75602 - Numerical Analysis and Geometric Modeling

Academic Year 2019/2020

  • Moduli: Carolina Vittoria Beccari (Modulo 1) Francesco Regonati (Modulo 2)
  • Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
  • Campus: Bologna
  • Corso: First cycle degree programme (L) in Industrial Design (cod. 8182)

Learning outcomes

The course aims at providing the theoretical foundations, the numerical-mathematical aspects and the main methodologies for the representation and manipulation of mathematical shapes. The training course provides a basis on numerical linear algebra and an introduction to the differential geometry of curves and surfaces in the bi- and tri- dimensional Euclidean space. Introduced tools 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

PART A (4 CFU) (Module 1)

1- Geometric vectors, operations, vector spaces.

Geometric vectors and geometric vector spaces V^n_o (n=1,2,3). Vector spaces; linearly independent vectors, bases and dimension; coordinates of a vector with respect to a basis. Numeric vector spaces R^n (n=1,2,3,...). Identification of V^n_o with R^n with respect to a basis (n=1,2,3). Length of a vector, cosine of the angle between two vectors; scalar product of two vectors; orientations of space, vector product of two vectors.

2- Euclidean plane and Euclidean space, linear analytic geometry.

Geometric vector spaces V^n_o and Euclidean spaces E^n (n=1,2,3). Parametric equation of the line through a point parallel to a vector and of the plane through a point parallel to two vecors. Cartesian equation of the line in E^2 and of the plane in E^3 through a point and orthogonal to a vector. In E^n, frames of reference and identification with R^n (n=1,2,3). Scalar product and vector product coordinate formulas. Parametric and cartesian equations of lines and planes in R^n (n=2,3); parallelism conditions. In R^3, mutual positions of two planes, of a plane and a line, and of two lines; skew lines. Normal cartesian equation of a line in the plane and of a plane in space. Distance between points, lines, planes; angle betweeen two rays.

3- Linear systems, matrix algebra, linear maps.

Linear systems of m equations in n unknowns. Matrices; product of matrices. Matrix representation Ax=b of a linear system. Invertible matrices and inverse matrix. Determinant of 2x2 and 3x3 matrices, signed areas and volumes; determinant of nxn matrices. Square linear systems Ax=b, existence and uniqueness of a solution and invertibility of A, solution x=(A^-1)b. Linear maps R^n -> R^m and their representation x -> Ax with mxn matrices A; composition of linear maps and product of matrices. Bijectivity, invertibility and inversion of linear maps of R^n in itself, determinant and inversion of nxn matrices.

4- Linear transformations and affine linear transformations of plane and space.

Linear maps of V^n_o in itself, identification with linear maps of R^n in itself and with nxn matrices (n=2,3); geometric meaning of the determinant. Rotation, orthogonal projection, reflection, scaling, shear; their matrices with respect to suitable and arbitrary bases. Affine linear maps of E^n in itself as composition of a translation after a linear map and their identification with maps of R^n in itself (n=2,3). Orthogonal projections, reflections.

5- Differential and integral calculus of real functions of one real variable.

Real functions of one real variable and their graphs; affine linear, quadratic, polynomial, rational, trigonometric, exponential and logarithmic functions. Operations on functions. Continuity and its implications. Derivative of a funtion at a point and its geometric meaning. Derivation rules. Riemann integral and its geometric meaning. Antiderivatives of a function on an interval. Fundamental theorem of calculus.

 

PART B (5 CFU) (Module 2)

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- Representation and geometric modeling of curves and surfaces

2.1- Bézier curves
Polynomial functions in the Bernstein basis. Bézier curves. Properties. Composition of Bézier curves. Rational Bézier curves. Conics as quadratic rational curves.

2.2- Spline curves

Polynomial splines. Spline curves. Rational splines (NURBS).

2.3- Surfaces
Bézier surfaces, spline surfaces, NURBS surfaces, and trimmed NURBS. Construction of NURBS surfaces: skinning, extrusion, ruled surfaces, sweeping.

3- Polynomial interpolation and approximation with parametric curves

Polynomial and spline interpolation and approximation. Lagrange and Hermite interpolation problems. Construction of a piecewise cubic Bézier curve with continuity C^1 continuity.

Readings/Bibliography

For part A the main reference is instructor's lecture notes, published weekly during the course on his web page.

Further (optional) reading: "Geometria analitica del piano e dello spazio", author: Abeasis Silvana, Zanichelli.

For part B, the reference is the course handout which will be made available at the beginning of the lessons (downloadable from the IOL platform).

Teaching methods

Lectures and exercises in computer lab. The exercises complement the theoretical part to stimulate understanding.

Assessment methods

PART A:

Testing is done through a final written exam consisting of exercises on the model of lesson exercises. To pass the exam a minimum score of 18 is required, obtained by adding the scores of the individual answers.

PART B:

Verification of learning takes place through a final written test to be performed in the lab consisting of exercises on the model of lessons and 3 theoretical questions. To pass the exam, a minimum score of 18 is required, obtained by adding the scores of the individual answers.

The final grade will be calculated as the weighted average of PART A and PART B.

Teaching tools

Handouts, slides, exercises.

Office hours

See the website of Carolina Vittoria Beccari

See the website of Francesco Regonati

SDGs

Quality education Partnerships for the goals

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