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Computer Graphics Introduction To Curves

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UNIT 2-Geometric Modeling.pptx

Unit 2 discusses geometric modeling techniques. It covers representation of curves using Hermite, Bezier, and B-spline curves. It also discusses surface modeling techniques including surface patches, Coons and bicubic patches, and Bezier and B-spline surfaces. Solid modeling techniques of CSG (Constructive Solid Geometry) and B-rep (Boundary Representation) are also introduced.

5_6221983039971394498.pptx

Unit-V discusses curves and fractals. Curves are continuous maps from a one-dimensional space to an n-dimensional space. Curves can be represented explicitly, implicitly, or parametrically. Splines are commonly used curves that are constructed by specifying control points and interpolating between them. Bezier curves are another type of curve defined by control points, where the first and last points are the endpoints and the other points control the tangents. Both splines and Bezier curves can be subdivided recursively to render smooth curves.

Bezier Curve and Spline Curve

- Bezier curves and spline curves are parametric curves used in computer graphics and design. Bezier curves were developed in the 1960s and use control points to define quadratic, cubic, or higher order curves. Spline curves use piecewise polynomials to represent complex curves and surfaces for data interpolation and smoothing.
- Both curves allow for local control of the shape and can interpolate or approximate data points. Bezier curves have properties like the first/last control points defining the endpoints, and the curve lying within the convex hull of all control points. Higher order curves are constructed by combining lower order curve sections.

Introduction to the curves

This document provides an introduction to different types of curves used in computer graphics. It discusses curve continuity, conic curves such as parabolas and hyperbolas, piecewise curves, parametric curves, spline curves, Bezier curves, B-spline curves, and applications of fractals. Key points covered include the four types of continuity, how conic curves are defined by discriminant functions, using control points to define piecewise, spline, Bezier and B-spline curves, and properties of Bezier curves such as passing through the first and last control points.

Curves wire frame modelling

The document discusses wireframe modeling in computer-aided design. It defines wireframe modeling as consisting only of points and curves without faces. A wireframe model uses a vertex table and edge table to define geometry. Advantages include ease of creation and low hardware/software requirements, while disadvantages include difficulty visualizing designs. The document also covers classifications of curves used in wireframe modeling like analytical, synthetic, and parametric curves. It provides examples of representing common curves like lines, circles, and splines parametrically.

Elhabian_curves10.pdf

This document discusses curves and surfaces. It begins by introducing parametric curves, which are defined by continuous functions of a parameter. Important properties of curves discussed include affine invariance and satisfying the convex hull property. Several interpolation techniques for constructing curves through given points are covered, including Lagrange interpolation and Bézier curves. Bézier curves are constructed using Bernstein polynomials and have desirable properties like the convex hull property. The document shows how to define piecewise continuous Bézier curves and discusses algorithms for evaluating and rendering Bézier curves.

Unit 2 curves & surfaces

Representation of curves- Hermite curve- Bezier curve- B-spline curves-rational curves-Techniques
for surface modeling – surface patch- Coons and bicubic patches- Bezier and B-spline surfaces.

UNIT 2- GEOMETRIC MODELLING

Representation of curves- Hermite curve- Bezier curve- B-spline curves-rational curves-Techniques for surface modeling – surface patch- Coons and bicubic patches- Bezier and B-spline surfaces. Solid modeling techniques- CSG andB-rep

UNIT 2-Geometric Modeling.pptx

Unit 2 discusses geometric modeling techniques. It covers representation of curves using Hermite, Bezier, and B-spline curves. It also discusses surface modeling techniques including surface patches, Coons and bicubic patches, and Bezier and B-spline surfaces. Solid modeling techniques of CSG (Constructive Solid Geometry) and B-rep (Boundary Representation) are also introduced.

5_6221983039971394498.pptx

Unit-V discusses curves and fractals. Curves are continuous maps from a one-dimensional space to an n-dimensional space. Curves can be represented explicitly, implicitly, or parametrically. Splines are commonly used curves that are constructed by specifying control points and interpolating between them. Bezier curves are another type of curve defined by control points, where the first and last points are the endpoints and the other points control the tangents. Both splines and Bezier curves can be subdivided recursively to render smooth curves.

Bezier Curve and Spline Curve

- Bezier curves and spline curves are parametric curves used in computer graphics and design. Bezier curves were developed in the 1960s and use control points to define quadratic, cubic, or higher order curves. Spline curves use piecewise polynomials to represent complex curves and surfaces for data interpolation and smoothing.
- Both curves allow for local control of the shape and can interpolate or approximate data points. Bezier curves have properties like the first/last control points defining the endpoints, and the curve lying within the convex hull of all control points. Higher order curves are constructed by combining lower order curve sections.

Introduction to the curves

This document provides an introduction to different types of curves used in computer graphics. It discusses curve continuity, conic curves such as parabolas and hyperbolas, piecewise curves, parametric curves, spline curves, Bezier curves, B-spline curves, and applications of fractals. Key points covered include the four types of continuity, how conic curves are defined by discriminant functions, using control points to define piecewise, spline, Bezier and B-spline curves, and properties of Bezier curves such as passing through the first and last control points.

Curves wire frame modelling

The document discusses wireframe modeling in computer-aided design. It defines wireframe modeling as consisting only of points and curves without faces. A wireframe model uses a vertex table and edge table to define geometry. Advantages include ease of creation and low hardware/software requirements, while disadvantages include difficulty visualizing designs. The document also covers classifications of curves used in wireframe modeling like analytical, synthetic, and parametric curves. It provides examples of representing common curves like lines, circles, and splines parametrically.

Elhabian_curves10.pdf

This document discusses curves and surfaces. It begins by introducing parametric curves, which are defined by continuous functions of a parameter. Important properties of curves discussed include affine invariance and satisfying the convex hull property. Several interpolation techniques for constructing curves through given points are covered, including Lagrange interpolation and Bézier curves. Bézier curves are constructed using Bernstein polynomials and have desirable properties like the convex hull property. The document shows how to define piecewise continuous Bézier curves and discusses algorithms for evaluating and rendering Bézier curves.

Unit 2 curves & surfaces

Representation of curves- Hermite curve- Bezier curve- B-spline curves-rational curves-Techniques
for surface modeling – surface patch- Coons and bicubic patches- Bezier and B-spline surfaces.

UNIT 2- GEOMETRIC MODELLING

Representation of curves- Hermite curve- Bezier curve- B-spline curves-rational curves-Techniques for surface modeling – surface patch- Coons and bicubic patches- Bezier and B-spline surfaces. Solid modeling techniques- CSG andB-rep

1516 contouring

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics

CAD - UNIT 2 (Geometric Modelling)

This document provides an overview of geometric modeling techniques used in computer aided design (CAD). It discusses representation of curves including Hermite curves, Bezier curves, B-spline curves, and rational curves. It also covers surface modeling techniques such as surface patches, Coons patches, and Bicubic patches. For solid modeling, it describes constructive solid geometry (CSG) and boundary representation (B-rep) techniques. CSG uses boolean operations on primitives to create models while B-rep defines models based on their bounding faces, edges and vertices.

ME6501 Unit 2 geometric modeling

This document provides an overview of geometric modeling techniques used in computer aided design (CAD). It discusses representation of curves including Hermite curves, Bezier curves, B-spline curves, and rational curves. It also covers surface modeling techniques such as surface patches, Coons patches, bicubic patches, Bezier surfaces, and B-spline surfaces. For solid modeling, it describes constructive solid geometry (CSG) and boundary representation (B-rep) techniques. CSG uses boolean operations on primitives to create models while B-rep models objects based on their bounding faces, edges, and vertices.

cg mod2.pdf

- Polygon fill areas are useful for describing picture components with a solid color or pattern. Common polygon shapes include triangles, rectangles, and octagons.
- Polygons can be classified as convex or concave. Concave polygons are more complicated to process than convex polygons.
- OpenGL provides functions like glBegin and glVertex for specifying polygon vertices and drawing polygon fill areas. Common primitives include GL_POLYGON, GL_TRIANGLES, GL_TRIANGLE_STRIP, and GL_TRIANGLE_FAN.

Surface models

This document discusses different types of surface models used in computer graphics, including:
- Plane, ruled, surface of revolution, tabulated, bilinear, Coons patch, and bicubic surfaces. Plane and ruled surfaces are linear, while surfaces of revolution and tabulated surfaces are axisymmetric. Bilinear surfaces are generated by interpolating 4 endpoints and are useful for finite element analysis. Coons patches interpolate 4 edge curves. Bicubic surfaces use parametric curves and interpolation of control points to define smooth surfaces.

10_1425_web_Lec_04_2D_Motion.pdf

This document discusses vectors and their application to motion in two and three dimensions. It introduces vectors as representations of displacement that have both magnitude and direction. Displacement, velocity and acceleration can all be described as vectors, allowing their addition and subtraction using graphical vector addition. Vector components in Cartesian coordinates are also discussed. The document uses examples of particle motion to illustrate average and instantaneous velocity and acceleration in two dimensions and the concept of relative velocity.

Curve modeling bezier curves

This document discusses Bézier curves and their properties. It begins by stating that traditional parametric curves are not very geometric and do not provide intuitive shape control. It then outlines desirable properties for curve design systems, including being intuitive, flexible, easy to use, providing a unified approach for different curve types, and producing invariant curves under transformations. The document proceeds to discuss Bézier, B-spline and NURBS curves which address these properties by allowing users to manipulate control points to modify curve shapes. Key properties of Bézier curves are described, including their basis functions and the fact that moving control points modifies the curve smoothly. Cubic Bézier curves are discussed in detail as a common parametric curve type, and

UNIT V.pptx

This document provides an introduction to digital electronics and digital systems. It discusses the differences between analog and digital systems, with digital systems using discrete binary values of 1 and 0 rather than continuous values. The advantages of digital systems include ease of programmability, lower costs, higher speeds and reliability. Number systems are also introduced, including binary, octal, hexadecimal and binary coded decimal. Techniques for minimizing boolean expressions using Karnaugh maps are described through examples. Finally, boolean algebra and the laws used for simplifying boolean expressions are covered.

Plastica 1º eso

The document defines basic geometry concepts like points, lines, line segments, rays, and angles. It also describes common geometry tools used for technical drawing like compasses, protractors, set squares, rulers, and erasers. The bulk of the document provides step-by-step instructions for performing basic geometry constructions like copying line segments, adding and subtracting line segments, constructing perpendicular bisectors of line segments, and constructing angle bisectors. It also defines circles and related terms like circumference and center.

Eg unit 1 2

The document discusses the syllabus for the course 20MEGO1 - Engineering Graphics. Module 1 covers curve constructions, orthographic projection principles, and drawing multiple views of objects. Specific topics include constructing conic sections, cycloids, and involutes; principles of orthographic projection; and projecting engineering components using first angle projection. Examples are provided for constructing a cycloid traced by a rolling circle, drawing the involute of a square and circle, and obtaining front and top views of objects.

Pictorial projection

This document discusses different methods of pictorial projection, including isometric and oblique projection. It provides details on how to construct and draw various geometric shapes and objects using these projection methods. Specifically, it describes rules for isometric projection, types of oblique projection, and several techniques for drawing circles and curves in both isometric and oblique views, such as the American method, ordinate method, and diagonal method. Examples are given for how to construct isometric drawings of solid cylinders, hollow cylinders, and step shafts using these circle construction techniques.

curve one

This document provides an overview of different techniques for representing polygon meshes and parametric curves. It discusses explicit representations of polygon meshes using vertex and edge lists, as well as parametric cubic curves including Hermite, Bezier, and B-spline curves. It describes how each technique represents the geometry and constraints, and properties such as continuity and invariance. Key topics covered include representations of polygon meshes, Hermite curves defined by endpoints and tangents, Bezier curves defined by control points, and B-spline curves having local control and smooth joins.

ch4.pptx

Windowing and clipping are techniques used in computer graphics to select and display portions of an image or drawing. Windowing refers to selecting a region or "window" to view. The viewport defines the area on the display device where the window will be mapped. Clipping determines which parts of an image or drawing are visible within the window by dividing elements into visible and invisible portions. Common clipping techniques include point, line, polygon and curve clipping. The Cohen-Sutherland and Liang-Barsky algorithms are used for line clipping, and Sutherland-Hodgeman for polygon clipping. Midpoint subdivision is another line clipping method.

Anti aliasing,area sampling,koch curve and c curve

This document discusses various topics related to computer graphics including anti-aliasing, area sampling, the Koch curve, and the C curve. Anti-aliasing is a technique used to reduce jagged edges by blending pixels. Area sampling is an anti-aliasing method that treats pixels as areas and calculates color based on object overlap. The Koch curve is a fractal curve generated by recursively altering line segments. The C curve replaces lines with two shorter lines at 90 degrees to form triangles at each iteration.

Construction class 9

This document discusses various geometric constructions that can be performed using only a compass and straightedge. It explains how to bisect angles and line segments, construct a 60 degree angle, and construct triangles given properties such as the base, a base angle, the sum or difference of the other sides, or the perimeter and two base angles. Constructions are performed through a series of defined steps using arcs drawn with a compass and straight lines drawn with a straightedge, without measuring lengths or angles numerically.

Construction of maths class 9th

This document discusses various geometric constructions that can be performed using only a compass and ruler. It explains how to bisect angles and line segments, construct a 60 degree angle, and construct triangles given different combinations of side lengths or angles. Specifically, it provides step-by-step instructions on how to construct a triangle if given its base, one base angle, and the sum of the other two sides; or given its base, a base angle, and the difference between the other two sides; or given its perimeter and two base angles.

Diagrams part 4

This document discusses different types of two-dimensional bar diagrams that can be used to represent data graphically, including squares, pie diagrams, and rectangle bar diagrams. It provides examples of how to construct these diagrams from sample data about sales over time and expenditures by schools. Students learn how to analyze and interpret these diagrams within their given contexts. The document also includes multiple choice questions to test comprehension.

Surface design and visible surfaces

This document discusses various techniques for modeling and rendering surfaces in 3D computer graphics, including visible surface determination. It describes several types of surfaces like bilinear, ruled, developable, Coons patch, and sweep surfaces. It also covers surfaces of revolution, quadratic surfaces, constructive solid geometry, Bezier surfaces, and Bspline surfaces. The document explains common visible surface algorithms like painter's algorithm and z-buffering used to determine which surfaces are visible and remove hidden surfaces from the rendered scene. It discusses concepts like coherence, bounding volumes, back face culling used to optimize visible surface calculations.

Lect14

B-splines are polynomial curves used for modeling curves and surfaces. They consist of curve segments whose polynomial coefficients depend on a few control points, allowing for local control of the shape. B-splines provide smooth joins between segments and have higher continuity than other curves like Bezier or Hermite curves. The shape of a B-spline is constrained within the convex hull of its control points. Knots divide the curve into segments and affect the smoothness. Uniform and non-uniform B-splines as well as manipulating knots and control points to control the shape are discussed.

Lec-4_Rendering-1.pdf

This document discusses algorithms for rendering lines and circles on a pixel-based display from vector representations. It begins by explaining the need to convert from continuous to discrete space. It then covers the scan conversion of lines using approaches like the digital differential analyzer algorithm and Bresenham's line algorithm. For circles, it discusses solving the circle equation but notes inefficiencies. It introduces the mid-point circle algorithm which uses the symmetry of circles and only calculates pixels in one eighth by evaluating the circle function at midpoints.

Web Technology LAB MANUAL for Undergraduate Programs

Web Technology LAB MANUAL for Undergraduate Programs

UNIVERSAL HUMAN VALUES- Harmony in the Nature

UNIVERSAL HUMAN VALUES- Harmony in the Nature

1516 contouring

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics

CAD - UNIT 2 (Geometric Modelling)

This document provides an overview of geometric modeling techniques used in computer aided design (CAD). It discusses representation of curves including Hermite curves, Bezier curves, B-spline curves, and rational curves. It also covers surface modeling techniques such as surface patches, Coons patches, and Bicubic patches. For solid modeling, it describes constructive solid geometry (CSG) and boundary representation (B-rep) techniques. CSG uses boolean operations on primitives to create models while B-rep defines models based on their bounding faces, edges and vertices.

ME6501 Unit 2 geometric modeling

This document provides an overview of geometric modeling techniques used in computer aided design (CAD). It discusses representation of curves including Hermite curves, Bezier curves, B-spline curves, and rational curves. It also covers surface modeling techniques such as surface patches, Coons patches, bicubic patches, Bezier surfaces, and B-spline surfaces. For solid modeling, it describes constructive solid geometry (CSG) and boundary representation (B-rep) techniques. CSG uses boolean operations on primitives to create models while B-rep models objects based on their bounding faces, edges, and vertices.

cg mod2.pdf

- Polygon fill areas are useful for describing picture components with a solid color or pattern. Common polygon shapes include triangles, rectangles, and octagons.
- Polygons can be classified as convex or concave. Concave polygons are more complicated to process than convex polygons.
- OpenGL provides functions like glBegin and glVertex for specifying polygon vertices and drawing polygon fill areas. Common primitives include GL_POLYGON, GL_TRIANGLES, GL_TRIANGLE_STRIP, and GL_TRIANGLE_FAN.

Surface models

This document discusses different types of surface models used in computer graphics, including:
- Plane, ruled, surface of revolution, tabulated, bilinear, Coons patch, and bicubic surfaces. Plane and ruled surfaces are linear, while surfaces of revolution and tabulated surfaces are axisymmetric. Bilinear surfaces are generated by interpolating 4 endpoints and are useful for finite element analysis. Coons patches interpolate 4 edge curves. Bicubic surfaces use parametric curves and interpolation of control points to define smooth surfaces.

10_1425_web_Lec_04_2D_Motion.pdf

This document discusses vectors and their application to motion in two and three dimensions. It introduces vectors as representations of displacement that have both magnitude and direction. Displacement, velocity and acceleration can all be described as vectors, allowing their addition and subtraction using graphical vector addition. Vector components in Cartesian coordinates are also discussed. The document uses examples of particle motion to illustrate average and instantaneous velocity and acceleration in two dimensions and the concept of relative velocity.

Curve modeling bezier curves

This document discusses Bézier curves and their properties. It begins by stating that traditional parametric curves are not very geometric and do not provide intuitive shape control. It then outlines desirable properties for curve design systems, including being intuitive, flexible, easy to use, providing a unified approach for different curve types, and producing invariant curves under transformations. The document proceeds to discuss Bézier, B-spline and NURBS curves which address these properties by allowing users to manipulate control points to modify curve shapes. Key properties of Bézier curves are described, including their basis functions and the fact that moving control points modifies the curve smoothly. Cubic Bézier curves are discussed in detail as a common parametric curve type, and

UNIT V.pptx

This document provides an introduction to digital electronics and digital systems. It discusses the differences between analog and digital systems, with digital systems using discrete binary values of 1 and 0 rather than continuous values. The advantages of digital systems include ease of programmability, lower costs, higher speeds and reliability. Number systems are also introduced, including binary, octal, hexadecimal and binary coded decimal. Techniques for minimizing boolean expressions using Karnaugh maps are described through examples. Finally, boolean algebra and the laws used for simplifying boolean expressions are covered.

Plastica 1º eso

The document defines basic geometry concepts like points, lines, line segments, rays, and angles. It also describes common geometry tools used for technical drawing like compasses, protractors, set squares, rulers, and erasers. The bulk of the document provides step-by-step instructions for performing basic geometry constructions like copying line segments, adding and subtracting line segments, constructing perpendicular bisectors of line segments, and constructing angle bisectors. It also defines circles and related terms like circumference and center.

Eg unit 1 2

The document discusses the syllabus for the course 20MEGO1 - Engineering Graphics. Module 1 covers curve constructions, orthographic projection principles, and drawing multiple views of objects. Specific topics include constructing conic sections, cycloids, and involutes; principles of orthographic projection; and projecting engineering components using first angle projection. Examples are provided for constructing a cycloid traced by a rolling circle, drawing the involute of a square and circle, and obtaining front and top views of objects.

Pictorial projection

This document discusses different methods of pictorial projection, including isometric and oblique projection. It provides details on how to construct and draw various geometric shapes and objects using these projection methods. Specifically, it describes rules for isometric projection, types of oblique projection, and several techniques for drawing circles and curves in both isometric and oblique views, such as the American method, ordinate method, and diagonal method. Examples are given for how to construct isometric drawings of solid cylinders, hollow cylinders, and step shafts using these circle construction techniques.

curve one

This document provides an overview of different techniques for representing polygon meshes and parametric curves. It discusses explicit representations of polygon meshes using vertex and edge lists, as well as parametric cubic curves including Hermite, Bezier, and B-spline curves. It describes how each technique represents the geometry and constraints, and properties such as continuity and invariance. Key topics covered include representations of polygon meshes, Hermite curves defined by endpoints and tangents, Bezier curves defined by control points, and B-spline curves having local control and smooth joins.

ch4.pptx

Windowing and clipping are techniques used in computer graphics to select and display portions of an image or drawing. Windowing refers to selecting a region or "window" to view. The viewport defines the area on the display device where the window will be mapped. Clipping determines which parts of an image or drawing are visible within the window by dividing elements into visible and invisible portions. Common clipping techniques include point, line, polygon and curve clipping. The Cohen-Sutherland and Liang-Barsky algorithms are used for line clipping, and Sutherland-Hodgeman for polygon clipping. Midpoint subdivision is another line clipping method.

Anti aliasing,area sampling,koch curve and c curve

This document discusses various topics related to computer graphics including anti-aliasing, area sampling, the Koch curve, and the C curve. Anti-aliasing is a technique used to reduce jagged edges by blending pixels. Area sampling is an anti-aliasing method that treats pixels as areas and calculates color based on object overlap. The Koch curve is a fractal curve generated by recursively altering line segments. The C curve replaces lines with two shorter lines at 90 degrees to form triangles at each iteration.

Construction class 9

This document discusses various geometric constructions that can be performed using only a compass and straightedge. It explains how to bisect angles and line segments, construct a 60 degree angle, and construct triangles given properties such as the base, a base angle, the sum or difference of the other sides, or the perimeter and two base angles. Constructions are performed through a series of defined steps using arcs drawn with a compass and straight lines drawn with a straightedge, without measuring lengths or angles numerically.

Construction of maths class 9th

This document discusses various geometric constructions that can be performed using only a compass and ruler. It explains how to bisect angles and line segments, construct a 60 degree angle, and construct triangles given different combinations of side lengths or angles. Specifically, it provides step-by-step instructions on how to construct a triangle if given its base, one base angle, and the sum of the other two sides; or given its base, a base angle, and the difference between the other two sides; or given its perimeter and two base angles.

Diagrams part 4

This document discusses different types of two-dimensional bar diagrams that can be used to represent data graphically, including squares, pie diagrams, and rectangle bar diagrams. It provides examples of how to construct these diagrams from sample data about sales over time and expenditures by schools. Students learn how to analyze and interpret these diagrams within their given contexts. The document also includes multiple choice questions to test comprehension.

Surface design and visible surfaces

This document discusses various techniques for modeling and rendering surfaces in 3D computer graphics, including visible surface determination. It describes several types of surfaces like bilinear, ruled, developable, Coons patch, and sweep surfaces. It also covers surfaces of revolution, quadratic surfaces, constructive solid geometry, Bezier surfaces, and Bspline surfaces. The document explains common visible surface algorithms like painter's algorithm and z-buffering used to determine which surfaces are visible and remove hidden surfaces from the rendered scene. It discusses concepts like coherence, bounding volumes, back face culling used to optimize visible surface calculations.

Lect14

B-splines are polynomial curves used for modeling curves and surfaces. They consist of curve segments whose polynomial coefficients depend on a few control points, allowing for local control of the shape. B-splines provide smooth joins between segments and have higher continuity than other curves like Bezier or Hermite curves. The shape of a B-spline is constrained within the convex hull of its control points. Knots divide the curve into segments and affect the smoothness. Uniform and non-uniform B-splines as well as manipulating knots and control points to control the shape are discussed.

Lec-4_Rendering-1.pdf

This document discusses algorithms for rendering lines and circles on a pixel-based display from vector representations. It begins by explaining the need to convert from continuous to discrete space. It then covers the scan conversion of lines using approaches like the digital differential analyzer algorithm and Bresenham's line algorithm. For circles, it discusses solving the circle equation but notes inefficiencies. It introduces the mid-point circle algorithm which uses the symmetry of circles and only calculates pixels in one eighth by evaluating the circle function at midpoints.

1516 contouring

1516 contouring

CAD - UNIT 2 (Geometric Modelling)

CAD - UNIT 2 (Geometric Modelling)

ME6501 Unit 2 geometric modeling

ME6501 Unit 2 geometric modeling

cg mod2.pdf

cg mod2.pdf

Surface models

Surface models

10_1425_web_Lec_04_2D_Motion.pdf

10_1425_web_Lec_04_2D_Motion.pdf

Curve modeling bezier curves

Curve modeling bezier curves

UNIT V.pptx

UNIT V.pptx

Plastica 1º eso

Plastica 1º eso

Eg unit 1 2

Eg unit 1 2

Pictorial projection

Pictorial projection

curve one

curve one

ch4.pptx

ch4.pptx

Anti aliasing,area sampling,koch curve and c curve

Anti aliasing,area sampling,koch curve and c curve

Construction class 9

Construction class 9

Construction of maths class 9th

Construction of maths class 9th

Diagrams part 4

Diagrams part 4

Surface design and visible surfaces

Surface design and visible surfaces

Lect14

Lect14

Lec-4_Rendering-1.pdf

Lec-4_Rendering-1.pdf

Web Technology LAB MANUAL for Undergraduate Programs

Web Technology LAB MANUAL for Undergraduate Programs

UNIVERSAL HUMAN VALUES- Harmony in the Nature

UNIVERSAL HUMAN VALUES- Harmony in the Nature

Study of Computer Hardware System using Block Diagram

Study of Computer Hardware System using Block Diagram

Computer System Output Devices Peripherals

Computer System Output Devices Peripherals

Computer system Input Devices Peripherals

Computer system Input Devices Peripherals

Computer system Input and Output Devices

Computer system Input and Output Devices

Introduction to COMPUTER’S MEMORY RAM and ROM

Introduction to COMPUTER’S MEMORY RAM and ROM

Introduction to Computer Hardware Systems

Introduction to Computer Hardware Systems

Fundamentals of Internet of Things (IoT) Part-2

Fundamentals of Internet of Things (IoT) Part-2

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Introduction to Artificial Intelligence ( AI)

Introduction to Artificial Intelligence ( AI)

Fundamentals of functions in C program.pptx

Fundamentals of functions in C program.pptx

Fundamentals of Structure in C Programming

Fundamentals of Structure in C Programming

INPUT AND OUTPUT STATEMENTS IN PROGRAMMING IN C

INPUT AND OUTPUT STATEMENTS IN PROGRAMMING IN C

Programming in C - Fundamental Study of Strings

Programming in C - Fundamental Study of Strings

Basics of Control Statement in C Languages

Basics of Control Statement in C Languages

Features and Fundamentals of C Language for Beginners

Features and Fundamentals of C Language

Basics of Programming Algorithms and Flowchart

Basics of Programming Algorithms and Flowchart

Computer Graphics Three-Dimensional Geometric Transformations

Computer Graphics Three-Dimensional Geometric Transformations

Computer Graphics - Windowing and Clipping

Computer Graphics - Windowing and Clipping

Web Technology LAB MANUAL for Undergraduate Programs

Web Technology LAB MANUAL for Undergraduate Programs

UNIVERSAL HUMAN VALUES- Harmony in the Nature

UNIVERSAL HUMAN VALUES- Harmony in the Nature

Study of Computer Hardware System using Block Diagram

Study of Computer Hardware System using Block Diagram

Computer System Output Devices Peripherals

Computer System Output Devices Peripherals

Computer system Input Devices Peripherals

Computer system Input Devices Peripherals

Computer system Input and Output Devices

Computer system Input and Output Devices

Introduction to COMPUTER’S MEMORY RAM and ROM

Introduction to COMPUTER’S MEMORY RAM and ROM

Introduction to Computer Hardware Systems

Introduction to Computer Hardware Systems

Fundamentals of Internet of Things (IoT) Part-2

Fundamentals of Internet of Things (IoT) Part-2

Fundamentals of Internet of Things (IoT)

Fundamentals of Internet of Things (IoT)

Introduction to Artificial Intelligence ( AI)

Introduction to Artificial Intelligence ( AI)

Fundamentals of functions in C program.pptx

Fundamentals of functions in C program.pptx

Fundamentals of Structure in C Programming

Fundamentals of Structure in C Programming

INPUT AND OUTPUT STATEMENTS IN PROGRAMMING IN C

INPUT AND OUTPUT STATEMENTS IN PROGRAMMING IN C

Programming in C - Fundamental Study of Strings

Programming in C - Fundamental Study of Strings

Basics of Control Statement in C Languages

Basics of Control Statement in C Languages

Features and Fundamentals of C Language for Beginners

Features and Fundamentals of C Language for Beginners

Basics of Programming Algorithms and Flowchart

Basics of Programming Algorithms and Flowchart

Computer Graphics Three-Dimensional Geometric Transformations

Computer Graphics Three-Dimensional Geometric Transformations

Computer Graphics - Windowing and Clipping

Computer Graphics - Windowing and Clipping

Beckhoff Programmable Logic Control Overview Presentation

This presentation is to describe the overview of PLC Beckhoff for beginners

Call For Paper -3rd International Conference on Artificial Intelligence Advan...

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DESIGN AND MANUFACTURE OF CEILING BOARD USING SAWDUST AND WASTE CARTON MATERI...

The need for ecofriendly materials as building materials in this century cannot be overemphasized

EV BMS WITH CHARGE MONITOR AND FIRE DETECTION.pptx

EV BMS WITH CHARGE MONITOR AND FIRE DETECTION

AI + Data Community Tour - Build the Next Generation of Apps with the Einstei...

AI + Data Community Tour - Build the Next Generation of Apps with the Einstei...Paris Salesforce Developer Group

Build the Next Generation of Apps with the Einstein 1 Platform.
Rejoignez Philippe Ozil pour une session de workshops qui vous guidera à travers les détails de la plateforme Einstein 1, l'importance des données pour la création d'applications d'intelligence artificielle et les différents outils et technologies que Salesforce propose pour vous apporter tous les bénéfices de l'IA.Height and depth gauge linear metrology.pdf

Height gauges may also be used to measure the height of an object by using the underside of the scriber as the datum. The datum may be permanently fixed or the height gauge may have provision to adjust the scale, this is done by sliding the scale vertically along the body of the height gauge by turning a fine feed screw at the top of the gauge; then with the scriber set to the same level as the base, the scale can be matched to it. This adjustment allows different scribers or probes to be used, as well as adjusting for any errors in a damaged or resharpened probe.

一比一原版(爱大毕业证书)爱荷华大学毕业证如何办理

原版一模一样【微信：741003700 】【(爱大毕业证书)爱荷华大学毕业证成绩单】【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】外观非常简单，由纸质材料制成，上面印有校徽、校名、毕业生姓名、专业等信息。
办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】格式相对统一，各专业都有相应的模板。通常包括以下部分：
校徽：象征着学校的荣誉和传承。
校名:学校英文全称
授予学位：本部分将注明获得的具体学位名称。
毕业生姓名：这是最重要的信息之一，标志着该证书是由特定人员获得的。
颁发日期：这是毕业正式生效的时间，也代表着毕业生学业的结束。
其他信息：根据不同的专业和学位，可能会有一些特定的信息或章节。
办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】价值很高，需要妥善保管。一般来说，应放置在安全、干燥、防潮的地方，避免长时间暴露在阳光下。如需使用，最好使用复印件而不是原件，以免丢失。
综上所述，办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】是证明身份和学历的高价值文件。外观简单庄重，格式统一，包括重要的个人信息和发布日期。对持有人来说，妥善保管是非常重要的。

openshift technical overview - Flow of openshift containerisatoin

openshift overview

一比一原版(uoft毕业证书)加拿大多伦多大学毕业证如何办理

原版一模一样【微信：741003700 】【(uoft毕业证书)加拿大多伦多大学毕业证成绩单】【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
办理(uoft毕业证书)加拿大多伦多大学毕业证【微信：741003700 】外观非常简单，由纸质材料制成，上面印有校徽、校名、毕业生姓名、专业等信息。
办理(uoft毕业证书)加拿大多伦多大学毕业证【微信：741003700 】格式相对统一，各专业都有相应的模板。通常包括以下部分：
校徽：象征着学校的荣誉和传承。
校名:学校英文全称
授予学位：本部分将注明获得的具体学位名称。
毕业生姓名：这是最重要的信息之一，标志着该证书是由特定人员获得的。
颁发日期：这是毕业正式生效的时间，也代表着毕业生学业的结束。
其他信息：根据不同的专业和学位，可能会有一些特定的信息或章节。
办理(uoft毕业证书)加拿大多伦多大学毕业证【微信：741003700 】价值很高，需要妥善保管。一般来说，应放置在安全、干燥、防潮的地方，避免长时间暴露在阳光下。如需使用，最好使用复印件而不是原件，以免丢失。
综上所述，办理(uoft毕业证书)加拿大多伦多大学毕业证【微信：741003700 】是证明身份和学历的高价值文件。外观简单庄重，格式统一，包括重要的个人信息和发布日期。对持有人来说，妥善保管是非常重要的。

Call Girls Chennai +91-8824825030 Vip Call Girls Chennai

Call Girls Chennai +91-8824825030 Vip Call Girls Chennai

Digital Twins Computer Networking Paper Presentation.pptx

A Digital Twin in computer networking is a virtual representation of a physical network, used to simulate, analyze, and optimize network performance and reliability. It leverages real-time data to enhance network management, predict issues, and improve decision-making processes.

Determination of Equivalent Circuit parameters and performance characteristic...

Includes the testing of induction motor to draw the circle diagram of induction motor with step wise procedure and calculation for the same. Also explains the working and application of Induction generator

A high-Speed Communication System is based on the Design of a Bi-NoC Router, ...

The Network on Chip (NoC) has emerged as an effective
solution for intercommunication infrastructure within System on
Chip (SoC) designs, overcoming the limitations of traditional
methods that face significant bottlenecks. However, the complexity
of NoC design presents numerous challenges related to
performance metrics such as scalability, latency, power
consumption, and signal integrity. This project addresses the
issues within the router's memory unit and proposes an enhanced
memory structure. To achieve efficient data transfer, FIFO buffers
are implemented in distributed RAM and virtual channels for
FPGA-based NoC. The project introduces advanced FIFO-based
memory units within the NoC router, assessing their performance
in a Bi-directional NoC (Bi-NoC) configuration. The primary
objective is to reduce the router's workload while enhancing the
FIFO internal structure. To further improve data transfer speed,
a Bi-NoC with a self-configurable intercommunication channel is
suggested. Simulation and synthesis results demonstrate
guaranteed throughput, predictable latency, and equitable
network access, showing significant improvement over previous
designs

Null Bangalore | Pentesters Approach to AWS IAM

#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
#Prerequisites:
- Basic understanding of AWS services and architecture
- Familiarity with cloud security concepts
- Experience using the AWS Management Console or AWS CLI.
- For hands on lab create account on [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
# Scenario Covered:
- Basics of IAM in AWS
- Implementing IAM Policies with Least Privilege to Manage S3 Bucket
- Objective: Create an S3 bucket with least privilege IAM policy and validate access.
- Steps:
- Create S3 bucket.
- Attach least privilege policy to IAM user.
- Validate access.
- Exploiting IAM PassRole Misconfiguration
-Allows a user to pass a specific IAM role to an AWS service (ec2), typically used for service access delegation. Then exploit PassRole Misconfiguration granting unauthorized access to sensitive resources.
- Objective: Demonstrate how a PassRole misconfiguration can grant unauthorized access.
- Steps:
- Allow user to pass IAM role to EC2.
- Exploit misconfiguration for unauthorized access.
- Access sensitive resources.
- Exploiting IAM AssumeRole Misconfiguration with Overly Permissive Role
- An overly permissive IAM role configuration can lead to privilege escalation by creating a role with administrative privileges and allow a user to assume this role.
- Objective: Show how overly permissive IAM roles can lead to privilege escalation.
- Steps:
- Create role with administrative privileges.
- Allow user to assume the role.
- Perform administrative actions.
- Differentiation between PassRole vs AssumeRole
Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)

Supermarket Management System Project Report.pdf

Supermarket management is a stand-alone J2EE using Eclipse Juno program.
This project contains all the necessary required information about maintaining
the supermarket billing system.
The core idea of this project to minimize the paper work and centralize the
data. Here all the communication is taken in secure manner. That is, in this
application the information will be stored in client itself. For further security the
data base is stored in the back-end oracle and so no intruders can access it.

Applications of artificial Intelligence in Mechanical Engineering.pdf

Historically, mechanical engineering has relied heavily on human expertise and empirical methods to solve complex problems. With the introduction of computer-aided design (CAD) and finite element analysis (FEA), the field took its first steps towards digitization. These tools allowed engineers to simulate and analyze mechanical systems with greater accuracy and efficiency. However, the sheer volume of data generated by modern engineering systems and the increasing complexity of these systems have necessitated more advanced analytical tools, paving the way for AI.
AI offers the capability to process vast amounts of data, identify patterns, and make predictions with a level of speed and accuracy unattainable by traditional methods. This has profound implications for mechanical engineering, enabling more efficient design processes, predictive maintenance strategies, and optimized manufacturing operations. AI-driven tools can learn from historical data, adapt to new information, and continuously improve their performance, making them invaluable in tackling the multifaceted challenges of modern mechanical engineering.

This study Examines the Effectiveness of Talent Procurement through the Imple...

In the world with high technology and fast
forward mindset recruiters are walking/showing interest
towards E-Recruitment. Present most of the HRs of
many companies are choosing E-Recruitment as the best
choice for recruitment. E-Recruitment is being done
through many online platforms like Linkedin, Naukri,
Instagram , Facebook etc. Now with high technology E-
Recruitment has gone through next level by using
Artificial Intelligence too.
Key Words : Talent Management, Talent Acquisition , E-
Recruitment , Artificial Intelligence Introduction
Effectiveness of Talent Acquisition through E-
Recruitment in this topic we will discuss about 4important
and interlinked topics which are

Presentation on Food Delivery Systems

This presentation is about Food Delivery Systems and how they are developed using the Software Development Life Cycle (SDLC) and other methods. It explains the steps involved in creating a food delivery app, from planning and designing to testing and launching. The slide also covers different tools and technologies used to make these systems work efficiently.

OOPS_Lab_Manual - programs using C++ programming language

This manual contains programs on object oriented programming concepts using C++ language.

Beckhoff Programmable Logic Control Overview Presentation

Beckhoff Programmable Logic Control Overview Presentation

Call For Paper -3rd International Conference on Artificial Intelligence Advan...

Call For Paper -3rd International Conference on Artificial Intelligence Advan...

DESIGN AND MANUFACTURE OF CEILING BOARD USING SAWDUST AND WASTE CARTON MATERI...

DESIGN AND MANUFACTURE OF CEILING BOARD USING SAWDUST AND WASTE CARTON MATERI...

EV BMS WITH CHARGE MONITOR AND FIRE DETECTION.pptx

EV BMS WITH CHARGE MONITOR AND FIRE DETECTION.pptx

AI + Data Community Tour - Build the Next Generation of Apps with the Einstei...

AI + Data Community Tour - Build the Next Generation of Apps with the Einstei...

Height and depth gauge linear metrology.pdf

Height and depth gauge linear metrology.pdf

一比一原版(爱大毕业证书)爱荷华大学毕业证如何办理

一比一原版(爱大毕业证书)爱荷华大学毕业证如何办理

openshift technical overview - Flow of openshift containerisatoin

openshift technical overview - Flow of openshift containerisatoin

一比一原版(uoft毕业证书)加拿大多伦多大学毕业证如何办理

一比一原版(uoft毕业证书)加拿大多伦多大学毕业证如何办理

Call Girls Chennai +91-8824825030 Vip Call Girls Chennai

Call Girls Chennai +91-8824825030 Vip Call Girls Chennai

Digital Twins Computer Networking Paper Presentation.pptx

Digital Twins Computer Networking Paper Presentation.pptx

Determination of Equivalent Circuit parameters and performance characteristic...

Determination of Equivalent Circuit parameters and performance characteristic...

A high-Speed Communication System is based on the Design of a Bi-NoC Router, ...

A high-Speed Communication System is based on the Design of a Bi-NoC Router, ...

Null Bangalore | Pentesters Approach to AWS IAM

Null Bangalore | Pentesters Approach to AWS IAM

Supermarket Management System Project Report.pdf

Supermarket Management System Project Report.pdf

Applications of artificial Intelligence in Mechanical Engineering.pdf

Applications of artificial Intelligence in Mechanical Engineering.pdf

This study Examines the Effectiveness of Talent Procurement through the Imple...

This study Examines the Effectiveness of Talent Procurement through the Imple...

1FIDIC-CONSTRUCTION-CONTRACT-2ND-ED-2017-RED-BOOK.pdf

1FIDIC-CONSTRUCTION-CONTRACT-2ND-ED-2017-RED-BOOK.pdf

Presentation on Food Delivery Systems

Presentation on Food Delivery Systems

OOPS_Lab_Manual - programs using C++ programming language

OOPS_Lab_Manual - programs using C++ programming language

- 1. Introduction To Curves C. P. Divate
- 2. Abstract • We present and study existing digital differential analyzer (DDA) algorithms for circle generation, including an improved two-step DDA algorithm which can be implemented solely in terms of elementary shifts, addition, and subtraction.
- 3. Introduction • Digital interpolation algorithms are widely used in machine tools with numerical control, graphics displays and plotters, and manipulation robots. Circles and circular arcs frequently appear in computer graphics, computer-controlled printing, and automated control; • One of most popular methods for generation of circles and arcs is known as digital differential analyzer (DDA). • In this chapter a new improved two-step algorithm for DDA circle generation is presented. • The accuracy of this method is higher than the accuracy of other known algorithms. • Because of its simplicity (it uses only elementary shift, addition, and subtraction); this method can also be used in numerical control, planning mechanisms, and so forth.
- 4. DDA Algorithms • The general class of DDA algorithms for circles generation is based on obvious trigonometric transformations describing rotation of vector R in the coordinate plane x and y • Advantages of DDA algorithms include simplicity and high speed in generating circle point coordinate xn+1, yn+1 .
- 6. Bezier Curve- Bezier Curve may be defined as- •Bezier Curve is parametric curve defined by a set of control points. •Two points are ends of the curve. •Other points determine the shape of the curve. The concept of Bezier curves was given by Pierre Bezier. Bezier Curve Example- The following curve is an example of a Bezier curve- Here, • This bezier curve is defined by a set of control points b0, b1, b2 and b3. • Points b0 and b3 are ends of the curve. • Points b1 and b2 determine the shape of the curve.
- 7. Bezier Curve Properties- Few important properties of a Bezier curve are- Property-01: Bezier curve is always contained within a polygon called as convex hull of its control points. Property-02: • Bezier curve generally follows the shape of its defining polygon. • The first and last points of the curve are coincident with the first and last points of the defining polygon. Property-03: • The degree of the polynomial defining the curve segment is one less than the total number of control points. Degree = Number of Control Points – 1
- 8. Bezier Curve Properties- Few important properties of a Bezier curve are- Property-04: • The order of the polynomial defining the curve segment is equal to the total number of control points. Order = Number of Control Points Property-05: • Bezier curve exhibits the variation diminishing property. • It means the curve do not oscillate about any straight line more often than the defining polygon.
- 9. Bezier Curve Equation- A bezier curve is parametrically represented by- Here, • t is any parameter where 0 <= t <= 1 • P(t) = Any point lying on the bezier curve • Bi = ith control point of the bezier curve • n = degree of the curve • Jn,i(t) = Blending function = C(n,i)ti(1-t)n-i where C(n,i) = n! / i!(n-i)!
- 10. Cubic Bezier Curve- • Cubic bezier curve is a bezier curve with degree 3. • The total number of control points in a cubic bezier curve is 4. Here, • This curve is defined by 4 control points b0, b1, b2 and b3. • The degree of this curve is 3. • So, it is a cubic bezier curve. Substituting n = 3 for a cubic bezier curve, we get-
- 11. Cubic Bezier Curve- • Cubic bezier curve is a bezier curve with degree 3. • The total number of control points in a cubic bezier curve is 4. Substituting n = 3 for a cubic bezier curve, we get- Expanding the above equation, we get- P (t) = B0J3,0(t) + B1J3,1(t) + B2J3,2(t) + B3J3,3(t) ………..(1) Now,
- 12. Cubic Bezier Curve- Expanding the above equation, we get- P (t) = B0J3,0(t) + B1J3,1(t) + B2J3,2(t) + B3J3,3(t) ………..(1) Now,
- 13. Cubic Bezier Curve- Expanding the above equation, we get- P (t) = B0J3,0(t) + B1J3,1(t) + B2J3,2(t) + B3J3,3(t) ………..(1) Now, Using (2), (3), (4) and (5) in (1), we get- P(t) = B0(1-t)3 + B13t(1-t)2 + B23t2(1-t) + B3t3 This is the required parametric equation for a cubic bezier curve.
- 14. Applications of Bezier Curve- Bezier curves have their applications in the following fields- 1. Computer Graphics- • Bezier curves are widely used in computer graphics to model smooth curves. • The curve is completely contained in the convex hull of its control points. • So, the points can be graphically displayed & used to manipulate the curve intuitively. 2. Animation- • Bezier curves are used to outline movement in animation applications such as Adobe Flash and synfig. • Users outline the wanted path in bezier curves. • The application creates the needed frames for the object to move along the path. • For 3D animation, bezier curves are often used to define 3D paths as well as 2D curves. 3. Fonts- • True type fonts use composite bezier curves composed of quadratic bezier curves. • Modern imaging systems like postscript, asymptote etc use composite bezier curves composed of cubic bezier curves for drawing curved shapes.
- 15. 2 Koch Curve What is Koch Curve? The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a mathematical curve and one of the earliest fractal curves to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled “On a continuous curve without tangents, constructible from elementary geometry” by the Swedish mathematician Helge von Koch. The progression for the area of the snowflake converges to 8/5 times the area of the original triangle, while the progression for the snowflake’s perimeter diverges to infinity. Consequently, the snowflake has a finite area bounded by an infinitely long line.
- 16. 3 Depth 1 1 line segment
- 17. 4 Depth 2 Contains 4 depth-1 curves (4 line segments)
- 18. 5 Depth 3 Contains 4 depth-2 curves (16 line segments)
- 19. 6 Depth 4 Contains 4 depth-3 curves (64 line segments)
- 20. 7 Depth 5 Contains 4 depth-4 curves (256 line segments)
- 21. Construction Step1: Draw an equilateral triangle. You can draw it with a compass or protractor, or just eyeball it if you don’t want to spend too much time drawing the snowflake. It’s best if the length of the sides are divisible by 3, because of the nature of this fractal. This will become clear in the next few steps.
- 22. Construction Step2: Divide each side in three equal parts. This is why it is handy to have the sides divisible by three.
- 23. Construction Step3: Draw an equilateral triangle on each middle part. Measure the length of the middle third to know the length of the sides of these new triangles.
- 24. Construction Step4: Divide each outer side into thirds. You can see the 2nd generation of triangles covers a bit of the first. These three line segments shouldn’t be parted in three.
- 25. Construction Step5: Draw an equilateral triangle on each middle part. Note how you draw each next generation of parts that are one 3rd of the mast one.
- 26. 8 Pseudocode Algorithm Tomake a Koch Curve: • Draw a straight line if depth is zero; otherwise draw four smaller Koch Curves.
- 27. Koch Curve The Koch curve is to divide a line into three equal segments and replace the middle segment with two lines of the same length. 12
- 28. • The Koch snowflake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows: divide the line segment into fore segments of equal length. draw an equilateral triangle that has the middle segment from step 1 as its base and points outward. remove the line segment that is the base of the triangle from step 2. 28 Koch Curve(Construction)
- 29. Koch Curve Start with a line.
- 32. Calculating the Projection l l y = √3/2l x = ½l m2 + (½ l)2 =l2 m2 + ¼ l2 =l2 √m2 = √¾ l2 m = √3/2l * 1/3 = √3l/6 1/3 length of each iteration