# 75602 - Numerical Analysis and Geometric Modeling

## 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) Notions of Mathematical Analysis

Real numbers; Function in one and more variables; Derivative, partial and vector derivatives, directional and total derivatives; Increasing and decreasing functions, maxima and minima.

2) Elements of linear algebra and Euclidean geometry

Reference Systems. Cartesian coordinates in the plane and space.
Geometric vectors in E^1, E^2, E^3. Basis of a vector space. basis. Sub-spaces and generators. Dimension. Properties and operations; scalar product, vector product. Matrices, matrix operations, determinant, rank of a matrix.
Linear spaces, bases and change of basis, linear combinations, affine combinations, convex combinations. Linear transformations between vector spaces. Linear transformations and matrices. Matrix of a linear transformation with respect to a basis.
Solution of linear systems using Gauss method. Solution of over-determined linear systems using normal equations.

Cartesian and parametric representations of lines and planes in R^3. Line and plane equation in explicit and implicit form. Distance of a point and a line. Vector orthogonal to a straight line. Distance of a point from a plane, plane orthogonal vector. Sheaf of planes, improper sheaf of planes, parallelism/orthogonality between lines and planes.
Planar geometric transformations: translation, rotation, reflection, shear, composition of transformations, inverse transformations (matrix form). 3D transformations. Homogeneous coordinates. Prospective projections.

PART B (5 CFUs) (Modules 2 and 3)

3) Elements of differential geometry

2D parametric curves, parametrization. Derivative of a parametric curve, regular curve, integral of a 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 from transformation of parametric curves.

4) Representation and geometric modeling of curves and surfaces

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

4.1) Spline curves

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

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

5) 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.

Geometria analitica del piano e dello spazio, autore: Abeasis Silvana, Zanichelli

## 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.