the tangent. Families of lines and conics. Parametric plane curves. Tangents and the use of derivatives.
2. INTRODUCTION TO VECTOR METHODS: Unit vectors. Scalar products, angles and orthogonality. Position vectors. Centroids and orthocentres. Vector equations of lines and planes.
3. GEOMETRY IN REAL 3-SPACE: Cartesian coordinates. Equations of lines and planes. The normal. Curves, their tangents and other special direction. Surfaces and their tangent planes. Polar coordinates.