# 75602 - Numerical Analysis and Geometric Modeling

### SDGs

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

## 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 - Euclidean vectors, operations; order, signed areas and volumes. Vector spaces, linear combinations. Algebraic description of geometric relations among vectors; coordinate systems for vectors.

1.2 - Vector spaces, linear independence, bases, coordinates, dimension. Numeric vector spaces Rn, bases recognition and coordinate computation. Linear systems of n equations in n unknowns with a unique solution. Determinants, properties, Cramer rule.

1.3 - Coordinate systems for points and vectors; compatibility with operations on vectors; signed areas, signed volumes and determinants. Plane: parametrric and cartesian equations of straight lines; incidence, parallelism. Space: parametrric and cartesian equations of straight lines and planes; incidence, parallelism; skew straight lines.

1.4 - Length and orthogonality, dot product of vectors; distance between points; cross product. Orthonormal coordinate systems; expressions for dot product and cross product. Plane: orthogonal projection on a straight line, point-line distance. Space: orthogonal projection on a straight line and on a plane, point-line and point-plane distance; minimum distance points on a pair of skew straight lines.

2 - Trasformations

2.1 - Affine maps of plane and space in itself and induced maps on vectors; compatibility with operations on vectors; variation of signed areas and signed volumes. Linear maps between vector spaces; linear maps with given values on a basis. Affine map, induced linear map, reconstrucion of the first from the second and the value on a point.

2.2 - Linear maps between numeric vector spaces Rn, operation of composition, bijective maps, inverse map. Matrices, operation of product rows by columns, invertible matrices, determinant and inverse matrix. Equivalence between the algebra of linear maps and the algebra of matrices.

2.3 - Representation of an affine map and its linear part with respect to a coordinte system; matrix of the linear part, geometric meaning of determinants. Affine maps of the plane and of the space in itself: dilations, projections, reflections, scalings, shears, rotations. Rappresentation in a suitable and in an arbitrary coordinate system.

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

Real functions of one real variable; kinematic interpretation; graphs; operations on functions; vector spaces of functions. Polynomial, rational, trigonometric, exponential and logarithmic functions. Continuity and its implications. Derivative of a function at a point; speed; tangent to the graph at a point. Derivation rules. Riemann integral of a function; definition and computation of areas. Antiderivatives of a function on an interval. 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, geometric and parametric continuity. Examples of curves. 3D curves in parametric form, curvature and torsion. Frenet frame.
Regular 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. Conics as quadratic rational curves.

2.2- Spline curves

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

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.

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: S. Abeasis, Geometria analitica del piano e dello spazio, Zanichelli; 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; often it will be asked to solve them in two ways or to verify the result.

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 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