The primary goals are to (1) learn about different parametric curve and surface schemes, (2) understand the advantages and disadvantages of different geometry representations, (3) gain practical experience by implementing several CAGD techniques and user interfaces and (4) to apply CAGD methods to practical applications. Interactive graphics techniques for defining and manipulating geometrical shapes used in computer animation, car body design, aircraft design, and architectural design.

*Learning Outcomes*

- Students are able to write graphics applications using Mathematica, Matlab, C/C++, OpenGL and GLUT.
- Students are able to write concisely structured and documented application programs.
- Students are able to work with analytical models for curves, surfaces and volumetric models.
- Students are able to implement a complete modeling package using standard CAGD concepts

*Prerequisites: Nill*

*Contents*

- Linear interpolation, Piecewise linear interpolation blossoms, Barycentric coordinates in the plane, The de Casteljau algorithm, Properties of Bezier curves, Bernstein polynomials
- Composite Bezier curves, Degree elevation, The variation diminishing property
- Degree reduction, Polynomial curve constructions
- Aitken’s Algorithm, Lagrange Polynomials, Lagrange interpolation, Cubic Hermite interpolation
- Point-normal interpolation, B-Spline curves, B-spline segments, Knot insertion, degree elevation
- Greville Abscissae, Smoothness
- Constructing Splines Curves, Greville interpolation, modifying B-Spline curves
- Cubic spline interpolation, the minimum property
- Piecewise cubic interpolation.
- Rational Bezier and B-Spline Curves,
- Rational Bezier curves,
- Rational Cubic B-spline curves.

*Recommended Books*

- Gerald F. (2002),
*Curves and surfaces for CAGD, a practical guide (*5th ed.). Massachusetts: Morgan Kaufmann Publishers. - Bartels, R. H., Bealty, J. C. and Beatty, J. C. (2006).
*An Introduction to spline for use in computer graphics and geometric modeling.*Massachusetts: Morgan Kaufmann Publishers.

*Suggested Books *

- Josef H., Dieter L. (1993).
*Fundamentals of computer aided geometric design.*Massachusetts: A K Peter, Ltd. - de Boor, C. (2001).
*A practical guide to splines*. New York: Springer Verlag - Wang, R. H. (2005).
*Multivariate spline functions and their applications (mathematics and its applications)*. Netherland: Science Press/ Kluwer Academic Publishers. - Recent Published papers

**RESEARCH PROJECT /PRACTICALS/LABS/ASSIGNMENTS**

The projects assigned in this course follow a modular approach and contribute different components to the development of an interactive curve and surface modeling system.

- Curve Modeling Techniques: Students will implement various curve interpolation and approximation techniques that allow the interactive specification of three-dimensional curves (e.g. Bezier, B-spline, rational curves).
- Surface modeling techniques: Students will implement various surface interpolation and approximation techniques that allow the interactive specification of three-dimensional surfaces (e.g. Bezier, B-spline, rational surfaces).
- Simple, 3D Modeling System: Students will integrate the curve and surface modules into a system that allows the user to interactively design and store simple, 3D geometries.

**ASSESSMENT CRITERIA**

Sessional: 20 (Presentation / Assignment 04, Attendance 08, Result Mid-Term 04, Quiz 04

Mid-Term Exam: 30

Final-Term Exam: 50

Monday--Tuesday 3:30pm-5: 00pm (MPhil-I (SS))

Commencement of Classes November 02, 2020

Mid Term Examination December 28, 2020

Final Term Examination March 01, 2021

Result Decleration March 12, 2021