This document discusses two-dimensional geometric transformations including translation, rotation, and scaling. It introduces how these transformations can be represented using matrix math. Transformations can be combined by multiplying the matrices together to efficiently apply multiple transformations at once. The document concludes by introducing homogeneous coordinates, which allow geometric transformations to be uniformly represented as matrix multiplications by expanding two-dimensional points and matrices to three dimensions.
Geometric transformations include translation, rotation, and scaling. Translation moves every point of an object by adding a translation vector. Rotation rotates the object around an axis by a certain angle. Scaling enlarges or shrinks the object by a scale factor. More complex transformations can be achieved by combining these basic transformations through composition.
1) 2-D geometric transformations allow manipulation of objects in 2-D space by changing their position, size, and orientation.
2) The basic geometric transformations are translation, rotation, scaling, reflection, and shear. Translation moves an object by shifting its coordinates. Rotation turns an object around a fixed point. Scaling enlarges or shrinks an object. Reflection produces a mirror image. Shear distorts an object.
3) Each transformation can be described by a matrix equation. The inverse of a transformation performs the opposite operation to return the object to its original state.
This document discusses various two-dimensional geometric transformations including translations, rotations, scaling, reflections, shears, and composite transformations. Translations move objects without deformation using a translation vector. Rotations rotate objects around a fixed point or pivot point. Scaling transformations enlarge or shrink objects using scaling factors. Reflections produce a mirror image of an object across an axis. Shearing slants an object along an axis. Composite transformations combine multiple basic transformations using matrix multiplication.
This document discusses 2D geometric transformations including translation, rotation, and scaling. It provides the mathematical definitions and matrix representations for each transformation. Translation moves an object along a straight path, rotation moves it along a circular path, and scaling changes its size. All transformations can be represented by 3x3 matrices using homogeneous coordinates to allow combinations of multiple transformations. The inverse of each transformation matrix is also defined.
2 d transformations and homogeneous coordinatesTarun Gehlot
1. 2D transformations like translation, rotation, and scaling can be represented using homogeneous coordinates and matrix multiplication.
2. Homogeneous coordinates add a third coordinate to allow a consistent matrix notation for all transformations.
3. Transformations can be combined by multiplying their matrices in the appropriate order.
1. 2D transformations include translation, rotation, scaling, and shearing. They can change an object's position, size, orientation, and shape.
2. Transformations are represented by matrices and applied through matrix multiplication. A sequence of transformations can be combined into a single matrix by multiplying the matrices.
3. Common 2D transformations include translating an object by adding offsets to x and y, rotating objects around an origin by adjusting x and y coordinates, and scaling by multiplying x and y by factors.
This document discusses various 2D transformations including translation, rotation, scaling, shearing, and their implementations using transformation matrices. Translation moves an object along a straight line and can be represented by a 3x3 matrix. Rotation rotates an object around a center point, with the standard rotation matrix rotating around the origin. Scaling changes the size of an object, which can cause undesirable movement. Shearing skews an object along an axis. The document also covers composing multiple transformations using matrix multiplication and how OpenGL applies transformations through a global model-view matrix.
This document discusses 2D geometric transformations including translation, rotation, scaling, and composite transformations. It provides definitions and formulas for each type of transformation. Translation moves objects by adding offsets to coordinates without deformation. Rotation rotates objects around an origin by a certain angle. Scaling enlarges or shrinks objects by multiplying coordinates by scaling factors. Composite transformations apply multiple transformations sequentially by multiplying their matrices. Homogeneous coordinates are also introduced to represent transformations in matrix form.
Geometric transformations include translation, rotation, and scaling. Translation moves every point of an object by adding a translation vector. Rotation rotates the object around an axis by a certain angle. Scaling enlarges or shrinks the object by a scale factor. More complex transformations can be achieved by combining these basic transformations through composition.
1) 2-D geometric transformations allow manipulation of objects in 2-D space by changing their position, size, and orientation.
2) The basic geometric transformations are translation, rotation, scaling, reflection, and shear. Translation moves an object by shifting its coordinates. Rotation turns an object around a fixed point. Scaling enlarges or shrinks an object. Reflection produces a mirror image. Shear distorts an object.
3) Each transformation can be described by a matrix equation. The inverse of a transformation performs the opposite operation to return the object to its original state.
This document discusses various two-dimensional geometric transformations including translations, rotations, scaling, reflections, shears, and composite transformations. Translations move objects without deformation using a translation vector. Rotations rotate objects around a fixed point or pivot point. Scaling transformations enlarge or shrink objects using scaling factors. Reflections produce a mirror image of an object across an axis. Shearing slants an object along an axis. Composite transformations combine multiple basic transformations using matrix multiplication.
This document discusses 2D geometric transformations including translation, rotation, and scaling. It provides the mathematical definitions and matrix representations for each transformation. Translation moves an object along a straight path, rotation moves it along a circular path, and scaling changes its size. All transformations can be represented by 3x3 matrices using homogeneous coordinates to allow combinations of multiple transformations. The inverse of each transformation matrix is also defined.
2 d transformations and homogeneous coordinatesTarun Gehlot
1. 2D transformations like translation, rotation, and scaling can be represented using homogeneous coordinates and matrix multiplication.
2. Homogeneous coordinates add a third coordinate to allow a consistent matrix notation for all transformations.
3. Transformations can be combined by multiplying their matrices in the appropriate order.
1. 2D transformations include translation, rotation, scaling, and shearing. They can change an object's position, size, orientation, and shape.
2. Transformations are represented by matrices and applied through matrix multiplication. A sequence of transformations can be combined into a single matrix by multiplying the matrices.
3. Common 2D transformations include translating an object by adding offsets to x and y, rotating objects around an origin by adjusting x and y coordinates, and scaling by multiplying x and y by factors.
This document discusses various 2D transformations including translation, rotation, scaling, shearing, and their implementations using transformation matrices. Translation moves an object along a straight line and can be represented by a 3x3 matrix. Rotation rotates an object around a center point, with the standard rotation matrix rotating around the origin. Scaling changes the size of an object, which can cause undesirable movement. Shearing skews an object along an axis. The document also covers composing multiple transformations using matrix multiplication and how OpenGL applies transformations through a global model-view matrix.
This document discusses 2D geometric transformations including translation, rotation, scaling, and composite transformations. It provides definitions and formulas for each type of transformation. Translation moves objects by adding offsets to coordinates without deformation. Rotation rotates objects around an origin by a certain angle. Scaling enlarges or shrinks objects by multiplying coordinates by scaling factors. Composite transformations apply multiple transformations sequentially by multiplying their matrices. Homogeneous coordinates are also introduced to represent transformations in matrix form.
This document summarizes different types of 2D transformations, including translation, rotation, scaling, reflection, and shearing. Translation involves moving an object by adding offsets to the x- and y-coordinates. Rotation rotates an object around an origin by a certain angle using trigonometric functions. Scaling resizes an object by multiplying the x- and y-coordinates by scaling factors. Each transformation can be represented using a transformation matrix. Examples are provided to demonstrate how to apply the transformations to change the coordinates of points on a geometric object.
The document discusses 3D geometric transformations including rotation and reflection. It explains that transformations move points in space and can be expressed through 4x4 matrices. Specifically, it covers rotating objects around axes, conventions for right-handed vs left-handed systems, and how to perform rotations around arbitrary axes through a series of translations and rotations. It also discusses reflecting objects across planes by treating it as a scaling by -1 along one axis and provides AutoCAD commands for rotations and reflections.
Two-dimensional transformations include translations, rotations, and scalings. Transformations manipulate objects by altering their coordinate descriptions without redrawing them. Matrices can represent linear transformations and are used to describe 2D transformations. Common 2D transformations include translation by adding offsets to coordinates, rotation by applying a rotation matrix, and scaling by multiplying coordinates by scaling factors. More complex transformations can be achieved by combining basic transformations through matrix multiplication in a specific order.
2 d transformations by amit kumar (maimt)Amit Kapoor
Transformations are operations that change the position, orientation, or size of an object in computer graphics. The main 2D transformations are translation, rotation, scaling, reflection, shear, and combinations of these. Transformations allow objects to be manipulated and displayed in modified forms without needing to redraw them from scratch.
This document is a project report by Aditi Patni on 2-D transformations for her Bachelor's degree. It defines 2-D transformations as changes in orientation, shape, size or position of an object that alter its coordinate values. There are two types of 2-D transformations: basic transformations of translation, rotation, and scaling; and derived transformations of reflection and shearing. The report provides detailed explanations and examples of how to perform each of these transformations using matrix representations.
3D transformations use homogeneous coordinates and 4x4 matrices similarly to 2D transformations. There are basic transformations like identity, scale, translation, and mirroring as well as rotations around the X, Y, and Z axes represented by matrices. To reverse a rotation of q degrees, apply the inverse rotation R(-q) which has the same cosine elements but flipped sine elements, making it the transpose of the original rotation matrix.
The document discusses how more complex geometric transformations can be performed by combining basic transformations through composition. It provides examples of how scaling and rotation can be done with respect to a fixed point by first translating the object to align the point with the origin, then performing the basic transformation, and finally translating back. Mirror reflection about a line is similarly described as a composite of translations and rotations.
The document discusses 2D geometric transformations including translation, rotation, scaling, and their matrix representations. It explains that transformations can be combined through matrix multiplication and represented as sequences of transformations applied to modeling coordinates to obtain world coordinates. Basic transformations include translation, scaling, rotation, and shearing, which can be expressed as 3x3 matrices using homogeneous coordinates.
OpenGL uses model-view and projection matrices to apply transformations like translation, rotation, and scaling. The document discusses constructing transformation matrices for different types of transformations, including translation, rotation around fixed points and arbitrary axes, scaling, and shearing. It also covers combining multiple transformations using matrix multiplication and storing transformations in the current transformation matrix.
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.
Overview of 2D and 3D Transformation, Translation, scaling, rotation, shearing, reflection, 3D transformation, rotation about arbitrary / pivot point, 3D rotation with x axis, 3D rotation with y axis, 3D rotation with x axis, viewing transformation, parallel projection, perspective projection
it is related to Computer Graphics Subject.in this ppt we describe what is 2D Transformation, Translation, Rotation, Scaling : Uniform Scaling,Non-uniform Scaling ;Reflection,Shear,Composite Transformations
1. The document discusses various 2D and 3D transformations including translation, scaling, rotation, reflection, and shearing.
2. It explains the basic transformation types including rigid-body, affine, and free-form and provides examples of each.
3. Key concepts covered include the homogeneous coordinate system, composition of transformations using matrices, and expressing transformations like translation, scaling, and rotation using matrices.
This document discusses 2D transformations including vector addition, dot products, cross products, translation, rotation, and scaling. Vector addition involves adding the x and y components of two vectors. The dot product of two vectors produces a scalar value based on the cosine of the angle between the vectors and can be used to determine if they are perpendicular. The cross product of two vectors produces a new vector perpendicular to both input vectors, determined by the right hand rule. Translation moves every point of an object by the same displacement amounts. Rotation rotates every point of an object by the same angle. Scaling enlarges or shrinks an object by a uniform scaling factor.
Computer Graphic - Transformations in 2D2013901097
The document discusses 2D geometric transformations using matrices. It defines a general transformation equation [B] = [T] [A] where [T] is the transformation matrix and [A] and [B] are the input and output point matrices. It then explains various transformation matrices for scaling, reflection, rotation and translation. It also discusses representing transformations in homogeneous coordinates using 3x3 matrices. Finally, it provides examples of applying multiple transformations and conditions when the order of transformations can be changed.
Transformations in OpenGL are not drawing
commands. They are retained as part of the
graphics state. When drawing commands are issued, the
current transformation is applied to the points
drawn. Transformations are cumulative.
2D Rotation- Transformation in Computer GraphicsSusmita
This document discusses 2D rotation in computer graphics. It provides the rotation equations to calculate new x- and y-coordinates when an object is rotated by an angle θ. As an example, it solves the problem of rotating a triangle with vertices at (0,0), (1,0), and (1,1) by 90 degrees counterclockwise. The new coordinates of the rotated triangle are found to be (0,0), (0,1), and (-1,1).
Two dimensional geometric transformationjapan vasani
This document discusses various 2D geometric transformations including translation, rotation, scaling, reflection, and shear. It provides the mathematical formulas to perform each transformation using homogeneous coordinates and matrix representations. It also describes how to perform a composite transformation using multiple basic transformations sequentially, such as translating, rotating and translating again. Finally, it includes an example of rotating a triangle about two different points to demonstrate composite transformations.
The document discusses 2D geometric transformations including translation, rotation, scaling, and matrix representations. It explains that transformations can be combined through matrix multiplication and represented by 3x3 matrices in homogeneous coordinates. Common transformations like translation, rotation, scaling and reflections are demonstrated.
This document provides 56 examples of geometric transformations including translation, rotation, and flipping of letters, shapes, and objects. Each example demonstrates a transformation or combination of transformations of a figure including letters T, O, F, Z, squares, rectangles, hexagons, houses, and stars.
The document discusses compositions of transformations, which involve performing two or more transformations sequentially on a figure. It provides examples of compositions, such as a glide reflection, which reflects a figure and then translates it. Compositions are described using notation listing the transformations from right to left. Most compositions are important to perform in the given order. Examples are provided to describe and graph compositions involving reflections across axes or lines combined with translations or rotations.
This document summarizes different types of 2D transformations, including translation, rotation, scaling, reflection, and shearing. Translation involves moving an object by adding offsets to the x- and y-coordinates. Rotation rotates an object around an origin by a certain angle using trigonometric functions. Scaling resizes an object by multiplying the x- and y-coordinates by scaling factors. Each transformation can be represented using a transformation matrix. Examples are provided to demonstrate how to apply the transformations to change the coordinates of points on a geometric object.
The document discusses 3D geometric transformations including rotation and reflection. It explains that transformations move points in space and can be expressed through 4x4 matrices. Specifically, it covers rotating objects around axes, conventions for right-handed vs left-handed systems, and how to perform rotations around arbitrary axes through a series of translations and rotations. It also discusses reflecting objects across planes by treating it as a scaling by -1 along one axis and provides AutoCAD commands for rotations and reflections.
Two-dimensional transformations include translations, rotations, and scalings. Transformations manipulate objects by altering their coordinate descriptions without redrawing them. Matrices can represent linear transformations and are used to describe 2D transformations. Common 2D transformations include translation by adding offsets to coordinates, rotation by applying a rotation matrix, and scaling by multiplying coordinates by scaling factors. More complex transformations can be achieved by combining basic transformations through matrix multiplication in a specific order.
2 d transformations by amit kumar (maimt)Amit Kapoor
Transformations are operations that change the position, orientation, or size of an object in computer graphics. The main 2D transformations are translation, rotation, scaling, reflection, shear, and combinations of these. Transformations allow objects to be manipulated and displayed in modified forms without needing to redraw them from scratch.
This document is a project report by Aditi Patni on 2-D transformations for her Bachelor's degree. It defines 2-D transformations as changes in orientation, shape, size or position of an object that alter its coordinate values. There are two types of 2-D transformations: basic transformations of translation, rotation, and scaling; and derived transformations of reflection and shearing. The report provides detailed explanations and examples of how to perform each of these transformations using matrix representations.
3D transformations use homogeneous coordinates and 4x4 matrices similarly to 2D transformations. There are basic transformations like identity, scale, translation, and mirroring as well as rotations around the X, Y, and Z axes represented by matrices. To reverse a rotation of q degrees, apply the inverse rotation R(-q) which has the same cosine elements but flipped sine elements, making it the transpose of the original rotation matrix.
The document discusses how more complex geometric transformations can be performed by combining basic transformations through composition. It provides examples of how scaling and rotation can be done with respect to a fixed point by first translating the object to align the point with the origin, then performing the basic transformation, and finally translating back. Mirror reflection about a line is similarly described as a composite of translations and rotations.
The document discusses 2D geometric transformations including translation, rotation, scaling, and their matrix representations. It explains that transformations can be combined through matrix multiplication and represented as sequences of transformations applied to modeling coordinates to obtain world coordinates. Basic transformations include translation, scaling, rotation, and shearing, which can be expressed as 3x3 matrices using homogeneous coordinates.
OpenGL uses model-view and projection matrices to apply transformations like translation, rotation, and scaling. The document discusses constructing transformation matrices for different types of transformations, including translation, rotation around fixed points and arbitrary axes, scaling, and shearing. It also covers combining multiple transformations using matrix multiplication and storing transformations in the current transformation matrix.
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.
Overview of 2D and 3D Transformation, Translation, scaling, rotation, shearing, reflection, 3D transformation, rotation about arbitrary / pivot point, 3D rotation with x axis, 3D rotation with y axis, 3D rotation with x axis, viewing transformation, parallel projection, perspective projection
it is related to Computer Graphics Subject.in this ppt we describe what is 2D Transformation, Translation, Rotation, Scaling : Uniform Scaling,Non-uniform Scaling ;Reflection,Shear,Composite Transformations
1. The document discusses various 2D and 3D transformations including translation, scaling, rotation, reflection, and shearing.
2. It explains the basic transformation types including rigid-body, affine, and free-form and provides examples of each.
3. Key concepts covered include the homogeneous coordinate system, composition of transformations using matrices, and expressing transformations like translation, scaling, and rotation using matrices.
This document discusses 2D transformations including vector addition, dot products, cross products, translation, rotation, and scaling. Vector addition involves adding the x and y components of two vectors. The dot product of two vectors produces a scalar value based on the cosine of the angle between the vectors and can be used to determine if they are perpendicular. The cross product of two vectors produces a new vector perpendicular to both input vectors, determined by the right hand rule. Translation moves every point of an object by the same displacement amounts. Rotation rotates every point of an object by the same angle. Scaling enlarges or shrinks an object by a uniform scaling factor.
Computer Graphic - Transformations in 2D2013901097
The document discusses 2D geometric transformations using matrices. It defines a general transformation equation [B] = [T] [A] where [T] is the transformation matrix and [A] and [B] are the input and output point matrices. It then explains various transformation matrices for scaling, reflection, rotation and translation. It also discusses representing transformations in homogeneous coordinates using 3x3 matrices. Finally, it provides examples of applying multiple transformations and conditions when the order of transformations can be changed.
Transformations in OpenGL are not drawing
commands. They are retained as part of the
graphics state. When drawing commands are issued, the
current transformation is applied to the points
drawn. Transformations are cumulative.
2D Rotation- Transformation in Computer GraphicsSusmita
This document discusses 2D rotation in computer graphics. It provides the rotation equations to calculate new x- and y-coordinates when an object is rotated by an angle θ. As an example, it solves the problem of rotating a triangle with vertices at (0,0), (1,0), and (1,1) by 90 degrees counterclockwise. The new coordinates of the rotated triangle are found to be (0,0), (0,1), and (-1,1).
Two dimensional geometric transformationjapan vasani
This document discusses various 2D geometric transformations including translation, rotation, scaling, reflection, and shear. It provides the mathematical formulas to perform each transformation using homogeneous coordinates and matrix representations. It also describes how to perform a composite transformation using multiple basic transformations sequentially, such as translating, rotating and translating again. Finally, it includes an example of rotating a triangle about two different points to demonstrate composite transformations.
The document discusses 2D geometric transformations including translation, rotation, scaling, and matrix representations. It explains that transformations can be combined through matrix multiplication and represented by 3x3 matrices in homogeneous coordinates. Common transformations like translation, rotation, scaling and reflections are demonstrated.
This document provides 56 examples of geometric transformations including translation, rotation, and flipping of letters, shapes, and objects. Each example demonstrates a transformation or combination of transformations of a figure including letters T, O, F, Z, squares, rectangles, hexagons, houses, and stars.
The document discusses compositions of transformations, which involve performing two or more transformations sequentially on a figure. It provides examples of compositions, such as a glide reflection, which reflects a figure and then translates it. Compositions are described using notation listing the transformations from right to left. Most compositions are important to perform in the given order. Examples are provided to describe and graph compositions involving reflections across axes or lines combined with translations or rotations.
The student is able to draw and identify compositions of transformations, which involve performing two or more transformations sequentially to a figure. Compositions are described using notation by listing the transformations from right to left. It is important to perform the transformations in the given order. Examples of compositions include reflecting a figure across an axis and then translating it, as well as reflecting across a line and rotating the figure.
This document discusses vectors, coordinate systems, and geometric transformations that are fundamental concepts in computer graphics. It provides examples of different coordinate systems and how to project points from one system to another. It also explains various 2D affine transformations like translation, scaling, rotation, shearing, and reflection through transformation matrices. Homogeneous coordinates are introduced as a technique to represent 2D points as 3D homogeneous coordinates to allow for general linear transformations.
The document discusses geometric transformations in computer graphics, including translation, scaling, and shearing. It defines these transformations and explains how they are used for modeling, viewing, and image manipulation. Matrix representations are presented as an efficient way to represent sequences of transformations by multiplying the matrices. Basic 2D transformations can be represented with 2x2 or 3x3 matrices, with the additional third row and column used to allow for homogeneous coordinates.
Cohen and Sutherland Algorithm for 7-8 marksRehan Khan
The document discusses the Cohen-Sutherland line clipping algorithm. It assigns a 4-bit region code to each endpoint of a line based on whether it is inside or outside the viewport boundaries. Lines that are completely inside or outside the viewport are accepted or rejected based on the region codes. For lines partially inside, it calculates the intersection points with the viewport edges using equations involving the line slope and endpoints.
The document discusses 2D geometric transformations including translation, rotation, scaling, and shearing. It introduces representing transformations using 2x2 matrices and homogeneous coordinates using 3x3 matrices. Transformations can be combined by matrix multiplication, allowing multiple transformations to be applied sequentially in an efficient manner.
This document discusses 2D transformations in computer graphics, including translation, rotation, and scaling. It introduces representing transformations using homogeneous coordinates and 3x3 matrices, allowing all transformations to be expressed as matrix multiplications. This provides a unified approach to combining multiple transformations by simply multiplying the corresponding matrices.
This document discusses simple harmonic motion and elasticity. It covers topics like Hooke's law, springs, oscillations, energy related to springs, pendulums, damping, driven harmonic motion and resonance. Several examples are provided to illustrate concepts like calculating restoring forces on springs, determining natural frequencies, and relating stress and strain using Hooke's law.
The document describes several algorithms for clipping lines and polygons to a clip rectangle in computer graphics, including:
- The Cohen-Sutherland algorithm which uses outcodes to determine if lines can be trivially accepted or rejected from the clip rectangle without intersection calculations.
- The Cyrus-Back algorithm which clips lines by solving the simultaneous equations for the intersections of the line with the clip rectangle edges.
- It also discusses parametric line clipping which finds the intersection parameters t along the line segment to determine where it enters and exits the clip rectangle edges.
The document discusses lexical analysis and regular languages. It begins with an overview of lexical analysis and its components, including regular languages defined via regular grammars, regular expressions, and finite state automata. It then covers the equivalence between these formalisms for describing regular languages and how to construct a nondeterministic finite automaton from a regular expression.
This document discusses waves and sound. It defines different types of waves like transverse, longitudinal, and periodic waves. It describes the characteristics of waves including amplitude, wavelength, period, and frequency. It discusses how the speed of waves on a string depends on the tension and linear mass density of the string. It also describes the nature and speed of sound waves in different media. It introduces concepts like the Doppler effect and applications of sound in medicine like ultrasound imaging and lithotripsy.
Software re-engineering involves examining, analyzing, and altering an existing software system to reconstitute it in a new form to improve maintainability and reliability. It occurs at both the business and software levels through approaches like big bang, incremental, or evolutionary. Risks include issues with tools, processes, applications, technology, strategies, and personnel.
Reengineering involves improving existing software or business processes by making them more efficient, effective and adaptable to current business needs. It is an iterative process that involves reverse engineering the existing system, redesigning problematic areas, and forward engineering changes by implementing a redesigned prototype and refining it based on feedback. The goal is to create a system with improved functionality, performance, maintainability and alignment with current business goals and technologies.
This document discusses work, energy, and power. It defines work, kinetic energy, gravitational potential energy, and average power. It describes the work-energy theorem, the principle of conservation of mechanical energy, and how energy can be transferred between different forms but not created or destroyed based on the principle of conservation of energy. Examples are provided to demonstrate how to calculate work, kinetic energy, gravitational potential energy, and changes in speed and energy based on these principles.
Lex is a tool that generates lexical analyzers (scanners) that are used to break input text streams into tokens. It allows rapid development of scanners by specifying patterns and actions in a lex source file. The lex source file contains three sections - definitions, translation rules, and user subroutines. The translation rules specify patterns and corresponding actions. Lex compiles the source file to a C program that performs the tokenization. Example lex programs are provided to tokenize input based on regular expressions and generate output.
The document discusses various 2D geometric transformations including translation, rotation, and scaling. Translation moves objects by adding offsets to coordinates. Rotation changes the orientation of objects by specifying an axis and angle of rotation. Scaling alters the size of objects by multiplying coordinates by scaling factors. These transformations can be represented by 3x3 matrices using homogeneous coordinates, allowing multiple transformations to be combined through matrix multiplication.
Physics - Chapter 6 - Momentum and CollisionsJPoilek
This document provides an overview of linear momentum and impulse. It defines momentum as the product of an object's mass and velocity (p=mv) and describes how momentum is a vector quantity. Impulse is defined as the change in momentum over time due to an external force (Impulse=Force x Time). The document explains how momentum is conserved in collisions and how the impulse-momentum theorem can be used to analyze collisions. It also distinguishes between perfectly elastic, perfectly inelastic, and inelastic collisions in terms of the objects' motions and changes to their kinetic energy before and after the collision.
Clipping Algorithm In Computer Graphicsstudent(MCA)
This document discusses window clipping techniques for computer graphics. It introduces point and line clipping, describing a brute force approach and the more efficient Cohen-Sutherland clipping algorithm. It then explains the Sutherland-Hodgman area clipping algorithm. Key concepts covered include using region codes to efficiently determine which lines and portions of lines need to be clipped to a window.
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 provides a summary of key concepts in algebra, geometry, trigonometry and their definitions. It includes formulas and properties for lines, polynomials, exponents, trig functions, triangles, circles, spheres, cones, cylinders, distance and the quadratic formula. Key topics covered are factoring, binomials, slope-intercept form, trig ratios, trig identities, trig reciprocals and the Pythagorean identities.
This document derives several important trigonometric identities by considering a right triangle with angle θ and using the Pythagorean theorem and definitions of trigonometric functions. It shows that sin2θ + cos2θ = 1, which can be rearranged to obtain other important identities relating sin, cos, tan, sec, and cosec.
This document discusses various 2D transformations in computer graphics including translation, rotation, and scaling. Translation moves an object by adding offsets to the x and y coordinates. Rotation uses trigonometric functions and rotation matrices to reposition objects around a central point. Scaling enlarges or shrinks objects by multiplying their coordinates by scaling factors. Homogeneous coordinates generalize these transformations into matrix operations.
with today's advanced technology like photoshop, paint etc. we need to understand some basic concepts like how they are cropping the image , tilt the image etc.
In our presentation you will find basic introduction of 2D transformation.
The document is notes for a lesson on tangent planes. It provides definitions of tangent lines and planes, formulas for finding equations of tangent lines and planes, and examples of applying these concepts. Specifically, it defines that the tangent plane to a function z=f(x,y) through the point (x0,y0,z0) has normal vector (f1(x0,y0), f2(x0,y0),-1) and equation f1(x0,y0)(x-x0) + f2(x0,y0)(y-y0) - (z-z0) = 0 or z = f(x0,y0) +
The document discusses trigonometric functions on the unit circle. It defines trig ratios for angles in each of the four quadrants using right triangles formed with the point (x,y) and the origin. Key identities presented are:
1) tanθ = sinθ/cosθ
2) sin2θ + cos2θ = 1
The signs of the trig functions depend on the quadrant, with trig ratios being positive in Quadrant I and changing appropriately in other quadrants based on the signs of x and y.
This document discusses how to find the x-intercept and y-intercept of a linear equation by setting one variable equal to 0 and solving for the other. It provides examples of finding intercepts from equations, graphing lines using intercepts, and identifying intercepts from a graph.
This document discusses quantum modes and the correspondence between classical and quantum mechanics. It provides three key principles of quantum mechanics: (1) quantum states are represented by ket vectors, (2) quantum observables are hermitian operators, and (3) the Schrodinger equation governs the causal evolution of quantum systems. It also outlines how classical quantities like position and momentum correspond to quantum operators and how they form Lie algebras through commutation relations. Representations of quantum mechanics are discussed through examples like the energy basis of the harmonic oscillator.
Interpolation techniques - Background and implementationQuasar Chunawala
This document discusses interpolation techniques, specifically Lagrange interpolation. It begins by introducing the problem of interpolation - given values of an unknown function f(x) at discrete points, finding a simple function that approximates f(x).
It then discusses using Taylor series polynomials for interpolation when the function value and its derivatives are known at a point. The error in interpolation approximations is also examined.
The main part discusses Lagrange interpolation - given data points (xi, f(xi)), there exists a unique interpolating polynomial Pn(x) of degree N that passes through all the points. This is proved using the non-zero Vandermonde determinant. Lagrange's interpolating polynomial is then introduced as a solution.
The document provides an overview of techniques for solving different types of ordinary differential equations (ODEs):
1. It describes prototypes and solution methods for various types of first-order ODEs, including separable, exact, homogeneous, Bernoulli, and linear.
2. It discusses techniques for solving second and higher-order linear ODEs with constant or Cauchy-Euler coefficients, including the auxiliary equation and using the variation of parameters or undetermined coefficients methods.
3. It mentions series solutions centered around x=0.
4. For homogeneous systems of ODEs, it outlines converting between system and matrix forms, finding eigenvalues and eigenvectors, and using the eigenstructure to solve
The document provides an overview of Calculus I taught by Professor Matthew Leingang at New York University. It outlines key topics that will be covered in the course, including different classes of functions, transformations of functions, and compositions of functions. The first assignments are due on January 31 and February 2, with first recitations on February 3. The document uses examples to illustrate concepts like linear functions, other polynomial functions, and trigonometric functions. It also explains how vertical and horizontal shifts can transform the graph of a function.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
1) The document discusses solving partial differential equations using variable separation. It presents the one-dimensional wave equation and solves it using variable separation.
2) It derives three cases for the solution based on whether k is positive, negative, or zero. It then presents the general solution as a summation involving sines and cosines.
3) It applies the general solution to two example problems of a vibrating string, finding the displacement as a function of position and time by satisfying the boundary conditions.
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2. 5.1 Basic Two-Dimensional
Geometric Transformations
Operations that are applied to the geometric
description of an object to change its
position, orientation, or size are called
geometric transformations.
Geometric transformations can be used to
describe how objects might move around in
a scene during an animation sequence or
simply to view them from another angle.
5. Two-Dimensional Translation
We perform a translation on a single coordinate
point by adding offsets to its coordinates so as to
generate a new coordinate position.
To translate a two-dimensional position, we add
translation distances, tx and ty to the original
coordinates (x,y) to obtain the new coordinate
position (x’,y’),
x = x+t
'
x
y = y + ty
'
6. The two-dimensional translation equations in the
matrix form
จาก x = x + tx
'
ให้
รูป
y
y' = y + ty x
P=
x'
P' = '
P' ( x' , y' ) y y
T t x
ty และ T =
P ( x, y )
t y
tx
x ' x t x
' = +
x
จะได้ y y t y
P = P +T
'
7. Two-Dimensional Rotation
We generate a rotation transformation of an
object by specifying a rotation axis and a
rotation angle.
A two-dimensional rotation of an object is
obtained by repositioning the object along a
circular path in the xy plane.
Parameters for the two-dimensional rotation
are
The rotation angle θ
A position (x,y) – rotation point (pivot
point)
8. The two-dimensional rotation
x ' = r cos(φ + θ ) = r cos φ cos θ − r sin φ sin θ
y ' = r sin(φ + θ ) = r cos φ sin θ + r sin φ cos θ
y
Polar coordinate system
x = r cos φ
O' ( x' , y' ) y = r sin φ
O ( x, y )
θ
φ
x
P(0,0) x ' = x cos θ − y sin θ
y ' = x sin θ + y cos θ
9. The two-dimensional rotation
x ' cos θ - sinθ x
' =
y
y
sin θ cosθ y
O' ( x' , y' ) O' = R • O
O ( x, y )
θ โดยที่
φ x x'
x O= O' = '
P(0,0) y y
cos θ - sinθ
และ R = Rotation matrix
sin θ cosθ
11. Rotation of a point about an arbitrary pivot position
x ' = ( x − xr ) cos θ − ( y − yr ) sin θ + xr
y ' = ( x − xr ) sin θ + ( y − yr ) cos θ + yr
y
O' ( x' , y' ) x ' cos θ - sinθ x − xr xr
' = y − y + y
O ( x, y ) y sin θ cosθ
r r
θ O ' = R • O* + P
φ
P ( xr , y r )
x
โดยที่
x − x'
O* = '
y − y
12. Two-Dimensional Scaling
To alter the size of an
object, we apply a
scaling transformation.
A simple two- x = x ⋅ sx
'
dimensional scaling
operation is performed
by multiplying object
y = y ⋅ sy
'
positions (x,y) by
scaling factors sx and
sy to produce the x ' sx 0 x
transformed ' =
coordinates (x’,y’). y 0 s y y
P = S•P
'
13. Any positive values can be assigned to the
scaling factors.
Values less than 1 reduce the size of
object;
Values greater than 1 produce
enlargements.
Uniform scaling – scaling values have the
same value
Differential scaling – unequal of the
scaling factor
14. Scaling relative to a chosen fixed point
x ' = x ⋅ s x + x f (1 − s x )
y ' = y ⋅ s y + y f (1 − s y )
y
P1
x ' s x 0 x 1 − s x 0 x f
• ' = + 0 1-s
Pf ( x f , y f ) P2 y 0 s y y
y y f
P3 P ' = S • P + S * • Pf
x
15. 5.2 Matrix Representations and
Homogeneous Coordinates
Many graphics applications involve
sequences of geometric transformations.
Hence we consider how the matrix
representations can be reformulated so that
such transformation sequence can be
efficiently processed.
Each of three basic two-dimensional
transformations (translation, rotation and
scaling) can be expressed in the general
matrix form
16. P and P ’ = column vectors,
coordinate position
M1 = 2 by 2 array containing P = M1 ⋅ P + M 2
'
multiplicative factors, for
translation M1 is the identity
matrix
M2 = two-element column matrix
containing translational terms, for
rotation or scaling M2 contains
the translational terms associated
with the pivot point or scaling
fixed point
17. P' = M1 ⋅ P + M 2
x ' 1 0 x t x
' = y + t Translation
y 0 1 y
x ' cos θ - sinθ x − xr xr
' = y − y + y Rotation
y sin θ cosθ
r r
x ' cos θ - sinθ x cos θ - sinθ xr xr
' = y − sin θ cosθ y + y
y sin θ cosθ
r r
x ' s x 0 x 1 − s x 0 x f
' = + 0 1-s Scaling
y 0 s y y
y y f
18. To produce a sequence of transformations
such as scaling followed by rotation then
translation, we could calculate the
transformed coordinates one step at a time.
A more efficient approach is to combine the
transformations so that the final coordinate
positions are obtained directly from the
initial coordinates, without calculating
intermediate coordinate values.
19. Homogeneous Coordinates
Multiplicative and translational terms for a
two-dimensional geometric transformations
can be combined into a single matrix if we
expand the representations to 3 by 3
matrices.
Then we can use the third column of a
transformation matrix for the translation
terms, and all transformation equations can
be expressed as matrix multiplications.
20. But to do so, we also need to expand the
matrix representation for a two-dimensional
coordinate position to a three-element column
matrix.
A standard technique for accomplishing this
is to expand each two-dimensional
coordinate-position representation (x,y) to a
three-element representation (xh,yh,h), called
homogeneous coordinates, where the
homogeneous parameter h is a nonzero
value such that
xh yh
x= , y=
h h
21. A convenient choice is simply to set h=1.
Each two-dimensional position is then
represented with homogeneous coordinate
(x,y,1).
The term “homogeneous coordinates” is
used in mathematics to refer to the effect of
this representation on Cartesian equations.
22.
23. x ' 1 0 t x x
'
y = 0 1 t y y Translation
0
1 0 1 1
P ' = T (t x , t y ) ⋅ P
x ' cos θ - sinθ 0 x
'
Rotation y = sin θ cosθ 0 y
1 0 0 1 1
x ' sx 0 0 x P ' = R (θ ) ⋅ P
'
y = 0 s y 0 y Scaling
1 0 0 1 1
P ' = S (sx , s y ) ⋅ P
24. 5.3 Inverse Transformations
1 0 - t x Inverse translation matrix
−1
T = 0 1 - t y
0 0 1
Inverse rotation matrix 1
s 0 0
cos θ sinθ 0 x
R −1 = − sin θ cosθ 0 −1 1
S = 0 0
0
0 1 sy
0 0 1
Inverse scaling matrix
25. 5.4 Two-Dimensional Composite
Transformations
Using matrix representations, we
can set up a sequence of
transformations as a composite
transformation matrix by
calculating the product of the
individual transformations.
Thus, if we ant to apply two
transformations to point position
P, the transformed location would
be calculated as
P = M 2 ⋅ M1 ⋅ P
'
P = M ⋅P
'
27. Composite Two-dimensional Translations
P ' = T (t 2 x , t 2 y ) ⋅ {T (t1x , t1 y ) ⋅ P}
P ' = {T (t 2 x , t 2 y ) ⋅ T (t1x , t1 y )} ⋅ P
composite transformation matrix
1 0 t 2 x 1 0 t1x 1 0 t1x + t 2 x
0 1 t 2 y ⋅ 0 1 t1 y = 0 1 t1 y + t 2 y
0 0 1 0 0 1 0 0 1
T (t 2 x , t 2 y ) ⋅ T (t1x , t1 y ) = T (t1x + t 2 x , t1 y + t 2 y )
28. Composite Two-dimensional Rotations
P ' = R (θ 2 ) ⋅ {R (θ1 ) ⋅ P}
P ' = {R (θ 2 ) ⋅ R (θ1 )} ⋅ P
composite transformation matrix
R (θ 2 ) ⋅ R (θ1 ) = R (θ1 + θ 2 )
29.
30. Composite Two-dimensional Scaling
composite transformation matrix
S 2 x 0 0 S1x 0 0 S1x ⋅ S 2 x 0 0
0 S 2 y 0 ⋅ 0 S1 y 0 = 0 S1 y ⋅ S 2 y 0
0 0 1 0 0 1 0 0 1
S ( s2 x , s2 y ) ⋅ S ( s1x , s1 y ) = S ( s1x ⋅ s2 x , s1 y ⋅ s2 y )
31. General Two-dimensional Pivot-
Point Rotation
A transformation sequence for rotating
an object about a sepcified pivot point
using the rotation matrix R(θ).
Translate the object so that the pivot-point
position is moved to the coordinate origin.
Rotate the object about the coordinate
origin.
Translate the object so that the pivot point
is returned to its original position.
33. 1 0 xr cos θ - sinθ 0 1 0 - xr
0 1 y ⋅ sin θ cosθ 0 ⋅ 0 1 - y
r r
0 0 1 0
0 1 0 0 1
cos θ - sinθ xr( 1- cos θ) + yr sin θ
= sin θ cosθ y ( 1- cos θ) − x sin θ
r r
0
0 1
T ( xr , yr ) ⋅ R (θ ) ⋅ T (− xr ,− yr ) = R ( xr , yr , θ )
34. General Two-dimensional Fixed-
Point Scaling
A transformation sequence to produce a two-
dimensional scaling with respect to a selected
fixed position (xf,yf).
Translate the object so that the fixed point
coincides with the coordinate origin.
Scale the object with respect to the coordinate
origin.
Use the inverse of the translation in step (1) to
return the object to its original position.
36. 1 0 x f s x 0 0 1 0 - x f
0 1 y f ⋅ 0 s y 0 ⋅ 0 1 - y f
0 0 1 0 0 1 0 0 1
s x 0 x f ( 1-s x )
= 1 s y y f ( 1-s y )
0 0 1
T ( x f , y f ) ⋅ S ( s x , s y ) ⋅ T (− x f ,− y f ) = S ( x f , y f , ( s x , s y ))
37. General Two-dimensional Scaling
Directions
We can scale an object in
other directions by rotating
the object to align the
y
desired scaling directions s2
with the coordinate axes
before applying the scaling
transformation.
Suppose we want to apply
scaling factors with values
specified by parameters s1 x
and s2 in the directions s1
shown in fig.
38. The composite matrix resulting from the product of
- rotation so that the directions for s1 and s2
coincide with the x and y axes
- scaling transformation S(s1,s2)
- opposite rotation to return points to their
original orientations
s1Cos 2θ + s2 Sin 2θ ( s2 − s1 )CosθSinθ 0
R (θ ) ⋅ S ( s1 , s2 ) ⋅ R (θ ) = ( s2 − s1 )CosθSinθ s1Sin 2θ + s2Cos 2θ
-1
0
0 0 1
39. y y
(2,2)
(1/2,3/2)
(0,1) (1,1)
(3/2,1/2)
(0,0) (1,0) x (0,0) x
s1= 1 s2=2
40. Matrix Concatenation Properties
Transformation products may not be
commutative.
The matrix product M2M1 is not equal to M1M2.
This means that if we want to translate and
rotate an object, we must be careful about the
order in which the composite matrix is
evaluated.
Reversing the order in which a sequence of
transformations is performed may affect the
transformed position of an object.
42. General Two-dimensional Composite
Transformations and Computational
Efficiency
x ' rs xx rs xy trs x x
'
y = rs yx rs yy trs y y
1 0 0 1 1
rs** are the multiplicative rotation-scaling terms
(rotation angles, scaling factors)
trsx and trsy are the translation terms
(translation distances, pivot-point and fixed-
point coordinates, rotation angles, scaling
parameters)
43. Example
Scale and
rotate about its centroid coordinates (xc,yc)
and
translate
T (t x , t y ) ⋅ R ( xc , yc , θ ) ⋅ S ( xc , yc , s x , s y ) =
s x cos θ - s y sin θ xc (1 − s x cos θ ) + yc s y sin θ + t x
s x sin θ s y cos θ yc (1 − s y cos θ ) − xc s x sin θ + t y
0 0 1
44. Two-dimensional Rigid-body Transformation
If a transformation matrix includes only
translation and rotation parameters, it is a
rigid-body transformation matrix.
rxx rxy trx
ryx ryy try
0 0 1
r** are the multiplicative rotation terms (rotation angles)
trx and try are the translation terms (translation
distances, pivot-point and fixed-point coordinates,
rotation angles)
45. 5.5 Other Two-Dimensional
Transformations
Reflection
For a two-dimensional reflection, the
image is generated relative to an axis
of reflection by rotating the object 180º
about the reflection axis.
48. Reflection about any reflection point in
the xy plane
Reflection point
− 1 0 0
0 - 1 0
0 0 1
49. Reflection about the reflection axis
y=x
0 1 0
1 0 0
0 0 1
1. Rotate the line y=x with respect to
the original through a 45 angle
2. Reflection with respect to the x axis
3. Rotate the line y=x black to its
original position with a
counterclockwise rotation through
45
50. Reflection about the reflection axis y=-x
0 - 1 0
- 1 0 0
0 0 1
1. Clockwise rotate the line y=-x with
respect to the original through a 45
angle
2. Reflection with respect to the y axis
3. Rotate the line y=-x black to its
original position with a y = -x
counterclockwise rotation through
45
51. Shear
A transformation that distorts the
shape of an object such that the
transformed shape appears as if the
object were composed of internal
layers that had been caused to slide
over each other is called a shear.
53. We can generate x-direction shears
relative to other reference lines
1 shx -shx ⋅ yref
0 1 0 transformation matrix
0 0 1
y y
(1,1) (2,1)
(0,1) (1,1)
shx = ½
yref = -1
(0,0) (1,0) (0,0) (1/2,0) (3/2,0)
yref = -1 yref = -1
54.
55. A y-direction shears relative to the line
x=xref is generated with
1 0 0
shy 1 -shy ⋅ xref transformation matrix
0 0 1
y y
(1,2)
(0,3/2)
(0,1) (1,1) (1,1)
shy = ½
(0,1/2)
xref = -1
xref = -1 (0,0) (1,0) xref = -1 (0,0)
56.
57. Shear operations can be expressed as
sequences of basic transformations.
The x-direction shear matrix
1 shx 0
0 1 0
0 0 1
can be represented as a composite
transformation involving a series of rotation
and scaling metrices.
58.
59. 5.6 Raster Methods for Geometric
Transformations
Functions that manipulate rectangular pixel
arrays are called raster operations and
moving a block of pixel values from one
position to another is termed a block
transfer, a bitblt or a pixblt.
All bit settings in the rectangular area are
copied as a block into another part of the
frame buffer.
We can erase the pattern at the original
location by assigning the background color
to all pixels within that block.
60.
61. Array rotations that are not multiples of 90º
1 2 3 12 11 10
4 3 6 9 12
5 6
2 5 8 11
9
8 7
7 8 9 6 5 4
1 4 7 10
10 11 12 3 2 1
(a) (b) (c)
The original array is shown in (a),
the positions of the array elements after a 90º
counterclockwise rotation are shown in (b), and
the position of the array elements after a 180º
rotation are shown in (c).
62. 5.7 OpenGL Raster Transformations
A translation of a rectangular array of pixel-
color values from one buffer area to another
can be accomplished in OpenGL as a copy
operation:
glCopyPixels(xmin, ymin, width, height, GL_COLOR);
This array of pixels is to be copied to a
rectangular area of a refresh buffer whose
lower-left corner is at the location specified
by the current raster position.
63. We can rotate a block of pixel-color values in 90-degree
increments by
first saving the block in an array,
glReadPixels(xmin, ymin, width, height, GL_RGB,
GL_UNSIGNED_BYTE, colorArray);
then rearranging the elements of the array and
placing it back in the refresh buffer.
glDrawPixels(width, height, GL_RGB, GL_UNSIGNED_BYTE,
colorArray);
We set the scaling facters with
glPixelZoom(sx,sy);
64. 5.8 Transformations between Two-
Dimensional Coordinate Systems
y’
y
x’
θ
yn
xn x
A Cartesian x’y’ system specified with coordinate origin
(xn,yn) and orientation angle θ in a Cartesian xy reference
frame.
65. To transform object descriptions from xy
coordinates to x’y’ coordinates, we set up a
transformation that superimposes the axes
onto the x’y’ axes.
This is done in two steps:
Translate so that the origin (xn,yn) of the x’y’
system is moved to the origin (0,0) of the xy
system.
Rotation the x’ axis onto the x axis.
66. 1 0 − xn
1)Translation 0 1 − y
T ( − xn , − y n ) = n
0 0 1
y’
y y
P y ’
x’
P
y
x’
y’
θ
yn
x’
θ
xn x x x
67. 2)clockwise rotation
cosθ sinθ 0
2.1) R(−θ ) = - sinθ cosθ 0
0
0 1
y y
y ’
y ’
P P
y y
x’
y’
x’
θ
x x x x
composite matrix M xy , x ' y ' = R (−θ ) ⋅ T (− xn ,− yn )
68. OR
2.2) An alternate method for
describing the orientation of the
x’y’ coordinate system is to
specify a vector V that indicates
the direction for the positive y’
axis.
We can specify vector V as a
point in the xy reference frame
relative to the origin of the xy
system, which we can convert to
y’
the unit vector,
y
V P V
x’
v= = (v x , v y )
V
yn θ
xn x
69. We obtain the unit vector u along the x’ axis by
applying a 90º clockwise rotation to vector v
u = (v y ,−v x ) = (u x , u y )
The matrix to rotate the x’y’ system into
coincidence with the xy system can be written as
u x u y 0
R = v x v y 0
0 0 1
70. OR
In the case that we
coordinates of P0 and P1
are known,
P1 - P0
v= = (v x , v y )
P1 - P0
y’
y
V P
x’
P1
yn θ
P0
xn x