Calculus for Computer Graphics

Students studying different branches of computer graphics have to be familiar with geometry, matrices, vectors, rotation transforms, quaternions, curves and surfaces and as computer graphics software becomes increasingly sophisticated, calculus is also being used to resolve its associated problems. ...

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Bibliographic Details
Main Author Vince, John
Format eBook
LanguageEnglish
Published Cham Springer Nature 2019
Springer International Publishing AG
Springer International Publishing
Springer
Edition2
Subjects
Computer Graphics
Computer graphics-Mathematics
Computer programming, programs, data
Computer Science
Mathematical Applications in Computer Science
Online AccessGet full text
ISBN3030113760
9783030113766
3030113752
9783030113759
DOI10.1007/978-3-030-11376-6

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Table of Contents:
  • Intro -- Preface -- Contents -- 1 Introduction -- 1.1 What is Calculus? -- 1.2 Where is Calculus Used in Computer Graphics? -- 1.3 Who Invented Calculus? -- 2 Functions -- 2.1 Introduction -- 2.2 Expressions, Variables, Constants and Equations -- 2.3 Functions -- 2.3.1 Continuous and Discontinuous Functions -- 2.3.2 Linear Functions -- 2.3.3 Periodic Functions -- 2.3.4 Polynomial Functions -- 2.3.5 Function of a Function -- 2.3.6 Other Functions -- 2.4 A Function's Rate of Change -- 2.4.1 Slope of a Function -- 2.4.2 Differentiating Periodic Functions -- 2.5 Summary -- 3 Limits and Derivatives -- 3.1 Introduction -- 3.2 Small Numerical Quantities -- 3.3 Equations and Limits -- 3.3.1 Quadratic Function -- 3.3.2 Cubic Equation -- 3.3.3 Functions and Limits -- 3.3.4 Graphical Interpretation of the Derivative -- 3.3.5 Derivatives and Differentials -- 3.3.6 Integration and Antiderivatives -- 3.4 Summary -- 3.5 Worked Examples -- 3.5.1 Limiting Value of a Quotient -- 3.5.2 Limiting Value of a Quotient -- 3.5.3 Derivative -- 3.5.4 Slope of a Polynomial -- 3.5.5 Slope of a Periodic Function -- 3.5.6 Integrate a Polynomial -- 4 Derivatives and Antiderivatives -- 4.1 Introduction -- 4.2 Differentiating Groups of Functions -- 4.2.1 Sums of Functions -- 4.2.2 Function of a Function -- 4.2.3 Function Products -- 4.2.4 Function Quotients -- 4.2.5 Summary: Groups of Functions -- 4.3 Differentiating Implicit Functions -- 4.4 Differentiating Exponential and Logarithmic Functions -- 4.4.1 Exponential Functions -- 4.4.2 Logarithmic Functions -- 4.4.3 Summary: Exponential and Logarithmic Functions -- 4.5 Differentiating Trigonometric Functions -- 4.5.1 Differentiating tan -- 4.5.2 Differentiating csc -- 4.5.3 Differentiating sec -- 4.5.4 Differentiating cot -- 4.5.5 Differentiating arcsin, arccos and arctan -- 4.5.6 Differentiating arccsc, arcsec and arccot
  • 11.5.2 Volume of a Cylinder -- 11.5.3 Volume of a Sphere -- 11.5.4 Volume of a Cone -- 11.6 Summary -- 11.6.1 Summary of Formulae -- 12 Vector-Valued Functions -- 12.1 Introduction -- 12.2 Differentiating Vector Functions -- 12.2.1 Velocity and Speed -- 12.2.2 Acceleration -- 12.2.3 Rules for Differentiating Vector-Valued Functions -- 12.3 Integrating Vector-Valued Functions -- 12.3.1 Velocity of a Falling Object -- 12.3.2 Position of a Moving Object -- 12.4 Summary -- 12.4.1 Summary of Formulae -- 13 Tangent and Normal Vectors -- 13.1 Introduction -- 13.2 Notation -- 13.3 Tangent Vector to a Curve -- 13.4 Normal Vector to a Curve -- 13.5 Gradient of a Scalar Field -- 13.5.1 Unit Tangent and Normal Vectors to a Line -- 13.5.2 Unit Tangent and Normal Vectors to a Parabola -- 13.5.3 Unit Tangent and Normal Vectors to a Circle -- 13.5.4 Unit Tangent and Normal Vectors to an Ellipse -- 13.5.5 Unit Tangent and Normal Vectors to a Sine Curve -- 13.5.6 Unit Tangent and Normal Vectors to a cosh Curve -- 13.5.7 Unit Tangent and Normal Vectors to a Helix -- 13.5.8 Unit Tangent and Normal Vectors to a Quadratic Bézier Curve -- 13.6 Unit Tangent and Normal Vectors to a Surface -- 13.6.1 Unit Normal Vectors to a Bilinear Patch -- 13.6.2 Unit Normal Vectors to a Quadratic Bézier Patch -- 13.6.3 Unit Tangent and Normal Vector to a Sphere -- 13.6.4 Unit Tangent and Normal Vectors to a Torus -- 13.7 Summary -- 13.7.1 Summary of Formulae -- 14 Continuity -- 14.1 Introduction -- 14.2 B-Splines -- 14.2.1 Uniform B-Splines -- 14.2.2 B-Spline Continuity -- 14.3 Derivatives of a Bézier Curve -- 14.4 Summary -- 15 Curvature -- 15.1 Introduction -- 15.2 Curvature -- 15.2.1 Curvature of a Circle -- 15.2.2 Curvature of a Helix -- 15.2.3 Curvature of a Parabola -- 15.2.4 Parametric Plane Curve -- 15.2.5 Curvature of a Graph -- 15.2.6 Curvature of a 2D Quadratic Bézier Curve
  • 4.5.7 Summary: Trigonometric Functions -- 4.6 Differentiating Hyperbolic Functions -- 4.6.1 Differentiating sinh, cosh and tanh -- 4.6.2 Differentiating cosech, sech and coth -- 4.6.3 Differentiating arsinh, arcosh and artanh -- 4.6.4 Differentiating arcsch, arsech and arcoth -- 4.6.5 Summary: Hyperbolic Functions -- 4.7 Summary -- 5 Higher Derivatives -- 5.1 Introduction -- 5.2 Higher Derivatives of a Polynomial -- 5.3 Identifying a Local Maximum or Minimum -- 5.4 Derivatives and Motion -- 5.5 Summary -- 5.5.1 Summary of Formulae -- 6 Partial Derivatives -- 6.1 Introduction -- 6.2 Partial Derivatives -- 6.2.1 Visualising Partial Derivatives -- 6.2.2 Mixed Partial Derivatives -- 6.3 Chain Rule -- 6.4 Total Derivative -- 6.5 Second-Order and Higher Partial Derivatives -- 6.6 Summary -- 6.6.1 Summary of Formulae -- 7 Integral Calculus -- 7.1 Introduction -- 7.2 Indefinite Integral -- 7.3 Standard Integration Formulae -- 7.4 Integration Techniques -- 7.4.1 Continuous Functions -- 7.4.2 Difficult Functions -- 7.4.3 Trigonometric Identities -- 7.4.4 Exponent Notation -- 7.4.5 Completing the Square -- 7.4.6 The Integrand Contains a Derivative -- 7.4.7 Converting the Integrand into a Series of Fractions -- 7.4.8 Integration by Parts -- 7.4.9 Integration by Substitution -- 7.4.10 Partial Fractions -- 7.5 Summary -- 8 Area Under a Graph -- 8.1 Introduction -- 8.2 Calculating Areas -- 8.3 Positive and Negative Areas -- 8.4 Area Between Two Functions -- 8.5 Areas with the y-Axis -- 8.6 Area with Parametric Functions -- 8.7 Bernhard Riemann -- 8.7.1 Domains and Intervals -- 8.7.2 The Riemann Sum -- 8.8 Summary -- 9 Arc Length and Parameterisation of Curves -- 9.1 Introduction -- 9.2 Lagrange's Mean-Value Theorem -- 9.3 Arc Length -- 9.3.1 Arc Length of a Straight Line -- 9.3.2 Arc Length of a Circle -- 9.3.3 Arc Length of a Parabola -- 9.3.4 Arc Length of y=x32
  • 15.2.7 Curvature of a 2D Cubic Bézier Curve -- 15.3 Summary -- 15.3.1 Summary of Formulae -- 16 Conclusion -- A Limit of (sinθ)/θ -- B Integrating cosnθ -- Index
  • 9.3.5 Arc Length of a Sine Curve -- 9.3.6 Arc Length of a Hyperbolic Cosine Function -- 9.3.7 Arc Length of Parametric Functions -- 9.3.8 Arc Length of a Circle -- 9.3.9 Arc Length of an Ellipse -- 9.3.10 Arc Length of a Helix -- 9.3.11 Arc Length of a 2D Quadratic Bézier Curve -- 9.3.12 Arc Length of a 3D Quadratic Bézier Curve -- 9.3.13 Arc Length Parameterisation of a 3D Line -- 9.3.14 Arc Length Parameterisation of a Helix -- 9.3.15 Positioning Points on a Straight Line Using a Square Law -- 9.3.16 Positioning Points on a Helix Curve Using a Square Law -- 9.3.17 Arc Length Using Polar Coordinates -- 9.4 Summary -- 9.4.1 Summary of Formulae -- References -- 10 Surface Area -- 10.1 Introduction -- 10.2 Surface of Revolution -- 10.2.1 Surface Area of a Cylinder -- 10.2.2 Surface Area of a Right Cone -- 10.2.3 Surface Area of a Sphere -- 10.2.4 Surface Area of a Paraboloid -- 10.3 Surface Area Using Parametric Functions -- 10.4 Double Integrals -- 10.5 Jacobians -- 10.5.1 1D Jacobian -- 10.5.2 2D Jacobian -- 10.5.3 3D Jacobian -- 10.6 Double Integrals for Calculating Area -- 10.7 Summary -- 10.7.1 Summary of Formulae -- 11 Volume -- 11.1 Introduction -- 11.2 Solid of Revolution: Disks -- 11.2.1 Volume of a Cylinder -- 11.2.2 Volume of a Right Cone -- 11.2.3 Volume of a Right Conical Frustum -- 11.2.4 Volume of a Sphere -- 11.2.5 Volume of an Ellipsoid -- 11.2.6 Volume of a Paraboloid -- 11.3 Solid of Revolution: Shells -- 11.3.1 Volume of a Cylinder -- 11.3.2 Volume of a Right Cone -- 11.3.3 Volume of a Sphere -- 11.3.4 Volume of a Paraboloid -- 11.4 Volumes with Double Integrals -- 11.4.1 Objects with a Rectangular Base -- 11.4.2 Rectangular Box -- 11.4.3 Rectangular Prism -- 11.4.4 Curved Top -- 11.4.5 Objects with a Circular Base -- 11.4.6 Cylinder -- 11.4.7 Truncated Cylinder -- 11.5 Volumes with Triple Integrals -- 11.5.1 Rectangular Box