Curves and Surfaces

Academic year
Subject Area
Language of Instruction
Mode of Delivery
ECTS Credits
1st Cycle Studies

Recommended Prerequisites

Infinitesimal Analysis III,  Linear Algebra and Analytic Geometry II.

Teaching Methods

Lectures of a mixed type (with the presentation of the theoretical material and its practice with resolution of exercises), where the basic structures and ideas are taught with practicing on concrete examples, illustrated with their motivating historical aspects and relevant applications. A significant part of the lectures is devoted to problem solving, in which the individual work as well as some group discussion is promoted. The available technological resources (software, internet) for visualization of curves and surfaces and their applications are also exploited in the class.

Learning Outcomes

Knowledge of the fundamental theoretical and practical aspects of the classical theories of curves and surfaces in the tridimensional space, including proofs of selected theorems, problem solving and relevant applications.

The main competencies to develop are: ability to formulate and solve problems; design and use of mathematical models for concrete cases; ability on computations; autonomous learning ability; logical argumentation; critical spirit; clear and rigorous (written and oral) formulation of reasoning; communication skills; use of the internet as a means of communication and source of information.

Work Placement(s)



1. Curves in R3. Regular curves. Arc length and arc-length parameterization. Curvature and torsion. Frenet-Serret frame. Frenet-Serret formulas. Curves not parameterized by arc length: curvature and torsion, Frenet-Serret apparatus. Plane curves. Fundamental theorem of curves. Examples and applications.

2. Surfaces in R3. Regular surfaces. Special classes of surfaces: quadratic surfaces, surfaces of revolution, generalized cylinder and cones, ruled surfaces. Tangent vector space and tangent plane. Normal vectors. Orientable surfaces. First fundamental form. Isometries, equiareal and conformal maps. Application to the computation of areas, lengths and angles. Second fundamental form. Principal, gaussian and median curvature. Elliptical, hyperbolic, parabolic and planar points.

Head Lecturer(s)

António Manuel Freitas Gomes Cunha Salgueiro

Assessment Methods

Exam (100%) or Midterm exam (100%): 100.0%


A. Pressley, Elementary differential geometry, Springer, 2001


B. O'Neill, Elementary differential geometry, 2nd ed., Academic Press, 1997

M. P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, 1976