Theory of Splines-II (MATH-436)

The goal of the course is to provide the students with a strong background in numerical approximation strategies and basic knowledge on the theory of splines that supports numerical algorithms. Interactive graphics techniques for defining and manipulating geometrical shapes used in computer animation, car body design, aircraft design, and architectural design. 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.

Learning Outcomes

This course gives the students the knowledge of problem classes, basic mathematical and numerical concepts, and software for the solution of engineering and scientific problems formulated as using data sets. After successful completion, students should be able to design, implement, and use interpolations for the computer solution of scientific problems involving problems generated by a set of data.

Contents

Parametric curves (scalar and vector case), Algebraic form.
Hermite form, control point form, Bernstein Bezier form.
Matrix forms of parametric curves.
Algorithms to compute B.B. form, Convex hull property.
Affine invariance property, Variation diminishing property.
Rational quadratic form.
Rational cubic form.
Tensor product surface, B.B. cubic patch.
Quadratic by cubic B.B. patch, B.B. quartic patch.
Splines, Cubic splines,
End conditions of cubic splines, Clamped conditions.
Natural conditions, second derivative conditions.
Periodic conditions.
Not a knot conditions,
General splines, Natural splines,
Periodic splines,
Truncated power function, Representation of spline in terms of truncated power functions,
Odd degree interpolating splines.

Recommended Books

Farin G. Curves and Surfaces for Computer-Aided Geometric Design A Practical Guide. 5th ed. (Academic Press, 2002.)
Faux I.D. Computational Geometry for Design and Manufacture. 1st ed. (Ellis Horwood, 1979.)

Suggested Books

Bartle H.R, Beatly C.J. An Introduction to Spline for use in Computer Graphics and Geometric Modeling. 4th ed. (Morgan Kaufmann, 2006 )
Boor C.D. A Practical Guide to Splines. Revised ed. (Springer Verlag, 2001.)

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 10, Attendance 05, Quiz 05)

Mid-Term Exam: 30

Final-Term Exam: 50