Last update: 08-04-2015
340070 - MADI-D2O43 - Mathematics for Design
Coordinating unit:
340 - EPSEVG - Vilanova i la Geltrú School of Engineering
Teaching unit:
743 - MA IV - Department of Applied Mathematics IV
Academic year:
2015
Degree:
BACHELOR'S DEGREE IN INDUSTRIAL DESIGN AND PRODUCT DEVELOPMENT ENGINEERING
(Syllabus 2009). (Teaching unit Compulsory)
BACHELOR'S DEGREE IN MECHANICAL ENGINEERING (Syllabus 2009). (Teaching unit Optional)
ECTS credits:
6
Teaching languages:
Catalan, Spanish
Teaching staff
Coordinator:
Josep González
Others:
Q1 Josep González, Julio Fernández
Degree competences to which the subject contributes
Specific:
1. G1. Ability to solve arithmetic problems related to engineering. Aptitude to apply knowledge concerning: linear
algebra, geometry, differential geometry, differential and integral calculus, numerical methods, stadistics technology.
Transversal:
2. SELF-DIRECTED LEARNING. Detecting gaps in one's knowledge and overcoming them through critical selfappraisal. Choosing the best path for broadening one's knowledge.
4. EFFECTIVE USE OF INFORMATI0N RESOURCES. Managing the acquisition, structure, analysis and display of
information from the own field of specialization. Taking a critical stance with regard to the results obtained.
Teaching methodology
There are large group classes, that deal with theoretical explanations, descriptions of selected examples and problem
solving. In the computer lab sessions, students perform simulations with computer software in order to solve casestudies.
Learning objectives of the subject
* To understand the concepts and techniques of classical geometry that are essential for CAGD:
- To use affine coordinates and transformations to move and transform the shape of plane and spacial geometric figures
- To handle with conics and quadric surfaces, as exemple of basic curves and surfaces
- To identify some affine, Euclidean and projective characteristics (barycentric coordinates, distance, cross ratio)
- To understand the following concepts of differential geometry: curvature, torsion and osculating circle of a curve;
tangent plane, normal vector and Dupin indicatrix of a surface
* To use the techniques of Bézier and B-splines in designing curves and surfaces:
- To deal with Bernstein polynomials for Bézier curves and surfaces
- To learn the de Casteljau Algorithm
- To understand the problem of geometric continuity for spline curves and surfaces
- To complete the knowledge of Calculus that is necessary to achieve the previous goals
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Last update: 08-04-2015
340070 - MADI-D2O43 - Mathematics for Design
Study load
Total learning time: 150h
Hours large group:
45h
30.00%
Hours medium group:
0h
0.00%
Hours small group:
15h
10.00%
Guided activities:
0h
0.00%
Self study:
90h
60.00%
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Universitat Politècnica de Catalunya
Last update: 08-04-2015
340070 - MADI-D2O43 - Mathematics for Design
Content
1. Affine and Euclidean Geometry
Learning time: 42h
Theory classes: 16h
Laboratory classes: 4h
Self study : 22h
Description:
1.1 Points and vectors
1.2 Review of linear spaces: linear combinations and basis
1.3 Affine space and affine references
1.4 Affine combinations, barycentric coordinates and ratio of 3 collinear points
1.5 Affine maps
1.6 Metric, norm and distance; Euclidean affine space
1.7 Orthonormal references
1.8 Euclidean motions
2. Bézier curves and B-splines curves
Learning time: 30h
Theory classes: 8h
Laboratory classes: 4h
Self study : 18h
Description:
2.1 Linear interpolation. Examples of curves
2.2 Bernstein Polynomials
2.3 Bézier curves. Properties
2.4 Algorithm of de Casteljau
2.5 Geometric continuity
2.6 B-spline curves
3. Rational curves
Learning time: 28h
Theory classes: 8h
Laboratory classes: 2h
Self study : 18h
Description:
3.1 Projective geometry: Conics, Projections, Crossratio
3.2 Rational Bézier curves and NURBS
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Last update: 08-04-2015
340070 - MADI-D2O43 - Mathematics for Design
4. Differential Geometry of curves
Learning time: 24h
Theory classes: 6h
Laboratory classes: 2h
Self study : 16h
Description:
4.1 Regular parametrizations
4.2 Curvature and torsion
4.3 Osculating circle and evolutes
4.4 Frenet frame
4.5 Geometric continuity
5. Surfaces
Learning time: 24h
Theory classes: 8h
Self study : 16h
Description:
5.1 Functions of two variables: continuity, differentiability, Jacobian matrix and chain rule
5.2 Differential geometry of surfaces: tangent plane, Gaussian curvature Dupin's indicatrix
5.3 Surfaces of revolution
5.4 Rectangular Bézier surfaces
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Universitat Politècnica de Catalunya
Last update: 08-04-2015
340070 - MADI-D2O43 - Mathematics for Design
Planning of activities
1: EXAM OF ITEMS 1 AND 2 (FIRST MIDTERM EXAM)
Description:
Exam: Problems and theoretical questions of topics 1 and 2
2: EXAM OF ITEMS 3, 4 AND 5 (SECOND MIDTERM EXAM)
Description:
Exam: Problems and theoretical questions of topics 3, 4 and 5
3: COMPUTER LAB
Description:
Students should apply basic techniques to handling and depict geometric objects (curves and surfaces), booth in
general and by Bézier techniques in particular. They will use the software MAPLE ®
4: FINAL EXAM
Description:
Exam: Problems and theoretical questions of topics 1, 2, 3 4 and 5
Qualification system
max((0.35 NA1+0.35 NA2+0.3 NA3),(0.7 NA4 + 0.3 NA3)
NA1: First midterm exam (activity 1)
NA2: Second midterm exam (activity 2)
NA3: Computer Lab reports for case studies (activity 3)
NA4: Final exam (activity 4)
The completion of activity 3 is a necessary condition to be evaluated
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Last update: 08-04-2015
340070 - MADI-D2O43 - Mathematics for Design
Bibliography
Basic:
Trias Pairó, Joan. Geometria per a la informàtica gràfica i CAD [on line]. Barcelona: Edicions UPC, 1999 [Consultation:
06/11/2012]. Available on: <http://hdl.handle.net/2099.3/36243>. ISBN 8483013541.
Farin, Gerald E. Curves and surfaces for computer aided geometric design : a practical guide [on line]. 5th ed. San Francisco
[etc.]: Morgan Kaufmann, 2002 [Consultation: 06/11/2012]. Available on:
<http://www.sciencedirect.com/science/book/9781558607378>. ISBN 1558607374.
Cordero Valle, Juan Manuel; Cortés Parejo, José. Curvas y superficies para modelado geométrico. Madrid: RA-MA, 2002. ISBN
8478975314.
Complementary:
Boehm, Wolfgang; Prautzsch, Hartmut. Geometric concepts for geometric design. Wellesley, Mass: A.K. Peters, 1994. ISBN
1568810040.
Gallier, Jean H. Geometric methods and applications : for computer science and engineering. New Yok [etc.]: SpringerVerlag, 2001. ISBN 0387950443.
Hoschek, Josef; Lasser, Dieter. Fundamentals of computer aided geometric design. Wellesley, Massachusetts: A. K. Peters,
1993. ISBN 1568810075.
Marsh, Duncan. Applied geometry for computer graphics and CAD [on line]. 2nd ed. London [etc.]: Springer, 2005
[Consultation: 06/11/2012]. Available on: <http://dx.doi.org/10.1007/b138823>. ISBN 1852338016.
Piegl, Les; Tiller, Wayne. The NURBS book. 2nd ed. Berlin [etc]: Springer, 1997. ISBN 3540615458.
Trias Pairó, Joan. Laboratori de geometria computacional. Barcelona: Edicions UPC, 2005. ISBN 8483018268.
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