Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Licenţă
Domeniul: Informatică
Programul de studii: Informatică - în limba engleză
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MMG0002	Geometry	

Semester	Hours: C+S+L	Category	Status	
2	3+2+0	fundamental	compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
ŢOPAN Liana Manuela	Ph.D.	Lect.	Geometry	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
The purpose of the course is to introduce the basic elements in analytic and differential geometry, with applications in computer graphics.	

IV. Course contents
I. Analytic Geometry
1.	Systems of Coordinates (for 2-dim. and 3-dim. Euclidean spaces)
2.	Vectors
3.	Two-Dimensional Analytic Geometry 
4.	Three-Dimensional Analytic Geometry
5.	Conics 
6.	Quadric Surfaces
7.	Revolution Surfaces
II. Geometric Transformations (reflexions, translations, scalings, projections)
III. Differential Geometry of Curves and Surfaces
1.	Plane curves. Tangent and normal lines, curvature, envelope, evolute
2.	Space Cuves. Tangent lines, normal planes, curvature and torsion
3.	Surfaces. Tangent planes, normal lines, curves on a surfaces, first and second fundamental forms, curvatures.	

V. Bibliography
1. Andrica, D., Ţopan, L., Analytic Geometry, Cluj University Press, 2004
2. Blaga, P.A., Lectures on Classical Differential Geometry, Risoprint, 2005
3. Eggerton, P.A., Hall, W.S., Computer Graphics. Mathematical First Steps, Prentice Hall, 1999	

VI. Thematic of didactic activities per weeks
Week 1. Vectors. Coordinates, operations, components, linearly dependence.
              Systems of coordinates: (plane and space) affine system, rectangular coordinates, polar  
              coordinates, cylindrical coordinates, sperical coordinates. Change of coordinates.
              Vectors. Dot product, projections, cross product, triple scalar product. [1 ch 1;  3 ch 2, 3, 4]
Week 2: Two-dimensional analytic geometry. Several equations of lines. Parallelism and  
              orthogonality. Intersection of lines. Bundle of lines.
              Three-dimensional analytic geometry. Analytic representation of planes. Analytic  
              representation of lines. [1 ch 2;  3 ch 5, 6]
Week 3. Three-dimensional analytic geometry. Relative positions of lines and planes. Angles. 
              Distances. [1 ch 3;  3 ch 6]
Week 4. Conics defined through canonical equations. Circles, ellipses, hyperbolas, parabolas. 
              Conics defined through a general equation. Methods for graphical representation. [1 ch 5;  3 
              ch7]
Week 5. Quadric surfaces. Ellipsoids, hyperboloids of one sheet, hyperboloids of two sheets, cones, 
              elliptic paraboloids, hyperbolic paraboloids, cylinders.
              Surfaces of revolution. Cylindrical surfaces, conical surfaces, conoidal surfaces, revolution 
              surfaces. [1 ch 6, 7]
Week 6. Two-dimensional transformations. Scaling. Reflecting through an axis. Transformation of 
              lines, of segments, midpoint transformation, transformation of parallel and intersecting lines. 
              [3 ch 9-9.1, 9.2] 
Week 7. Two-dimensional transformations. Rotations about the origin. Combined transformation. 
              Transformation of the unit square. Conformal transformations. [3 ch 9-9.4, 9.5, ch 10]
Week 8. Two-dimensional transformations. Translations and homogeneous coordinates. Rotations 
               about un arbitrary point. Reflexions through an arbitrary line. [3 ch 10, 11, 12]
Week 9. Two-dimensional transformations. Projections. Overall scaling. Points at infinity. 
              Three-dimensional transformations. Scaling, shearing. Rotation about a coordinate axis. 
              Reflection through a coordinate plane. Translation. [3 ch 13]
Week 10. Three-dimensional transformations. Rotations about an axis parallel to a coordinate axis. 
               Rotations about an arbitrary axis. Reflections through an arbitrary plane. Orthographic 
               projections. Axonometric projections (trimetric projections, dimetric projections, isometric 
               projections). Perspective transformations.  [3 ch 14]
Week 11. Plane curves. Arc length and unit speed reparametrization. The tangent and the normal to a 
               plane curve. The Frenet frame. The curvature.  [2 ch 1-1.4.1, 1.5.1, ch 2-2.3]
Week 12. Plane curves. The Frenet formulae. The signed curvature. Intrinsic equations. Envelopes of 
                plane curves. The evolute of a plane curve. [2 ch 2-2.3, 2.4, 2.5, 2.6]
Week 13. Space curves. Arc length and unit speed reparametrization. The tangent line and the normal 
                 plane for a space curve. The curvature of a space curve. The Frenet frame, the Frenet  
                 equations. The torsion. [2 ch 1-1.4.2., 1.5, 1.6, 1.7, 1.8, 1.9, 1.10, 1.14]
Week 14. Surfaces. Cuves on a surface. The tangent plane and the normal line to a surface. The first 
                fundamental form. The length of a segment of curve on a surface. The angle of two curves on 
                a surface. The second fundamental form of an oriented surface. Normal curvature of a 
            curve on a surface. Principal directions, principal curvatures.  Total and mean curvature. [2 ch 4]	

VII. Didactic methods used
Exposition, didactic proof, problematization, algorithmization, exercise	

VIII. Assessment
Viva voce final exam. 	

IX. Additional references
1.	Davies, A., Samuels, P., An introduction to computational geometry for curves and surfaces, Clarendon Press, Oxford, 1996
2.	Pintea, C., Geometrie. Presa Universitară Clujeană, 2001
3.	Raussen, M., Elementary differential geometry: Curves and surfaces, 2003
4.	Rogers, D.F., Adams, J.A., Mathematical elements for computer graphics, McGrow-Hill Publishing Company, 1990	

	Date,	22.04.2007						Course responsible,
								Lect. ŢOPAN Liana Manuela, Ph.D.