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- Geometric vectors, operations, vector spaces.

Geometric vectors and geometric vector spaces V^n (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 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.

Euclidean spaces E^n and vector spaces V^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. Frames of reference and identification of E^n and V^n 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 equations of lines and planes. Distance between points, lines, planes; angle betweeen two rays.

3- Linear systems, matrix algebra, linear maps.

Summation; linear equation. 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 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 in itself, identification with linear maps of R^n in itself and with nxn matrices with respect to a basis (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 and their identification with maps of R^n in itself with respect to a frame of reference (n=2,3).

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

Real functions of one real variable and their graphs; vector spaces of functions. Affine linear, quadratic, polynomial, rational, trigonometric, exponential and logarithmic functions. Continuity and its implications. Derivative of a funtion at a point and its geometric meaning. Derivation rules. Riemann integral of a function and its geometric meaning. 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- Representation and geometric modeling of curves and surfaces

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 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 with parametric curves

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

Readings/Bibliography

For part A the main reference are instructor's lecture notes and exercises, published weekly during the course on IOL.

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

For part B, the main reference is lecture notes which will be made available at the beginning of this module of the course (downloadable from the IOL platform).

Teaching methods

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

Assessment methods

In order to pass the exam a student must pass the exam on each of the two parts; the grade of exam is the weighted average of the grades of the exams on the two parts.

FIRST PART:

Testing is done through a final written exam consisting of exercises and questions on the model of those delivered during the course. Upon request by the instructor or by the student, the written exam could be followed by an oral exam. To pass the exam a minimum score of 18 is required.

 

SECOND PART:

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.

Teaching tools

Handouts, slides, exercises.

Office hours

See the website of Carolina Vittoria Beccari

See the website of Francesco Regonati