# Engineer

Topics: Interpolation, Non-uniform rational B-spline, B-spline Pages: 25 (7636 words) Published: February 7, 2013
September 3, 2012 NURBS CURVE MATHEMATICAL REPRESENTATION AND NURBS BASE INTERPOLATION TECHNIQUES FOR CNC PROGRAMMING
Biruk Mamo Wodaj
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ABSTRACT
This term paper is prepared to partially fulfill a requirement of graduate studies program in industrial engineering for the course „FLEXIBLE MANUFACTURING SYSTEMS‟. The term paper studies on NURBS (Non-Uniform Rational B-Splines) curve Mathematical Representation and NURBS based Interpolations Techniques for CNC programming. Because of its importance in both high-speed machining and tool-path generation, approximating a contour by NURBS format is a potential function in CAD/CAM/CNC systems. NURBS curve is one of the most commonly used tools in CAD systems and geometric modeling for its various specialties, which mean that its shape is locally adjustable as well as its continuity order and it, can represent a conic curve precisely. Over the recent years, different curves and surface representation forms have been proposed. Currently, B-Splines and NURBS (Non-Uniform B-Splines ) are the most popular mathematical forms. NURBS offer a unified mathematical form not only for representation of free-form curves and surfaces but also for precise representation of close form shapes such as lines, conics, quadrics and surfaces of revolution. In the following sections, there is an introduction, a detail analysis of NURBS mathematical representation and NURBS base interpolations techniques discussed. Finally the study provides a conclusion. September 3, 2012 NURBS CURVE MATHEMATICAL REPRESENTATION AND NURBS BASE INTERPOLATION TECHNIQUES FOR CNC PROGRAMMING

Biruk Mamo Wodaj
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ACKNOWLEDGMENT
I would like to deeply praise the Holy Lord, God for allowing me passing all of this moment; nothing is possible without you. Special thanks are given to our instructor Dr. Tafesse Gebresenbet for the opportunity he provide in the preparation of a term paper about NURBS Curves Mathematical Representation and NURBS base Interpolation Techniques for CNC Programming. Finally I would like to say thank you to my parents and my friends for their support and encouragement. Thanks,

Biruk Mamo, September 2012
September 3, 2012 NURBS CURVE MATHEMATICAL REPRESENTATION AND NURBS BASE INTERPOLATION TECHNIQUES FOR CNC PROGRAMMING
Biruk Mamo Wodaj
Page 3 of 33
ACKNOWLEDGMENT ............................................................................................................... 2 LIST OF TABLES ......................................................................................................................... 4 LIST OF FIGURES ....................................................................................................................... 5 CHAPTER ONE ............................................................................................................................ 6 1. NON-UNIFORM RATIONAL B-SPLINE CURVES (NURBS) ........................................... 6 1.1. INTRODUCTION .............................................................................................................................. 6 1.2. HISTORY OF NURBS ...................................................................................................................... 7 1.3. INTRODUCTION TO NURBS ......................................................................................................... 7 1.4. CHARACTERSTICS OF NURBS ..................................................................................................... 8 CHAPTER TWO ......................................................................................................................... 13 2. NURBS CURVE MATHEMATICAL REPRESENTATION ............................................. 13 2.1. NURBS BASIS FUNCTIONS ......................................................................................................... 13 2.2 NURBS CURVE MATHEMATICAL REPRESENTATION...