Lecture 2- Curvature and Torsion of Curves | Calculus - Engineering Mathematics PDF Download

 Page 1


      Curvature and Torsion of Curves  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
  
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Curvature and Torsion of Curves 
Course Developer: Vivek N Sharma 
Department/College: Assistant Professor, Department of 
Mathematics, S.G.T.B. Khalsa College, University of Delhi 
 
 
 
 
 
  
Page 2


      Curvature and Torsion of Curves  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
  
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Curvature and Torsion of Curves 
Course Developer: Vivek N Sharma 
Department/College: Assistant Professor, Department of 
Mathematics, S.G.T.B. Khalsa College, University of Delhi 
 
 
 
 
 
  
      Curvature and Torsion of Curves  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
Table of Contents  
        
1. Learning Outcomes 
 
2. Introduction                                                                     
 
3. Curvature of a Plane Curve         
 
4. The Principal Unit Normal for a Plane Curve     
  
5. Circle of Curvature of a Plane Curve      
  
6. Curves in Space: Curvature & Normal Vectors      
 
7. Unit Binormal Vector for a Space Curve 
       
8. Torsion of a Space Curve 
   
9. Functions of Several Variables: Introduction 
10. Graphs & Level Curves 
11. Summary and Important Formulae 
12. Exercises  
13. Glossary and Further Reading 
14. Solutions for Exercises  
 
Page 3


      Curvature and Torsion of Curves  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
  
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Curvature and Torsion of Curves 
Course Developer: Vivek N Sharma 
Department/College: Assistant Professor, Department of 
Mathematics, S.G.T.B. Khalsa College, University of Delhi 
 
 
 
 
 
  
      Curvature and Torsion of Curves  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
Table of Contents  
        
1. Learning Outcomes 
 
2. Introduction                                                                     
 
3. Curvature of a Plane Curve         
 
4. The Principal Unit Normal for a Plane Curve     
  
5. Circle of Curvature of a Plane Curve      
  
6. Curves in Space: Curvature & Normal Vectors      
 
7. Unit Binormal Vector for a Space Curve 
       
8. Torsion of a Space Curve 
   
9. Functions of Several Variables: Introduction 
10. Graphs & Level Curves 
11. Summary and Important Formulae 
12. Exercises  
13. Glossary and Further Reading 
14. Solutions for Exercises  
 
      Curvature and Torsion of Curves  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
1. Learning Outcomes 
After studying this unit, you will be able to 
 
• state the concept of curvature of a plane curve. 
• calculate the curvature of various curves in plane and space. 
• explain the concept of torsion and binormal vectors for space 
 curves. 
• calculate torsion & binormal vectors of various space curves. 
• describe the meaning of a function of more than one variable. 
• analyse visually a function of two or three variables. 
• explain the concept of a level curve of a function of two or more            
        variables. 
 
2. Introduction: 
Geometric understanding of mathematics holds high importance in scientific 
analysis. It enables us to dig deeply about the question at hand. In this unit, 
we shall be studying the geometric properties of various plane and space 
curves.  
A very important aspect of drawing any curve is the amount of bending or 
twisting of the curve around any point. This is a very basic question 
regarding any curve we aim to draw on a plane or in space. And 
mathematicians have answered this question quite comfortably using 
calculus. They have formulated the notion of curvature of a plane curve and 
torsion of a space curve. These are the properties that we shall be learning 
in this unit.  
Curvature of a plane curve tells us how much does the curve bend or turn 
around a point. Torsion of a space curve reveals the tendency of the curve to 
move “away” from the plane. Since a plane curve always remains in the 
plane (that is it never escapes the plane to enter the space, its torsion is 
Page 4


      Curvature and Torsion of Curves  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
  
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Curvature and Torsion of Curves 
Course Developer: Vivek N Sharma 
Department/College: Assistant Professor, Department of 
Mathematics, S.G.T.B. Khalsa College, University of Delhi 
 
 
 
 
 
  
      Curvature and Torsion of Curves  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
Table of Contents  
        
1. Learning Outcomes 
 
2. Introduction                                                                     
 
3. Curvature of a Plane Curve         
 
4. The Principal Unit Normal for a Plane Curve     
  
5. Circle of Curvature of a Plane Curve      
  
6. Curves in Space: Curvature & Normal Vectors      
 
7. Unit Binormal Vector for a Space Curve 
       
8. Torsion of a Space Curve 
   
9. Functions of Several Variables: Introduction 
10. Graphs & Level Curves 
11. Summary and Important Formulae 
12. Exercises  
13. Glossary and Further Reading 
14. Solutions for Exercises  
 
      Curvature and Torsion of Curves  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
1. Learning Outcomes 
After studying this unit, you will be able to 
 
• state the concept of curvature of a plane curve. 
• calculate the curvature of various curves in plane and space. 
• explain the concept of torsion and binormal vectors for space 
 curves. 
• calculate torsion & binormal vectors of various space curves. 
• describe the meaning of a function of more than one variable. 
• analyse visually a function of two or three variables. 
• explain the concept of a level curve of a function of two or more            
        variables. 
 
2. Introduction: 
Geometric understanding of mathematics holds high importance in scientific 
analysis. It enables us to dig deeply about the question at hand. In this unit, 
we shall be studying the geometric properties of various plane and space 
curves.  
A very important aspect of drawing any curve is the amount of bending or 
twisting of the curve around any point. This is a very basic question 
regarding any curve we aim to draw on a plane or in space. And 
mathematicians have answered this question quite comfortably using 
calculus. They have formulated the notion of curvature of a plane curve and 
torsion of a space curve. These are the properties that we shall be learning 
in this unit.  
Curvature of a plane curve tells us how much does the curve bend or turn 
around a point. Torsion of a space curve reveals the tendency of the curve to 
move “away” from the plane. Since a plane curve always remains in the 
plane (that is it never escapes the plane to enter the space, its torsion is 
      Curvature and Torsion of Curves  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
zero. Both these properties help us visualize a curve clearly and explain its 
various properties. 
Curvature and torsion, both, are the local properties of a curve. A local 
property of a curve is the one which explains the geometry of the curve only 
“around a point”. For instance, a curve may bend “too much” around a point 
but just “too little” around another point. Therefore, the curvature of the 
curve can be zero around one point, but may be very large around another 
point. This explains why the curvature of a curve is a local property of the 
curve. These geometric considerations forced mathematicians to use calculus 
to understand the notion of curvature. Its further development led to the 
fascinating branch of mathematics called “Differential Geometry”. 
We shall first study curvature of plane curves and shall then extend this 
concept to space curves. 
 
3. Curvature of a Plane Curve: 
Intuitively, curvature explains how much does a curve bend (or turn) around 
a point. But whenever a curve turns, the tangent vector at that point 
changes its direction. And if a curve does not turn around a point, the 
tangent vector at that point will not change its direction around that point. 
In other words, as a particle moves along a smooth curve in a plane, the 
vector 
 
 ???? = 
???? ???? ???????? = Unit Tangent Vector of the curve  
 
turns as the curve bends. This is revealed in the figure overleaf. 
 
Page 5


      Curvature and Torsion of Curves  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
  
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Curvature and Torsion of Curves 
Course Developer: Vivek N Sharma 
Department/College: Assistant Professor, Department of 
Mathematics, S.G.T.B. Khalsa College, University of Delhi 
 
 
 
 
 
  
      Curvature and Torsion of Curves  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
Table of Contents  
        
1. Learning Outcomes 
 
2. Introduction                                                                     
 
3. Curvature of a Plane Curve         
 
4. The Principal Unit Normal for a Plane Curve     
  
5. Circle of Curvature of a Plane Curve      
  
6. Curves in Space: Curvature & Normal Vectors      
 
7. Unit Binormal Vector for a Space Curve 
       
8. Torsion of a Space Curve 
   
9. Functions of Several Variables: Introduction 
10. Graphs & Level Curves 
11. Summary and Important Formulae 
12. Exercises  
13. Glossary and Further Reading 
14. Solutions for Exercises  
 
      Curvature and Torsion of Curves  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
1. Learning Outcomes 
After studying this unit, you will be able to 
 
• state the concept of curvature of a plane curve. 
• calculate the curvature of various curves in plane and space. 
• explain the concept of torsion and binormal vectors for space 
 curves. 
• calculate torsion & binormal vectors of various space curves. 
• describe the meaning of a function of more than one variable. 
• analyse visually a function of two or three variables. 
• explain the concept of a level curve of a function of two or more            
        variables. 
 
2. Introduction: 
Geometric understanding of mathematics holds high importance in scientific 
analysis. It enables us to dig deeply about the question at hand. In this unit, 
we shall be studying the geometric properties of various plane and space 
curves.  
A very important aspect of drawing any curve is the amount of bending or 
twisting of the curve around any point. This is a very basic question 
regarding any curve we aim to draw on a plane or in space. And 
mathematicians have answered this question quite comfortably using 
calculus. They have formulated the notion of curvature of a plane curve and 
torsion of a space curve. These are the properties that we shall be learning 
in this unit.  
Curvature of a plane curve tells us how much does the curve bend or turn 
around a point. Torsion of a space curve reveals the tendency of the curve to 
move “away” from the plane. Since a plane curve always remains in the 
plane (that is it never escapes the plane to enter the space, its torsion is 
      Curvature and Torsion of Curves  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
zero. Both these properties help us visualize a curve clearly and explain its 
various properties. 
Curvature and torsion, both, are the local properties of a curve. A local 
property of a curve is the one which explains the geometry of the curve only 
“around a point”. For instance, a curve may bend “too much” around a point 
but just “too little” around another point. Therefore, the curvature of the 
curve can be zero around one point, but may be very large around another 
point. This explains why the curvature of a curve is a local property of the 
curve. These geometric considerations forced mathematicians to use calculus 
to understand the notion of curvature. Its further development led to the 
fascinating branch of mathematics called “Differential Geometry”. 
We shall first study curvature of plane curves and shall then extend this 
concept to space curves. 
 
3. Curvature of a Plane Curve: 
Intuitively, curvature explains how much does a curve bend (or turn) around 
a point. But whenever a curve turns, the tangent vector at that point 
changes its direction. And if a curve does not turn around a point, the 
tangent vector at that point will not change its direction around that point. 
In other words, as a particle moves along a smooth curve in a plane, the 
vector 
 
 ???? = 
???? ???? ???????? = Unit Tangent Vector of the curve  
 
turns as the curve bends. This is revealed in the figure overleaf. 
 
      Curvature and Torsion of Curves  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
 
Figure 1: The unit tangent vector ???? =
???? ???? ???????? at the point ???? turns along the curve 
as the point ???? proceeds along the curve. 
 
Since ???? is a unit vector, its length remains constant and only its direction 
changes as the particle moves along the curve. Now we are in a position to 
define the curvature of the smooth curve. 
 
3.1 Definition of Curvature: The rate at which ???? turns per unit of length 
along the curve is called the curvature of the curve. Thus, if ???? is the unit 
tangent vector of a smooth curve ???? ( ???? ), the curvature function ???? ( ???? ) of the 
curve is 
 ???? ( ???? ) =|
???? ???? ???????? ( ???? )|. 
 
Value Addition : Remarks 
• |
???? ???? ???????? | = 0 
• If |
???? ???? ???????? | = 0, the curve does not turn at all as the particle passes through 
the point ???? & the curvature of the curve at ???? is zero. 
• If |
???? ???? ???????? | is large, ???? turns sharply as the particle passes through the point ???? . 
• If |
???? ???? ???????? | is close to zero, ???? turns more slowly as the particle passes through 
the point ???? and the curvature of the curve at ???? is smaller.