75602 - Numerical Analysis and Geometric Modeling

Learning outcomes

The course aims at providing the theoretical foundations and discussing the numerical-mathematical aspects and the main methodologies for the representation and manipulation of mathematical shapes. The course outline provides the basics on numerical linear algebra and an introduction to the differential geometry of curves and surfaces in bi- and tri- dimensional Euclidean space. These notions will be applied to the geometric modeling of curves, surfaces and solids, the heart of computer design systems. The course includes a laboratory activity where the MATLAB software is used.

Course contents

FIRST PART (4 CFU) (Module 2)

1 - Linear Algebra and Analytic Geometry

1.1 - 1D, 2D, 3D geometric vector spaces. Algebraic description of geometric relations between vectors. Bases. Orientation, signed areas and volumes.

1.2 - Real vector spaces; linear independence, bases, coordinates, dimension. Vector spaces Rn (n=1,2,3,4,...). Linear systems of n equations in n unknowns with a unique solution. Determinants; properties, Cramer's rule.

1.3 - Points and vectors. 1D, 2D, 3D affine spaces. Frames, coordinates. Parametric and cartesian coordinates of straight lines and planes; incidence, parallelism; skew lines. Projections.

1.4 - Dot product. 1D, 2D, 3D Euclidean vector spaces; length, orthogonality, angles; 3D vector product. Orthonormal bases.

1.5 - 1D, 2D, 3D Euclidean spaces. Frames, coordinates. Orthogonal projections on straight lines and on planes. Distances between points, lines, planes.

2 - Trasformations

2.1 - Linear mapping between geometric vector spaces; construction starting from a basis and a basis image. Concatenation.

2.2 - Linear mapping between vector spaces Rn; construction starting from a basis and a basis image. Concatenation, bijectivity, inversion. Matrices, product row by column, invertibility, matrix inverse. Equivalence between the algebra of linear mappings and the algebra of matrices.

2.3 - Affine mapping of an affine space into intself, induced linear mapping; variation of signed areas and volumes. 2D, 3D translation, dilation, reflection, projection, scaling, shear. Matrix representation with respect to adapted frames and arbitrary frames. Concatenation, invertibility, inversion.

2.4 - Isometry of an Euclidean space into itself. 2D, 3D rotations and their matrix descriptions.

3 - Differential and integral calculus of real functions of one real variable

Real functions of one real variable; kinematic interpretation; graph. Function vector spaces. Polynomial, rational, trigonometric functions. Continuity, derivability and derivative of a function at one point. Derivation rules. Antiderivatives of a function on an interval. Riemann integral. Fundamental theorem of calculus.

SECOND PART (5 CFU) (Module 1)

1- Elements of differential geometry

2D parametric curves, parametrization. Derivative of a parametric curve, regular curve, length of a curve, tangent vector and curvature, normal vector. Examples of curves. 3D curves in parametric form, curvature and torsion. Frenet frame.

Parametric surfaces, tangent plane, normal vector, principal curvatures, mean curvature and Gaussian curvature. Generating surfaces through transformation of parametric curves.

 

2 - Numerical methods for curves and surfaces and geometric modeling

 

2.1- Bézier curves

Polynomial functions in the Bernstein basis. Bézier curves and their properties. Composition of Bézier curves. Rational Bézier curves.

 

2.2- Spline curves

Polynomial spline space, B-spline basis and construction of spline curves. Rational splines (NURBS). Circular arcs in NURBS form.

 

2.3- Surfaces

Bézier, spline and NURBS surfaces. Methods for the generation of NURBS surfaces from curves: skinning, extrusion, ruled surfaces, sweeping.

 

3- Polynomial interpolation with parametric curves

Polynomial and piecewise polynomial (spline) interpolation. Lagrange and Hermite interpolation problems. Construction of a piecewise cubic Bézier curve with C1 continuity.

Readings/Bibliography

FIRST PART: the main reference are instructor's lecture notes and exercises, published weekly during the course on https://virtuale.unibo.it. Further optional reading: G. Farin and D. Hansford, Practical linear algebra - a geometry toolbox, CRC Press

SECOND PART: the main reference is lecture notes which will be made available at the beginning of this module and selected exercises solved by the instructor during the laboratory sessions. The available material can be downloaded from virtuale.unibo.it.

Teaching methods

FIRST PART: Lectures. Exercises will be given weekly, that will be corrected by a tutor.

SECOND PART: Lectures and exercises in computer lab. The exercises complement the theoretical part to stimulate understanding. The software used during the laboratory sessions is Matlab. 

Assessment methods

To pass the final exam it is necessary to obtain a sufficient evaluation (>=18/30) on each of the two parts; the final mark is computed as the weighted average of the score of the two parts.

FIRST PART: The exam consists of a written test and, only after request made by the lecturer or by the student, an oral test.

The written test concerns exercises akin to those assigned during the course; it lasts 2h. The exercises must be solved writing and justifying the steps of the solution.

The possible oral test is crucial for passimg the exam and for the grade; it consists in a discussion of the written test and related topics; it lasts at least 30'.

SECOND PART: Learning assessment takes place through a final written test performed in the laboratory (2 hours) and consisting of three practical exercises and three theoretical questions. The final score is the sum of the scores relative to the individual exercises and questions. To pass the exam, a minimum score of 18 is required.

Teaching tools

Handouts, slides, exercises.

Office hours

See the website of Carolina Vittoria Beccari

See the website of Francesco Regonati