The document describes the DeBoor-Cox calculation, which relates the analytical and geometric definitions of B-spline curves. It begins by defining a B-spline curve analytically as a weighted sum of normalized B-spline blending functions and control points. The blending functions are defined recursively. DeBoor and Cox showed that starting from this analytical definition, one can derive the geometric definition of a B-spline curve as a pyramid of control points. Their calculation demonstrated the relationship between the two common definitions of B-splines.
Interpolation is a method to estimate values between known data points. It can be used in computer science for tasks like image reconstruction, CAD modeling, and graphics. Secret sharing schemes also use interpolation to distribute secret information among participants in a way that some minimum number of participants are needed to reconstruct the secret. Lagrange interpolation specifically involves defining a polynomial that passes through known data points, which can then be evaluated at other points to interpolate unknown values. While simple, it provides no error checking and mistakes could occur in calculations.
The document discusses B-spline curves, which are defined using control points, knots, and basis functions. B-spline curves allow for local control of the curve shape and independent specification of the degree and number of control points. The curve is computed as a weighted sum of basis functions evaluated over the knot vector. The shape of the B-spline curve can be modified by changing the control points, knots, or degree of the curve.
The Cyrus-Beck algorithm is used for line clipping against non-rectangular convex polygons. It uses a parametric equation to find the intersection point of the line with the polygon boundary. The algorithm calculates the time values for the line endpoints at each polygon edge, then uses those times in the parametric equation to find the clipped line segment P'0 and P'1 that is visible within the polygon clipping window.
The Cohen-Sutherland algorithm divides the plane into 9 regions and uses 4-bit codes to encode whether each endpoint of a line segment is left, right, above, or below the clipping window. It then uses the endpoint codes to either trivially accept or reject the line segment, or perform clipping by calculating the intersection point of the line with the window boundary and replacing the outside endpoint. The main steps are to assign codes to endpoints, AND the codes to check for trivial acceptance or rejection, clip by replacing outside endpoints if needed, and repeating for other line segments.
Clipping is a technique that identifies parts of an image that are inside or outside a defined clipping region or window. There are different types of clipping including point, line, polygon, curve, and text clipping. The Cohen-Sutherland algorithm is commonly used for line clipping. It assigns 4-bit codes to line endpoints to determine if they are fully inside, outside, or intersect the clipping window boundary. Intersecting line segments are then subdivided and clipped. Midpoint subdivision is another algorithm that divides partially visible lines at their midpoint into shorter segments.
This document summarizes the scan-line rendering algorithm. It maintains two tables - an edge table containing line coordinates and surface pointers, and a polygon table containing surface properties. For each scan line, all intersecting surfaces are examined to determine the visible surface. Depths are calculated to set surface flags and populate the image buffer with intensity values from the visible surface. Coherence between scan lines is exploited to reuse prior visibility calculations where edge intersections remain the same.
Interpolation is a method to estimate values between known data points. It can be used in computer science for tasks like image reconstruction, CAD modeling, and graphics. Secret sharing schemes also use interpolation to distribute secret information among participants in a way that some minimum number of participants are needed to reconstruct the secret. Lagrange interpolation specifically involves defining a polynomial that passes through known data points, which can then be evaluated at other points to interpolate unknown values. While simple, it provides no error checking and mistakes could occur in calculations.
The document discusses B-spline curves, which are defined using control points, knots, and basis functions. B-spline curves allow for local control of the curve shape and independent specification of the degree and number of control points. The curve is computed as a weighted sum of basis functions evaluated over the knot vector. The shape of the B-spline curve can be modified by changing the control points, knots, or degree of the curve.
The Cyrus-Beck algorithm is used for line clipping against non-rectangular convex polygons. It uses a parametric equation to find the intersection point of the line with the polygon boundary. The algorithm calculates the time values for the line endpoints at each polygon edge, then uses those times in the parametric equation to find the clipped line segment P'0 and P'1 that is visible within the polygon clipping window.
The Cohen-Sutherland algorithm divides the plane into 9 regions and uses 4-bit codes to encode whether each endpoint of a line segment is left, right, above, or below the clipping window. It then uses the endpoint codes to either trivially accept or reject the line segment, or perform clipping by calculating the intersection point of the line with the window boundary and replacing the outside endpoint. The main steps are to assign codes to endpoints, AND the codes to check for trivial acceptance or rejection, clip by replacing outside endpoints if needed, and repeating for other line segments.
Clipping is a technique that identifies parts of an image that are inside or outside a defined clipping region or window. There are different types of clipping including point, line, polygon, curve, and text clipping. The Cohen-Sutherland algorithm is commonly used for line clipping. It assigns 4-bit codes to line endpoints to determine if they are fully inside, outside, or intersect the clipping window boundary. Intersecting line segments are then subdivided and clipped. Midpoint subdivision is another algorithm that divides partially visible lines at their midpoint into shorter segments.
This document summarizes the scan-line rendering algorithm. It maintains two tables - an edge table containing line coordinates and surface pointers, and a polygon table containing surface properties. For each scan line, all intersecting surfaces are examined to determine the visible surface. Depths are calculated to set surface flags and populate the image buffer with intensity values from the visible surface. Coherence between scan lines is exploited to reuse prior visibility calculations where edge intersections remain the same.
Identify those parts of a scene that are visible from a chosen viewing position.
Visible-surface detection algorithms are broadly classified according to whether
they deal with object definitions directly or with their projected images.
These two approaches are called object-space methods and image-space methods, respectively
An object-space method compares
objects and parts of objects to each other within the scene definition to determine which surfaces, as a whole, we should label as visible.
In an image-space algorithm, visibility is decided point by point at each pixel position on the projection plane.
This lecture covers various solution methods for unconstrained optimization problems, including:
1) Line search methods like dichotomous search and the Fibonacci and golden section methods for one-dimensional problems.
2) Newton's method and the false position method for curve fitting to minimize functions in one dimension.
3) Descent methods for multidimensional problems like the gradient method, which follows the negative gradient direction at each step, and how scaling can improve convergence rates.
This document discusses bottom-up parsing and handle pruning. It begins with an example grammar and string. Bottom-up parsing works by repeatedly reducing substrings matching production bodies until reaching the start symbol. A handle is a substring matching a production body whose reduction moves backward through a rightmost derivation. Handle pruning is removing nonterminal children from a parse tree, allowing reconstruction of a rightmost derivation by working backward from handles. The document provides steps for using handle pruning to rebuild a rightmost derivation in reverse from a given string. An example demonstrates finding the handles in a string and reducing based on the corresponding productions to reach the start symbol.
This document discusses methods for identifying and removing hidden surfaces when rendering 3D scenes to create a realistic 2D image. It describes two approaches: object-space methods that compare whole objects, and image-space methods that decide visibility point-by-point. It focuses on the depth buffer/z-buffer method, which processes surfaces one point at a time, comparing depth values to determine visibility and store the color of visible points. It also discusses using scan line coherence to solve hidden surfaces one scan line at a time from top to bottom.
3D transformation in computer graphicsSHIVANI SONI
This document discusses different types of 2D and 3D transformations that are used in computer graphics, including translation, rotation, scaling, shearing, and reflection. It provides the mathematical equations and transformation matrices used to perform each type of transformation on 2D and 3D points and objects. Key types of rotations discussed are roll (around z-axis), pitch (around x-axis), and yaw (around y-axis). Homogeneous coordinates are introduced for representing 3D points.
This document provides an overview of 3D transformations, including translation, rotation, scaling, reflection, and shearing. It explains that 3D transformations generalize 2D transformations by including a z-coordinate and using homogeneous coordinates and 4x4 transformation matrices. Each type of 3D transformation is defined using matrix representations and equations. Rotation is described for each coordinate axis, and reflection is explained for each axis plane. Shearing is introduced as a way to modify object shapes, especially for perspective projections.
2D transformations can be represented by matrices and include translations, rotations, scalings, and reflections. Translations move objects by adding a translation vector. Rotations rotate objects around the origin by pre-multiplying the point coordinates with a rotation matrix. Scaling enlarges or shrinks objects by multiplying the point coordinates with scaling factors. Composite transformations represent multiple transformations applied in sequence, with the overall transformation represented as the matrix product of the individual transformations. The order of transformations matters as matrix multiplication is not commutative.
This document discusses techniques for modeling curves and surfaces in computer graphics. It introduces three common representations of curves and surfaces: explicit, implicit, and parametric forms. It focuses on parametric polynomial forms, specifically discussing cubic polynomial curves, Hermite curves, Bezier curves, B-splines, and NURBS. It also covers rendering curves and surfaces by evaluating polynomials, recursive subdivision of Bezier polynomials, and ray casting for implicit surfaces like quadrics. Finally, it discusses mesh subdivision techniques like Catmull-Clark and Loop subdivision for generating smooth surfaces.
The depth buffer method is used to determine visibility in 3D graphics by testing the depth (z-coordinate) of each surface to determine the closest visible surface. It involves using two buffers - a depth buffer to store the depth values and a frame buffer to store color values. For each pixel, the depth value is calculated and compared to the existing value in the depth buffer, and if closer the color and depth values are updated in the respective buffers. This method is implemented efficiently in hardware and processes surfaces one at a time in any order.
A Bezier curve is a mathematically defined curve used in graphic applications. It is defined by four points: two anchors at the initial and terminal positions and two handles that control the shape of the curve. The curve can be altered by moving the handles. Bezier curves are commonly used to piece together complex curves from simpler component curves. Matching endpoints ensures continuity, while aligning handles ensures smooth tangency between connecting segments.
This presentation gives a brief idea about Interpolation. Methods of interpolating with equally/unequally spaced intervals. Please note that not all the methods are being covered in this presentation. Topics like extrapolation and inverse interpolation have also been kept aside for another ppt.
Visible surface detection in computer graphicanku2266
Visible surface detection aims to determine which parts of 3D objects are visible and which are obscured. There are two main approaches: object space methods compare objects' positions to determine visibility, while image space methods process surfaces one pixel at a time to determine visibility based on depth. Depth-buffer and A-buffer methods are common image space techniques that use depth testing to handle occlusion.
The document discusses different methods for 3D display and projection. It describes parallel projection, where lines of sight are parallel, and perspective projection, where lines converge at vanishing points. The key types of projection are outlined as parallel (orthographic and oblique) and perspective. Orthographic projection uses perpendicular lines, while oblique projection uses arbitrary angles. Perspective projection creates realistic size variation with distance and can have one, two, or three vanishing points.
This document discusses various applications of interpolation in computer science and engineering. It describes interpolation as a method of constructing new data points within the range of a known discrete data set. Some examples of interpolation applications mentioned include estimating population values, image processing through transformations like resizing and rotation, zooming digital images using different interpolation functions, and ray tracing in computer graphics. Numerical integration techniques like the trapezoidal rule, Simpson's rule, and Romberg's method are also briefly covered.
B. SC CSIT Computer Graphics Unit 2 By Tekendra Nath YogiTekendra Nath Yogi
The document discusses various 2D and 3D geometric transformations including translation, rotation, scaling, and their implementations using matrix representations and homogeneous coordinates. It provides examples of translating, rotating, and scaling points and polygons. It also covers composite transformations, inverse transformations, and applications to 2D and 3D viewing. Homework examples are provided to practice applying the different transformation types.
A Bezier curve is defined by four points that determine its shape and movement. It is commonly used in computer graphics, animation, and fonts to model smooth curves. Bezier curves can be pieced together and their control points adjusted to ensure continuity between sections. They allow complex curves to be generated from simple components. Bezier curves are widely applied in fields like computer graphics, animation, and font design due their ability to efficiently represent smooth curves.
This document discusses various algorithms for polygon scan conversion and filling, including:
- The scan line polygon fill algorithm which determines pixel color by calculating polygon edge intersections with scan lines and using an odd-even rule.
- Methods for handling special cases like horizontal edges and vertex intersections.
- Using a sorted edge table and active edge list to incrementally calculate edge intersections across scan lines.
- Flood fill and depth/z-buffer algorithms for hidden surface removal when rendering overlapping polygons.
Reflection, Scaling, Shear, Translation, and RotationSaumya Tiwari
The algorithm takes input coordinates for a 2D or 3D point and applies various linear transformations - reflection, scaling, shear, translation, and rotation. For reflections, it calculates the reflected coordinates across lines or planes through different axes. For scaling and translation, it multiplies/adds the input coordinates with scaling/translation factors. For rotation, it uses rotation matrices to calculate the rotated coordinates around different axes. It prints the transformed coordinates after applying each transformation.
The document discusses algorithms and data structures, focusing on binary search trees (BSTs). It provides the following key points:
- BSTs are an important data structure for dynamic sets that can perform operations like search, insert, and delete in O(h) time where h is the height of the tree.
- Each node in a BST contains a key, and pointers to its left/right children and parent. The keys must satisfy the BST property - all keys in the left subtree are less than the node's key, and all keys in the right subtree are greater.
- Rotations are a basic operation used to restructure trees during insertions/deletions. They involve reassigning child
Cubic curves are commonly used in graphics because curves of lower order commonly have too little flexibility, while curves of higher order are usually considered unnecessarily complex and make it easy to introduce undesired wiggles.
B-spline surfaces are constructed using a grid of control points and blending functions. The blending functions are polynomials defined over intervals in each parametric direction. This allows a B-spline surface to be defined by blending between control points. Properties include continuity and local influence of control points. A surface can be subdivided by inserting knots or raising the degree. Rational B-spline surfaces use homogeneous coordinates to represent common surfaces like planes and spheres. NURBS are the basis for CAD standards like IGES.
This document discusses Hermite bicubic surface patches, which are a type of synthetic surface. Hermite bicubic surface patches connect four corner data points using a bicubic equation involving the corner points, corner tangent vectors, and corner twist vectors. The document provides the parametric equation for a Hermite bicubic surface patch and explains how boundary conditions and geometry are defined. It also discusses how Hermite bicubic surface patches provide C1 continuity between patches while maintaining C2 continuity within each patch when blending patches.
Identify those parts of a scene that are visible from a chosen viewing position.
Visible-surface detection algorithms are broadly classified according to whether
they deal with object definitions directly or with their projected images.
These two approaches are called object-space methods and image-space methods, respectively
An object-space method compares
objects and parts of objects to each other within the scene definition to determine which surfaces, as a whole, we should label as visible.
In an image-space algorithm, visibility is decided point by point at each pixel position on the projection plane.
This lecture covers various solution methods for unconstrained optimization problems, including:
1) Line search methods like dichotomous search and the Fibonacci and golden section methods for one-dimensional problems.
2) Newton's method and the false position method for curve fitting to minimize functions in one dimension.
3) Descent methods for multidimensional problems like the gradient method, which follows the negative gradient direction at each step, and how scaling can improve convergence rates.
This document discusses bottom-up parsing and handle pruning. It begins with an example grammar and string. Bottom-up parsing works by repeatedly reducing substrings matching production bodies until reaching the start symbol. A handle is a substring matching a production body whose reduction moves backward through a rightmost derivation. Handle pruning is removing nonterminal children from a parse tree, allowing reconstruction of a rightmost derivation by working backward from handles. The document provides steps for using handle pruning to rebuild a rightmost derivation in reverse from a given string. An example demonstrates finding the handles in a string and reducing based on the corresponding productions to reach the start symbol.
This document discusses methods for identifying and removing hidden surfaces when rendering 3D scenes to create a realistic 2D image. It describes two approaches: object-space methods that compare whole objects, and image-space methods that decide visibility point-by-point. It focuses on the depth buffer/z-buffer method, which processes surfaces one point at a time, comparing depth values to determine visibility and store the color of visible points. It also discusses using scan line coherence to solve hidden surfaces one scan line at a time from top to bottom.
3D transformation in computer graphicsSHIVANI SONI
This document discusses different types of 2D and 3D transformations that are used in computer graphics, including translation, rotation, scaling, shearing, and reflection. It provides the mathematical equations and transformation matrices used to perform each type of transformation on 2D and 3D points and objects. Key types of rotations discussed are roll (around z-axis), pitch (around x-axis), and yaw (around y-axis). Homogeneous coordinates are introduced for representing 3D points.
This document provides an overview of 3D transformations, including translation, rotation, scaling, reflection, and shearing. It explains that 3D transformations generalize 2D transformations by including a z-coordinate and using homogeneous coordinates and 4x4 transformation matrices. Each type of 3D transformation is defined using matrix representations and equations. Rotation is described for each coordinate axis, and reflection is explained for each axis plane. Shearing is introduced as a way to modify object shapes, especially for perspective projections.
2D transformations can be represented by matrices and include translations, rotations, scalings, and reflections. Translations move objects by adding a translation vector. Rotations rotate objects around the origin by pre-multiplying the point coordinates with a rotation matrix. Scaling enlarges or shrinks objects by multiplying the point coordinates with scaling factors. Composite transformations represent multiple transformations applied in sequence, with the overall transformation represented as the matrix product of the individual transformations. The order of transformations matters as matrix multiplication is not commutative.
This document discusses techniques for modeling curves and surfaces in computer graphics. It introduces three common representations of curves and surfaces: explicit, implicit, and parametric forms. It focuses on parametric polynomial forms, specifically discussing cubic polynomial curves, Hermite curves, Bezier curves, B-splines, and NURBS. It also covers rendering curves and surfaces by evaluating polynomials, recursive subdivision of Bezier polynomials, and ray casting for implicit surfaces like quadrics. Finally, it discusses mesh subdivision techniques like Catmull-Clark and Loop subdivision for generating smooth surfaces.
The depth buffer method is used to determine visibility in 3D graphics by testing the depth (z-coordinate) of each surface to determine the closest visible surface. It involves using two buffers - a depth buffer to store the depth values and a frame buffer to store color values. For each pixel, the depth value is calculated and compared to the existing value in the depth buffer, and if closer the color and depth values are updated in the respective buffers. This method is implemented efficiently in hardware and processes surfaces one at a time in any order.
A Bezier curve is a mathematically defined curve used in graphic applications. It is defined by four points: two anchors at the initial and terminal positions and two handles that control the shape of the curve. The curve can be altered by moving the handles. Bezier curves are commonly used to piece together complex curves from simpler component curves. Matching endpoints ensures continuity, while aligning handles ensures smooth tangency between connecting segments.
This presentation gives a brief idea about Interpolation. Methods of interpolating with equally/unequally spaced intervals. Please note that not all the methods are being covered in this presentation. Topics like extrapolation and inverse interpolation have also been kept aside for another ppt.
Visible surface detection in computer graphicanku2266
Visible surface detection aims to determine which parts of 3D objects are visible and which are obscured. There are two main approaches: object space methods compare objects' positions to determine visibility, while image space methods process surfaces one pixel at a time to determine visibility based on depth. Depth-buffer and A-buffer methods are common image space techniques that use depth testing to handle occlusion.
The document discusses different methods for 3D display and projection. It describes parallel projection, where lines of sight are parallel, and perspective projection, where lines converge at vanishing points. The key types of projection are outlined as parallel (orthographic and oblique) and perspective. Orthographic projection uses perpendicular lines, while oblique projection uses arbitrary angles. Perspective projection creates realistic size variation with distance and can have one, two, or three vanishing points.
This document discusses various applications of interpolation in computer science and engineering. It describes interpolation as a method of constructing new data points within the range of a known discrete data set. Some examples of interpolation applications mentioned include estimating population values, image processing through transformations like resizing and rotation, zooming digital images using different interpolation functions, and ray tracing in computer graphics. Numerical integration techniques like the trapezoidal rule, Simpson's rule, and Romberg's method are also briefly covered.
B. SC CSIT Computer Graphics Unit 2 By Tekendra Nath YogiTekendra Nath Yogi
The document discusses various 2D and 3D geometric transformations including translation, rotation, scaling, and their implementations using matrix representations and homogeneous coordinates. It provides examples of translating, rotating, and scaling points and polygons. It also covers composite transformations, inverse transformations, and applications to 2D and 3D viewing. Homework examples are provided to practice applying the different transformation types.
A Bezier curve is defined by four points that determine its shape and movement. It is commonly used in computer graphics, animation, and fonts to model smooth curves. Bezier curves can be pieced together and their control points adjusted to ensure continuity between sections. They allow complex curves to be generated from simple components. Bezier curves are widely applied in fields like computer graphics, animation, and font design due their ability to efficiently represent smooth curves.
This document discusses various algorithms for polygon scan conversion and filling, including:
- The scan line polygon fill algorithm which determines pixel color by calculating polygon edge intersections with scan lines and using an odd-even rule.
- Methods for handling special cases like horizontal edges and vertex intersections.
- Using a sorted edge table and active edge list to incrementally calculate edge intersections across scan lines.
- Flood fill and depth/z-buffer algorithms for hidden surface removal when rendering overlapping polygons.
Reflection, Scaling, Shear, Translation, and RotationSaumya Tiwari
The algorithm takes input coordinates for a 2D or 3D point and applies various linear transformations - reflection, scaling, shear, translation, and rotation. For reflections, it calculates the reflected coordinates across lines or planes through different axes. For scaling and translation, it multiplies/adds the input coordinates with scaling/translation factors. For rotation, it uses rotation matrices to calculate the rotated coordinates around different axes. It prints the transformed coordinates after applying each transformation.
The document discusses algorithms and data structures, focusing on binary search trees (BSTs). It provides the following key points:
- BSTs are an important data structure for dynamic sets that can perform operations like search, insert, and delete in O(h) time where h is the height of the tree.
- Each node in a BST contains a key, and pointers to its left/right children and parent. The keys must satisfy the BST property - all keys in the left subtree are less than the node's key, and all keys in the right subtree are greater.
- Rotations are a basic operation used to restructure trees during insertions/deletions. They involve reassigning child
Cubic curves are commonly used in graphics because curves of lower order commonly have too little flexibility, while curves of higher order are usually considered unnecessarily complex and make it easy to introduce undesired wiggles.
B-spline surfaces are constructed using a grid of control points and blending functions. The blending functions are polynomials defined over intervals in each parametric direction. This allows a B-spline surface to be defined by blending between control points. Properties include continuity and local influence of control points. A surface can be subdivided by inserting knots or raising the degree. Rational B-spline surfaces use homogeneous coordinates to represent common surfaces like planes and spheres. NURBS are the basis for CAD standards like IGES.
This document discusses Hermite bicubic surface patches, which are a type of synthetic surface. Hermite bicubic surface patches connect four corner data points using a bicubic equation involving the corner points, corner tangent vectors, and corner twist vectors. The document provides the parametric equation for a Hermite bicubic surface patch and explains how boundary conditions and geometry are defined. It also discusses how Hermite bicubic surface patches provide C1 continuity between patches while maintaining C2 continuity within each patch when blending patches.
Parametric surfaces represent complex surfaces using vector functions of two variables and multiple surface patches. Bilinear and Coons surfaces are constructed using interpolation between boundary curves. Coons bicubic surfaces use cubic splines for boundary curves and blending functions, allowing greater flexibility. Shape is controlled by modifying corner vertices, tangent vectors, and twist vectors. Developable surfaces can be unfolded without stretching and have zero Gaussian curvature everywhere.
This document discusses coordinate systems and mapping between world coordinates and screen coordinates in OpenGL. It explains that:
1) Objects are defined using world coordinates, while screens use pixel coordinates, so OpenGL maps between these spaces.
2) The world window defines the region of the world coordinates that will be drawn, and the viewport defines the screen region it will be drawn to.
3) OpenGL uses a linear transformation to map world to screen coordinates, defined by scaling and translation constants A, B, C, and D that are calculated based on the world window and viewport sizes.
The document discusses the window-to-viewport transformation process which maps a 2D scene to device coordinates. It involves developing formulas to proportionally map points from the world window to the viewport. The scale and translation factors that relate the world window and viewport coordinates are defined. Changing the viewport position, using multiple viewports, or adjusting the viewport dimensions allows manipulating the displayed scene. Distortion may occur if the aspect ratios of the world window and viewport differ.
Bezier surfaces are parametric surfaces used in computer graphics and CAD/CAM. They are based on Bernstein polynomials and control points. A Bezier surface is defined by a grid of control points that determine the shape of the surface. Changing control points modifies the shape globally. B-spline surfaces allow for more local control and ensure continuity between patches. Coons patches interpolate between four boundary curves to generate a smooth surface. Sculptured surfaces are used for complex, free-form shapes and consist of blended parametric surface patches.
Curves And Surfaces Representation And ApplicationDiaa ElKott
The document discusses curves and surfaces from mathematical and manufacturing perspectives. It introduces common curve types like circles and their parametric, implicit, and explicit representations. It also discusses curve properties like tangents and normals. Additionally, it covers the manufacturing process and sources of error at different stages from design to inspection. Representation of curves and surfaces is important for both analysis and manufacturing applications.
The document discusses window to viewport transformation. It defines a window as a world coordinate area selected for display and a viewport as a rectangular region of the screen selected for displaying objects. Window to viewport mapping requires transforming coordinates from the window to the viewport. This involves translation, scaling and another translation. Steps include translating the window to the origin, resizing it based on the viewport size, and translating it to the viewport position. An example transforms a sample window to a viewport through these three steps.
Group technology (GT) and cellular manufacturing involve organizing production around part families and machine cells. GT involves standardizing similar tasks, storing information about recurring problems, and performing similar activities together. Cellular manufacturing reduces flow times, inventories, and allows for rush orders. It involves analyzing machine-component relationships and grouping machines and parts into cells. Benefits of GT and cellular manufacturing include reduced setup times, work in process, defects, and costs as well as improved material flow and customer service.
This document provides an overview of flexible manufacturing systems (FMS). It defines FMS as an automated machine cell consisting of interconnected processing workstations and automated material handling. It discusses the history and purpose of FMS in optimizing manufacturing cycle times and reducing costs. The basic components of FMS are described as workstations, automated material handling systems, and computer control systems. The document outlines different types of FMS layouts and how flexibility is achieved. It provides examples of FMS applications and discusses the advantages of FMS in improving efficiency and reducing production time, while also noting the high expenses associated with implementation.
Concurrent engineering is a business strategy that replaces the traditional sequential product development process with one where tasks are done in parallel and all aspects of a product's development are considered early on. It focuses on optimizing resource distribution to ensure efficient product development. Concurrent engineering requires collaboration across individuals, groups, departments and organizations to develop a product rapidly and responsively to changing market needs. It is a complex strategic plan that demands full corporate commitment and strong leadership to implement continuous process improvements.
Find the compact trigonometric Fourier series for the periodic signal.pdfarihantelectronics
Find the compact trigonometric Fourier series for the periodic signal x(t) and sketch the
amplitude and phase spectrum for first 4 frequency components. By inspection of the spectra,
sketch the exponential Fourier spectra. By inspection of spectra in part b), write the exponential
Fourier series for x(t)
Solution
ECE 3640 Lecture 4 – Fourier series: expansions of periodic functions. Objective: To build upon
the ideas from the previous lecture to learn about Fourier series, which are series representations
of periodic functions. Periodic signals and representations From the last lecture we learned how
functions can be represented as a series of other functions: f(t) = Xn k=1 ckik(t). We discussed
how certain classes of things can be built using certain kinds of basis functions. In this lecture we
will consider specifically functions that are periodic, and basic functions which are
trigonometric. Then the series is said to be a Fourier series. A signal f(t) is said to be periodic
with period T0 if f(t) = f(t + T0) for all t. Diagram on board. Note that this must be an everlasting
signal. Also note that, if we know one period of the signal we can find the rest of it by periodic
extension. The integral over a single period of the function is denoted by Z T0 f(t)dt. When
integrating over one period of a periodic function, it does not matter when we start. Usually it is
convenient to start at the beginning of a period. The building block functions that can be used to
build up periodic functions are themselves periodic: we will use the set of sinusoids. If the period
of f(t) is T0, let 0 = 2/T0. The frequency 0 is said to be the fundamental frequency; the
fundamental frequency is related to the period of the function. Furthermore, let F0 = 1/T0. We
will represent the function f(t) using the set of sinusoids i0(t) = cos(0t) = 1 i1(t) = cos(0t + 1)
i2(t) = cos(20t + 2) . . . Then, f(t) = C0 + X n=1 Cn cos(n0t + n) The frequency n0 is said to be
the nth harmonic of 0. Note that for each basis function associated with f(t) there are actually two
parameters: the amplitude Cn and the phase n. It will often turn out to be more useful to
represent the function using both sines and cosines. Note that we can write Cn cos(n0t + n) = Cn
cos(n) cos(n0t) Cn sin(n)sin(n0t). ECE 3640: Lecture 4 – Fourier series: expansions of periodic
functions. 2 Now let an = Cn cos n bn = Cn sin n Then Cn cos(n0t + n) = an cos(n0t) + bn
sin(n0t) Then the series representation can be f(t) = C0 + X n=1 Cn cos(n0t + n) = a0 + X n=1 an
cos(n0t) + bn sin(n0t) The first of these is the compact trigonometric Fourier series. The second
is the trigonometric Fourier series.. To go from one to the other use C0 = a0 Cn = p a 2 n + b 2 n
n = tan1 (bn/an). To complete the representation we must be able to compute the coefficients.
But this is the same sort of thing we did before. If we can show that the set of functions
{cos(n0t),sin(n0t)} is in fact an orthogonal set, then we can use the same.
Bezier Curves, properties of Bezier Curves, Derivation for Quadratic Bezier Curve, Blending function specification for Bezier curve:, B-Spline Curves, properties of B-spline Curve?
Matrices can be used to represent 2D geometric transformations such as translation, scaling, and rotation. Translations move objects by adding offsets to x and y coordinates. Scaling enlarges or shrinks objects by multiplying x and y values. Rotations turn objects by applying trigonometric functions to coordinates. Homogeneous coordinates allow these transformations to be represented uniformly by matrix multiplication. Transformations can be combined by multiplying their matrices in sequence, with the order affecting the result. Only rigid transformations involving solely rotations and translations preserve angles and lengths, while more general affine transformations including scaling can distort objects.
This document discusses arithmetic functions in digital circuits. It begins by introducing iterative combinational circuits and binary adders such as half adders, full adders, and ripple carry adders. It then covers binary subtraction using two's complement representation and signed binary numbers. The document explains various signed number representations including signed magnitude and signed complement. It provides examples of addition and subtraction for signed integers in these representations.
1. The document discusses the basis functions used in 1D and 2D finite element analysis.
2. For 1D elements, it presents the linear, quadratic and cubic basis functions derived from imposing interpolation conditions on linear, quadratic and cubic shape functions respectively.
3. For 2D elements, it derives the linear and quadratic basis functions for triangular and rectangular elements by transforming the element coordinates and imposing interpolation conditions on linear and quadratic Ansatz shape functions.
This document discusses image interpolation and resampling. It explains that interpolation is needed when transforming images, such as scaling or rotating, as the transformed coordinates are not usually integers. Various interpolation methods are covered, including constant, linear, and cubic interpolation using B-spline functions. Examples are provided to demonstrate cubic interpolation to estimate values between given sample points. The document also discusses properties of interpolation, such as using nearby sample values and avoiding discontinuities.
Matrices can be used to represent 2D geometric transformations such as translation, scaling, and rotation. Translations are represented by adding a translation vector to coordinates. Scaling is represented by multiplying coordinates by scaling factors. Rotations are represented by premultiplying coordinates with a rotation matrix. Multiple transformations can be combined by multiplying their respective matrices. Using homogeneous coordinates allows all transformations to be uniformly represented as matrix multiplications.
Approximating offset curves using B ´ ezier curves with high accuracyIJECEIAES
In this paper, a new method for the approximation of offset curves is presented using the idea of the parallel derivative curves. The best uniform approximation of degree 3 with order 6 is used to construct a method to find the approximation of the offset curves for B ´ ezier curves. The proposed method is based on the best uniform approximation, and therefore; the proposed method for constructing the offset curves induces better outcomes than the existing methods.
1) Sorting algorithms are useful for keeping data ordered to aid other algorithms like searching. Finding the optimal order for matrix multiplication can greatly reduce computation time when transformations don't need to be performed immediately.
2) The inverse of a matrix is unique. The product of two lower triangular matrices is lower triangular. The inverse of a lower triangular matrix is also lower triangular.
3) Gaussian elimination preserves the inverse of a matrix up to row operations on the inverse. If a matrix has only real elements, then its inverse and cofactors must also be real.
1. The document provides definitions and identities relating to theoretical computer science topics like asymptotic analysis, series, recurrences, and discrete structures.
2. It includes definitions for big O, Omega, and Theta notation used to describe the asymptotic behavior of functions.
3. Recurrences, like the master method, are presented for analyzing the runtime of divide-and-conquer algorithms.
4. Discrete structures like combinations, Stirling numbers, and trees are also covered, along with their properties and relationships.
The document discusses various 2-D orthogonal and unitary transforms that can be used to represent digital images, including:
1. The discrete Fourier transform (DFT) which transforms an image into the frequency domain and has properties like energy conservation and fast computation via FFT.
2. The discrete cosine transform (DCT) which has good energy compaction properties and is close to the optimal Karhunen-Loeve transform.
3. The discrete sine transform (DST) which is real, symmetric, and orthogonal like the DCT.
4. The Hadamard transform which uses only ±1 values and has a fast computation, and the Haar transform which is a simpler wavelet transform
This document discusses recurrence relations and methods for solving recurrences. It introduces recurrence relations and examples. It covers the substitution method, iteration method, and Master Theorem for solving recurrences. The Master Theorem is a technique for solving divide-and-conquer recurrences to determine asymptotic tight bounds. Examples are provided to demonstrate applying these techniques.
CS330-Lectures Statistics And Probabilitybryan111472
Here are the key steps to multiply two large integers efficiently:
1. Use a divide-and-conquer approach called the Karatsuba algorithm. It recursively breaks the multiplication into smaller subproblems.
2. On inputs of size n, the Karatsuba algorithm uses 3 multiplications of size approximately n/2 instead of the 4 multiplications of the grade school method.
3. This reduction saves a multiplicative factor, resulting in an asymptotic running time of O(n^{log_2 3}) ≈ O(n^{1.585}), which is faster than the O(n^2) time of the grade school method.
4. The main idea is to express the products of
The purpose of this note is to elaborate in how far a 4 factor affine model can generate an incomplete bond market together with the flexibility of a 3 factor flexible affine cascade structure model.
This document discusses Fourier analysis and periodic functions. It defines periodic functions and introduces Fourier series representation of periodic functions as the sum of sinusoidal components. The key points are:
1) A periodic function can be represented by an infinite series of sinusoids of harmonically related frequencies, known as Fourier series representation.
2) The coefficients of the Fourier series (a0, an, bn) can be calculated from the integrals of the function over one period.
3) Periodic functions may exhibit odd symmetry, even symmetry, or half-wave symmetry, which determines which coefficients (an or bn) are zero.
4) Examples demonstrate the Fourier analysis of specific periodic waveforms, such as a
The document provides homework problems and solutions related to signals processing. It includes problems on determining the frequency and period of signals, properties of even and odd signals, sampling of continuous signals, and periodicity of sums of signals. It also provides detailed solutions and explanations to each problem.
This document discusses using persistent homology to analyze the topological structure of proteins and relate it to protein compressibility. It summarizes that researchers modeled protein molecules as alpha filtrations to obtain multi-scale insight into their tunnel and cavity structures. The persistence diagrams of the alpha filtrations capture the sizes and robustness of these features in a compact way. The researchers found a clear linear correlation between their topological measure and experimentally determined protein compressibility values.
linear transformation and rank nullity theorem Manthan Chavda
In these notes, I will present everything we know so far about linear transformations.
This material comes from sections in the book, and supplemental that
I talk about in class.
This document discusses different methods of spline interpolation. Spline interpolation is a technique for constructing smooth curves that pass through a set of points. Linear spline interpolation connects points with straight lines, while quadratic and cubic spline interpolation use polynomials to generate smoother curves. Natural cubic spline interpolation minimizes curvature and ensures the resulting spline is twice continuously differentiable. Bezier spline interpolation uses Bernstein polynomials to define curves based on control points, allowing for easy modification of the curve shape.
The document defines unit tangent and normal vectors for vector-valued functions. The unit tangent vector T(t) is the normalized derivative of the vector function C(t). The principal unit normal vector N is defined as the derivative of T(t) normalized. In 2D, there are two normal vectors to the tangent vector C'(t); the document provides formulas to determine them. Examples are given to demonstrate calculating T(t) and N for a given vector function at a specified time.
This document discusses fractals and provides examples of fractal images and algorithms. It defines fractals as self-similar mathematical expressions that approach infinity in an organized way. Examples of natural fractals mentioned include clouds, coastlines, and trees. The Mandelbrot set is presented as a famous fractal generated using iteration and recursion. Algorithms for generating the Koch curve and Koch snowflake fractals are described.
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.
This document contains code for a login system program with the following functionality:
1. It allows users to register with a username and password, which are encrypted and stored in a database.
2. Users can log in by entering their username and password, which are decrypted and validated against the database.
3. Registered users can be viewed by decrypting the usernames retrieved from the database.
4. The database contents can be displayed.
The program uses a switch case to prompt the user for their choice of these options and take the appropriate action.
This document discusses components of an interactive graphics system and different display techniques. It asks the reader to discuss the role of the display processor, device controller, frame buffer and CRT in an interactive graphics system. It also asks the reader to distinguish between raster scan display and random scan display techniques.
This document provides an introduction to computer graphics. It begins by defining computer graphics as using computers to generate and manipulate visual images and discusses how computer graphics has evolved from traditional technical drawings. The document then outlines several key applications of computer graphics, including presentation graphics, painting/drawing, photo editing, scientific visualization, image processing, education/training/entertainment, simulations, and animation/games. It also describes common graphics hardware components like input/output and display devices. The overall purpose is to introduce the field of computer graphics and discuss its uses and technologies.
This document discusses graphics hardware components. It describes various graphics input devices like the mouse, joystick, light pen etc. and how they are either analog or digital. It then covers commonly used graphics output devices like CRT displays, plasma displays, LCDs and 3D viewing systems. It provides details on the internal components and working of CRT displays. It also discusses graphics storage formats and the architecture of raster and random graphics systems.
This document provides a course plan for a Computer Graphics Lab course offered at the University of Petroleum & Energy Studies during the sixth semester from January 2015 to July 2015. The course aims to provide an overview of computer graphics and 3D modeling. The course outline includes 9 units covering topics like 2D and 3D transformations, curves, hidden surfaces, color models, and image processing. The document also lists 6 experiments related to scan conversion, filling algorithms, 2D transformations, viewing and clipping. It provides guidelines for students regarding attendance policy, passing criteria, and communication protocols.
This document provides a course plan for the Computer Graphics course offered at the University of Petroleum & Energy Studies during the sixth semester from January 2015 to July 2015. The course aims to provide an overview of the basics of computer graphics and 3D modeling. It will cover topics such as 2D and 3D transformations, curves, hidden surface removal, color models, and shading techniques. Students will implement graphics algorithms and design 2D-3D objects. Their performance will be evaluated through assignments, tests, quizzes, and a semester-end project. Lectures will be supplemented with presentations and examples to aid understanding. Relevant textbooks and online resources have also been suggested.
This document contains code snippets for implementing three different Android UI elements: AutocompleteTextView, Spinner, and ToggleButton.
The AutocompleteTextView snippet shows how to set up an AutocompleteTextView widget with a string array adapter to provide autocomplete suggestions.
The Spinner snippet demonstrates how to populate a spinner dropdown with string options from an array, set a click listener to handle item selection, and display a toast notification with the selected item.
The ToggleButton snippet sets up two toggle buttons and a button to display the on/off states of the toggle buttons in a toast message when clicked.
The document discusses algorithms for drawing lines and circles on a discrete pixel display. It begins by describing what characteristics an "ideal line" would have on such a display. It then introduces several algorithms for drawing lines, including the simple line algorithm, digital differential analyzer (DDA) algorithm, and Bresenham's line algorithm. The Bresenham algorithm is described in detail, as it uses only integer calculations. Next, a simple potential circle drawing algorithm is presented and its shortcomings discussed. Finally, the more accurate and efficient mid-point circle algorithm is introduced. This algorithm exploits the eight-way symmetry of circles and only calculates points in one octant.
1. On-Line Geometric Modeling Notes
DEFINITION OF A B-SPLINE CURVE
Kenneth I. Joy
Visualization and Graphics Research Group
Department of Computer Science
University of California, Davis
Overview
These notes present the direct definition of the B-Spline curve. This definition is given in two ways: first
by an analytical definition using the normalized B-spline blending functions, and then through a geometric
definition.
The B-Spline Curve – Analytical Definition
A B-spline curve P(t), is defined by
P(t) =
n
i=0
PiNi,k(t)
where
• the {Pi : i = 0, 1, ..., n} are the control points,
• k is the order of the polynomial segments of the B-spline curve. Order k means that the curve is made
up of piecewise polynomial segments of degree k − 1,
• the Ni,k(t) are the “normalized B-spline blending functions”. They are described by the order k and
by a non-decreasing sequence of real numbers
{ti : i = 0, ..., n + k} .
2. normally called the “knot sequence”. The Ni,k functions are described as follows
Ni,1(t) =
1 if u ∈ [ti, ti+1),
0 otherwise.
(1)
and if k > 1,
Ni,k(t) =
t − ti
ti+k−1 − ti
Ni,k−1(t) +
ti+k − t
ti+k − ti+1
Ni+1,k−1(t) (2)
• and t ∈ [tk−1, tn+1).
We note that if, in equation (2), either of the N terms on the right hand side of the equation are zero,
or the subscripts are out of the range of the summation limits, then the associated fraction is not evaluated
and the term becomes zero. This is to avoid a zero-over-zero evaluation problem. We also direct the readers
attention to the “closed-open” interval in the equation (1).
The order k is independent of the number of control points (n + 1). In the B-Spline curve, unlike
the B´ezier Curve, we have the flexibility of using many control points, and restricting the degree of the
polymonial segments.
The B-Spline Curve – Geometric Definition
Given a set of Control Points {P0, P1, ..., Pn}, an order k, and a set of knots {t0, t1, ..., tn+k}, the
B-Spline curve of order k is defined to be
P(t) = P
(k−1)
l (t) if u ∈ [tl, tl+1)
where
P
(j)
i (t) =
(1 − τj
i )P
(j−1)
i−1 (t) + τj
i P
(j−1)
i (t) if j > 0,
Pi if j = 0.
and
τj
i =
t − ti
ti+k−j − ti
2
3. It is useful to view the geometric construction as the following pyramid
...
Pl−k+1
P
(1)
l−k+2
Pl−k+2 P
(2)
l−k+3
P
(1)
l−k+3 ·
Pl−k+3 ·
Pl−k+4
· P
(k−2)
l−1
· P
(k−1)
l
· P
(k−2)
l
·
Pl−2 P
(2)
l−1
P
(1)
l−1 ·
Pl−1 P
(2)
l
P
(1)
l
Pl
...
Any P in this pyramid is calculated as a convex combination of the two P functions immediately to it’s left.
All contents copyright (c) 1996, 1997, 1998, 1999
Computer Science Department, University of California, Davis
All rights reserved.
3
4. On-Line Geometric Modeling Notes
THE UNIFORM B-SPLINE BLENDING FUNCTIONS
Kenneth I. Joy
Visualization and Graphics Research Group
Department of Computer Science
University of California, Davis
Overview
The normalized B-spline blending functions are defined recursively by
Ni,1(t) =
1 if u ∈ [ti, ti+1)
0 otherwise
(1)
and if k > 1,
Ni,k(t) =
t − ti
ti+k−1 − ti
Ni,k−1(t) +
ti+k − t
ti+k − ti+1
Ni+1,k−1(t) (2)
where {t0, t1, ..., tn+k} is a non-decreasing sequence of knots, and k is the order of the curve.
These functions are difficult to calculate directly for a general knot sequence. However, if the knot
sequence is uniform, it is quite straightforward to calculate these functions – and they have some suprising
properties.
Calculating the Blending Functions using a Uniform Knot Sequence
Assume that {t0, t1, t2, ..., tn} is a uniform knot sequence, i.e., {0, 1, 2, ..., n}. This will simplify the
calculation of the blending functions, as ti = i.
Blending Functions for k = 1
5. if k = 1, then by using equation (1), we can write the normalized blending functions as
Ni,1(t) =
1 if u ∈ [i, i + 1)
0 otherwise
(3)
These are shown together in the following figure, where we have plotted N0,1, N1,1, N2,1, and N3,1 respec-
tively, over five of the knots. Note the white circle at the end of the line where the functions value is 1. This
represents the affect of the “open-closed” interval found in equation (1).
2
9. For the case k = 3, we again use equation (2) to obtain
N0,3(t) =
t − t0
t2 − t0
N0,2(t) +
t3 − t
t3 − t1
N1,2(t)
=
t
2
N0,2(t) +
3 − t
2
N1,2(t)
=
t2
2 if 0 ≤ t 1
t2
2 (2 − t) + 3−t
2 (t − 1) if 1 ≤ t 2
(3−t)2
2 if 2 ≤ t 3
0 otherwise
=
t2
2 if 0 ≤ t 1
−3+6t−2t2
2 if 1 ≤ t 2
(3−t)2
2 if 2 ≤ t 3
0 otherwise
and by nearly identical calculations,
N1,3(t) =
t − t1
t3 − t1
N1,2(t) +
t4 − t
t4 − t2
N2,2(t)
=
t − 1
2
N1,2(t) +
4 − t
2
N2,2(t)
=
(t−1)2
2 if 1 ≤ t 2
−11+10t−2t2
2 if 2 ≤ t 3
(4−t)2
2 if 3 ≤ t 4
0 otherwise
These curves are shown in the following figure. They are clearly piecewise quadratic curves, each made up
of three parabolic segments that are joined at the knot values (we have placed tick marks on the curves to
show where they join).
6
11. Summary
In the case of the uniform knot sequence, the blending functions are fairly easy to calculate, are shifted
versions of each other, and have support over a simple interval determined by the knots. These characteristics
are unique to the uniform blending functions.
For other special characteristics see the notes that describe writing the uniform blending functions as a
convolution, and the notes that describe the two-scale relation for uniform B-splines
All contents copyright (c) 1996, 1997, 1998, 1999, 2000
Computer Science Department, University of California, Davis
All rights reserved.
8
12. On-Line Geometric Modeling Notes
THE DEBOOR-COX CALCULATION
Kenneth I. Joy
Visualization and Graphics Research Group
Department of Computer Science
University of California, Davis
Overview
In 1972, Carl DeBoor and M.G. Cox independently discovered the relationship between the analytic
and geometric definitions of B-splines. Starting with the definition of the normalized B-spline blending
functions, these two researchers were able to develop the geometric definition of the B-spline. It is this
calculation that is discussed in this paper.
Definition of the B-Spline Curve
A B-spline curve P(t), is defined by
P(t) =
n
i=0
PiNi,k(t) (1)
where
• the {Pi : i = 0, 1, ..., n} are the control points,
• k is the order of the polynomial segments of the B-spline curve. Order k means that the curve is made
up of piecewise polynomial segments of degree k − 1,
• the Ni,k(t) are the “normalized B-spline blending functions”. They are described by the order k and
by a non-decreasing sequence of real numbers
{ti : i = 0, ..., n + k} .
13. normally called the “knot sequence”. The Ni,k functions are described as follows
Ni,1(t) =
1 if t ∈ [ti, ti+1),
0 otherwise.
(2)
and if k 1,
Ni,k(t) =
t − ti
ti+k−1 − ti
Ni,k−1(t) +
ti+k − t
ti+k − ti+1
Ni+1,k−1(t) (3)
• the parameter t ranges throughout the interval [tk−1, tn+1).
We note that if, in equation (3), either of the N terms on the right hand side of the equation are zero, or
the subscripts are out of the range of the summation limits, then the associated fraction is not evaluated and
the term becomes zero. This is to avoid a zero-over-zero evaluation problem.
The order k is independent of the number of control points (n + 1). In the B-Spline curve, unlike
the B´ezier Curve, we have the flexibility of using many control points, and restricting the degree of the
polymonial segments.
The DeBoor-Cox Calculation
In the DeBoor-Cox calculation, we substitute the definition of Ni,k(t) given in equation (3), into the
right-hand side of
P(t) =
n
i=0
PiNi,k(t)
and simplify. This will give us the definition of P(t) in terms of Ni,k−1, which is of lower degree. The
general idea is to continue this process until the sum is written with Ni,1 functions, which we can evaluate
easily.
So here we go. If t ∈ [tk−1, tn+1), then by substituting equation (3) into the equation (1), we have
2
14. P(t) =
n
i=0
PiNi,k(t)
=
n
i=0
Pi
t − ti
ti+k−1 − ti
Ni,k−1(t) +
ti+k − t
ti+k − ti+1
Ni+1,k−1(t)
Distributing the sums, we obtain
=
n
i=0
Pi
t − ti
ti+k−1 − ti
Ni,k−1(t) +
n
i=0
Pi
ti+k − t
ti+k − ti+1
Ni+1,k−1(t)
We now separate out those unique terms of each sum, N0,k−1 and Nn+1,k−1, giving
= P0
t − t0
tk−1 − t0
N0,k−1(t) +
n
i=1
Pi
t − ti
ti+k−1 − ti
Ni,k−1(t)
+
n−1
i=0
Pi
ti+k − t
ti+k − ti+1
Ni+1,k−1(t) + Pn
tn+k − t
tn+k − tn+1
Nn+1,k−1(t)
Now since the support of a B-spline blending function Ni,k(t) is the interval [ti, ti+k], we have that
N0,k−1 is non-zero only if t ∈ [t0, tk−1), which is outside the interval [tk−1, tn+1) (where P(t) is defined).
Thus, N0,k−1(t) ≡ 0. Also Nn+1,k−1 is non-zero only if t ∈ [tn+1, tn+k−1), which is outside the interval
3
15. [tk−1, tn+1). Thus, Nn+1,k−1(t) ≡ 0 and we have
P(t) =
n
i=1
Pi
t − ti
ti+k−1 − ti
Ni,k−1(t) +
n−1
i=0
Pi
ti+k − t
ti+k − ti+1
Ni+1,k−1(t)
If we change the summation limits, we get
=
n
i=1
Pi
t − ti
ti+k−1 − ti
Ni,k−1(t) +
n
i=1
Pi−1
ti+k−1 − t
ti+k−1 − ti
Ni,k−1(t)
Combining the two sums, and rewriting, we obtain
=
n
i=1
ti+k−1 − t
ti+k−1 − ti
Pi−1 +
t − ti
ti+k−1 − ti
Pi Ni,k−1(t)
If we denote
P
(1)
i (t) =
ti+k−1 − t
ti+k−1 − ti
Pi−1 +
t − ti
ti+k−1 − ti
Pi
then the above result may be written
P(t) =
n
i=1
P
(1)
i (t)Ni,k−1(t)
Now we have written the summation terms of equation (1) in terms of blending functions of lower degree.
Of course, we have transferred some of the complexity to the P
(1)
i s, but we retain a similar form with control
points P
(1)
i s weighted by blending functions, and we can repeat this calculation again.
Once again then, repeating the calculation and manipulating the sums, we obtain
P(t) =
n
i=2
P
(2)
i (t)Ni,k−2(t)
where
P
(2)
i (t) =
ti+k−2 − t
ti+k−2 − ti
P
(1)
i−1(t) +
t − ti
ti+k−2 − ti
P
(1)
i (t)
4
16. (Note the appearance of the 2s on the right-hand side of the equation.)
If we continue with this process again, we will manipulate the sum so that the blending functions have
order k − 3. Then again with give us k − 4, and eventually we will obtain blending functions of order 1. We
are lead to the following result: If we define
P
(j)
i (t) =
(1 − τj
i )P
(j−1)
i−1 (t) + τj
i P
(j−1)
i (t) if j 0
Pi if j = 0
(4)
where
τj
i =
t − ti
ti+k−j − ti
Then, if t is in the interval [tl, tl+1), we have
P(t) = P
(k−1)
l (t)
This can be shown by continuing the DeBoor-Cox calculation k − 1 times. When complete, we arrive
at the formula
P(t) =
n
i=k−1
P
(k−1)
i (t)Ni,1(t)
where P
(j)
i (t) is given in equation (4). (Note the algebraic simplification that the τ’s provide.) If t ∈
[tl, tl+1], then then the only nonzero term of the sum is the lth term, which is one, and the sum must equal
P
(k−1)
l (t).
This enables us to define the geometrical definition of the B-spline curve.
Geometric Definition of the B-Spline Curve
Given a set of Control Points
P0, P1, ..., Pn
5
17. and a set of knots
t0, t1, ..., tn+k
The B-Spline curve of order k is defined to be
P(t) = P
(k−1)
l (t)if t ∈ [tl, tl+1)
where
P
(j)
i (t) =
(1 − τj
i )P
(j−1)
i−1 (t) + τj
i P
(j−1)
i (t) if j 0
Pi if j = 0
and
τj
i =
t − ti
ti+k−j − ti
6
18. It is useful to view the geometric construction as the following pyramid
...
Pl−k+1
P
(1)
l−k+2
Pl−k+2 P
(2)
l−k+3
P
(1)
l−k+3 ·
Pl−k+3 ·
Pl−k+4
· P
(k−2)
l−1
· P
(k−1)
l
· P
(k−2)
l
·
Pl−2 P
(2)
l−1
P
(1)
l−1 ·
Pl−1 P
(2)
l
P
(1)
l
Pl
...
Any P
(j)
i in this pyramid is calculated as a convex combination of the two P functions immediately to it’s
left.
Summary
The DeBoor-Cox calculation is a fundamental result in the geometric modeling field. It was used to
exhibit the relation between the analytic definition of the B-Spline curve and the geometric definition of the
curve. The geometric definition of the curve, because of its computational stability has become the primary
technique by which points on these curves are calculated.
7
19. All contents copyright (c) 1996, 1997, 1998, 1999, 2000
Computer Science Department, University of California, Davis
All rights reserved.
8
20. On-Line Geometric Modeling Notes
THE SUPPORT OF A
NORMALIZED B-SPLINE BLENDING FUNCTION
Kenneth I. Joy
Visualization and Graphics Research Group
Department of Computer Science
University of California, Davis
Overview
A B-spline blending function Ni,k(t) has compact support. This means that the function is zero outside
of some interval. In these notes, We find this interval explicitly in terms of the knot sequence.
The Support of the Function
Given an order k, and a knot sequence {t0, t1, t2, ..., tn+k}, the normalized B-spline blending function
Ni,k(t) is positive if and only if t ∈ [ti, ti+k).
21. We can show that this is true by considering the following pyramid structure.
Ni,1
. Ni+1,1
. .
. .
Ni,k−2 .
Ni,k−1
Ni,k Ni+1,k−2
Ni+1,k−1
Ni+2,k−2 .
. .
. .
. Ni+k−2,1
Ni+k−1,1
The definition of the normalized blending function Ni,k as a weighted sum of Ni,k−1(t) and Ni+1,k−1(t).
Thus for any of the N functions in the pyramid, it is a weighted sum of the two items immediately to its
right. If we follow the pyramid to its right edge, we see that the only blending functions Nj,1 that contribute
to Ni,k are those with i ≤ j ≤ i + k − 1, and these function are collectively nonzero when t ∈ [ti, ti+k).
Summary
A B-spline blending function has compact support. The support of this function depends on the knot
sequence and always covers an interval of containing several knots – containing k + 1 knots if the curve is
or order k.
All contents copyright (c) 1996, 1997, 1998, 1999, 2000
Computer Science Department, University of California, Davis
All rights reserved.
2