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.
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.
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 different types of video display devices, focusing on cathode ray tubes (CRTs). It describes how CRTs work using an electron gun, deflection plates, and phosphor-coated screen to produce images. Color CRT monitors are also covered, explaining how they produce color using either beam penetration or shadow mask methods. Other display types mentioned include direct view storage tubes, flat panel displays, and their key differences from CRTs.
The document discusses the 2D viewing pipeline. It describes how a 3D world coordinate scene is constructed and then transformed through a series of steps to 2D device coordinates that can be displayed. These steps include converting to viewing coordinates using a window-to-viewport transformation, then mapping to normalized and finally device coordinates. It also covers techniques for clipping objects and lines that fall outside the viewing window including Cohen-Sutherland line clipping and Sutherland-Hodgeman polygon clipping.
a spline is a flexible strip used to produce a smooth curve through a designated set of points.
Polynomial sections are fitted so that the curve passes through each control point, Resulting curve is said to interpolate the set of control points.
The document discusses 2D viewing and clipping techniques in computer graphics. It describes how clipping is used to select only a portion of an image to display by defining a clipping region. It also discusses 2D viewing transformations which involve operations like translation, rotation and scaling to map coordinates from a world coordinate system to a device coordinate system. It specifically describes the Cohen-Sutherland line clipping algorithm which uses region codes to quickly determine if lines are completely inside, outside or intersect the clipping region to optimize the clipping calculation.
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.
1. The presentation discusses different types of projections including parallel and perspective projections. Parallel projection involves projectors that are parallel, while perspective projection involves projectors that converge at a point.
2. Within parallel projection, there are orthographic and oblique projections. Orthographic projection uses perpendicular projectors, while oblique projection uses projectors that are not perpendicular. Specific types of oblique projection include cavalier and cabinet.
3. The presentation also derives the equations for parallel and oblique projections. It compares parallel and perspective projections, noting differences in properties like size preservation and foreshortening.
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.
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 different types of video display devices, focusing on cathode ray tubes (CRTs). It describes how CRTs work using an electron gun, deflection plates, and phosphor-coated screen to produce images. Color CRT monitors are also covered, explaining how they produce color using either beam penetration or shadow mask methods. Other display types mentioned include direct view storage tubes, flat panel displays, and their key differences from CRTs.
The document discusses the 2D viewing pipeline. It describes how a 3D world coordinate scene is constructed and then transformed through a series of steps to 2D device coordinates that can be displayed. These steps include converting to viewing coordinates using a window-to-viewport transformation, then mapping to normalized and finally device coordinates. It also covers techniques for clipping objects and lines that fall outside the viewing window including Cohen-Sutherland line clipping and Sutherland-Hodgeman polygon clipping.
a spline is a flexible strip used to produce a smooth curve through a designated set of points.
Polynomial sections are fitted so that the curve passes through each control point, Resulting curve is said to interpolate the set of control points.
The document discusses 2D viewing and clipping techniques in computer graphics. It describes how clipping is used to select only a portion of an image to display by defining a clipping region. It also discusses 2D viewing transformations which involve operations like translation, rotation and scaling to map coordinates from a world coordinate system to a device coordinate system. It specifically describes the Cohen-Sutherland line clipping algorithm which uses region codes to quickly determine if lines are completely inside, outside or intersect the clipping region to optimize the clipping calculation.
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.
1. The presentation discusses different types of projections including parallel and perspective projections. Parallel projection involves projectors that are parallel, while perspective projection involves projectors that converge at a point.
2. Within parallel projection, there are orthographic and oblique projections. Orthographic projection uses perpendicular projectors, while oblique projection uses projectors that are not perpendicular. Specific types of oblique projection include cavalier and cabinet.
3. The presentation also derives the equations for parallel and oblique projections. It compares parallel and perspective projections, noting differences in properties like size preservation and foreshortening.
This slide contain description about the line, circle and ellipse drawing algorithm in computer graphics. It also deals with the filled area primitive.
This document discusses various 3D transformations including translation, rotation, scaling, reflection, and shearing. It provides the transformation matrices for each type of 3D transformation. It also discusses combining multiple transformations through composite transformations by multiplying the matrices in sequence from right to left.
The document discusses line drawing algorithms in computer graphics. It defines a line segment and provides equations to determine the slope and y-intercept of a line given two endpoints. It then introduces the Digital Differential Analyzer (DDA) algorithm, an incremental scan conversion method that calculates the next point on the line based on the previous point's coordinates and the line's slope. The algorithm involves less floating point computation than directly using the line equation at each step. An example demonstrates applying DDA to scan convert a line between two points. Limitations of DDA include the processing costs of rounding and floating point arithmetic as well as accumulated round-off error over long line segments.
The document describes the Breshenham's circle generation algorithm. It explains that the algorithm uses a decision parameter to iteratively select pixels along the circumference of a circle. It provides pseudocode for the algorithm, which initializes x and y values, calculates a decision parameter, and increments x while decrementing y at each step, plotting points based on the decision parameter. An example of applying the algorithm to generate a circle with radius 5 is also provided.
Polygon clipping involves taking a polygon and clipping it against another shape to produce one or more smaller polygons. The Sutherland-Hodgman algorithm handles polygon clipping by testing each edge of the clipping polygon against each edge of the clip shape. There are four cases for how an edge can be clipped - wholly inside, exit, wholly outside, enter - and the algorithm saves or discards vertices based on these cases. Repeatedly clipping against each edge of the clip shape handles all cases and produces the final clipped polygon(s).
Transformation:
Transformations are a fundamental part of the computer graphics. Transformations are the movement of the object in Cartesian plane.
Types of transformation
Why we use transformation
3D Transformation
3D Translation
3D Rotation
3D Scaling
3D Reflection
3D Shearing
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.
Jack Bresenham developed an efficient algorithm for drawing lines on a raster display. The Bresenham's line algorithm uses only integer arithmetic to determine the next pixel to plot, allowing fast computation. It works by calculating a decision parameter to choose either the upper or lower pixel as it moves from the starting to ending point of the line. The algorithm guarantees connected lines and plots each point exactly once for accurate rendering compared to other methods.
There are two main types of projections: perspective and parallel. In perspective projection, lines converge to a single point called the center of projection, creating the illusion of depth. In parallel projection, lines remain parallel as they are projected onto the view plane. Perspective projection is more realistic but parallel projection preserves proportions. Perspective projections can be one-point, two-point, or three-point depending on the number of principal vanishing points. Orthographic projections use perpendicular lines while oblique projections are at an angle. Common parallel projections include isometric, dimetric, trimetric, cavalier and cabinet views.
The document discusses composite transformations, which involve performing two or more transformations in sequence. It provides examples that two successive translations can be represented as a single translation, and two successive rotations can be represented as a single rotation. It also explains that scaling an object with respect to a fixed point can be achieved through a sequence of translations, scaling around the origin, and inverse translations, as represented by a composite matrix.
Projection is the transformation of a 3D object into a 2D plane by mapping points from the 3D object to the projection plane. There are two main types of projection: perspective projection and parallel projection. Perspective projection uses lines that converge to a single point, while parallel projection uses parallel lines. Perspective projection includes one-point, two-point, and three-point perspectives. Parallel projection includes orthographic projection, which projects lines perpendicular to the plane, and oblique projection, where lines are parallel but not perpendicular to the plane.
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 document discusses the 3D viewing pipeline which transforms 3D world coordinates to 2D viewport coordinates through a series of steps. It then describes parallel and perspective projections. Parallel projection preserves object scale and shape while perspective projection does not due to foreshortening effects. Perspective projection involves projecting 3D points along projection rays to a view plane based on a center of projection. Other topics covered include vanishing points, different types of perspective projections, and how viewing parameters affect the view volume and object positioning in the view plane coordinates.
This document discusses various attributes that can be used to modify the appearance of graphical primitives like lines and curves when displaying them, including line type (solid, dashed, dotted), width, color, fill style (hollow, solid, patterned), and fill color/pattern. It describes how these attributes are specified in applications and how different rendering techniques like rasterization can be used to display primitives with various attribute settings.
The document discusses two algorithms for filling polygons: boundary fill and flood fill. Boundary fill starts at a point inside the polygon and fills pixels until it reaches the boundary color. Flood fill replaces all pixels of a specified interior color with a fill color. Both can be implemented with 4-connected or 8-connected pixels. Flood fill colors the entire area but uses more memory, while boundary fill stops at the boundary and is more efficient.
Three key points about advanced computer graphics and 3D viewing:
1. 3D viewing involves establishing a viewing coordinate system and transforming 3D world coordinates to 2D viewing coordinates using translations and rotations. Projections like parallel and perspective then project the viewing coordinates onto a 2D view plane.
2. Common projections used in 3D viewing are parallel projections, which project lines parallel to the view plane, and perspective projections, which simulate how the human eye sees and cause objects to appear smaller with distance.
3. Viewing pipelines involve modeling, transformations between coordinate systems, projections, clipping to a view volume, and normalization before rendering the 2D image. Technologies like OpenGL help specify common operations like projections, view
The document discusses different techniques for filling polygons, including boundary fill, flood fill, and scan-line fill methods. It provides details on how each technique works, such as using a seed point and filling neighboring pixels for boundary fill, replacing all pixels of a selected color for flood fill, and drawing pixels between edge intersections for each scan line for scan-line fill. Examples are given to illustrate the filling process for each method.
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.
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.
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.
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 slide contain description about the line, circle and ellipse drawing algorithm in computer graphics. It also deals with the filled area primitive.
This document discusses various 3D transformations including translation, rotation, scaling, reflection, and shearing. It provides the transformation matrices for each type of 3D transformation. It also discusses combining multiple transformations through composite transformations by multiplying the matrices in sequence from right to left.
The document discusses line drawing algorithms in computer graphics. It defines a line segment and provides equations to determine the slope and y-intercept of a line given two endpoints. It then introduces the Digital Differential Analyzer (DDA) algorithm, an incremental scan conversion method that calculates the next point on the line based on the previous point's coordinates and the line's slope. The algorithm involves less floating point computation than directly using the line equation at each step. An example demonstrates applying DDA to scan convert a line between two points. Limitations of DDA include the processing costs of rounding and floating point arithmetic as well as accumulated round-off error over long line segments.
The document describes the Breshenham's circle generation algorithm. It explains that the algorithm uses a decision parameter to iteratively select pixels along the circumference of a circle. It provides pseudocode for the algorithm, which initializes x and y values, calculates a decision parameter, and increments x while decrementing y at each step, plotting points based on the decision parameter. An example of applying the algorithm to generate a circle with radius 5 is also provided.
Polygon clipping involves taking a polygon and clipping it against another shape to produce one or more smaller polygons. The Sutherland-Hodgman algorithm handles polygon clipping by testing each edge of the clipping polygon against each edge of the clip shape. There are four cases for how an edge can be clipped - wholly inside, exit, wholly outside, enter - and the algorithm saves or discards vertices based on these cases. Repeatedly clipping against each edge of the clip shape handles all cases and produces the final clipped polygon(s).
Transformation:
Transformations are a fundamental part of the computer graphics. Transformations are the movement of the object in Cartesian plane.
Types of transformation
Why we use transformation
3D Transformation
3D Translation
3D Rotation
3D Scaling
3D Reflection
3D Shearing
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.
Jack Bresenham developed an efficient algorithm for drawing lines on a raster display. The Bresenham's line algorithm uses only integer arithmetic to determine the next pixel to plot, allowing fast computation. It works by calculating a decision parameter to choose either the upper or lower pixel as it moves from the starting to ending point of the line. The algorithm guarantees connected lines and plots each point exactly once for accurate rendering compared to other methods.
There are two main types of projections: perspective and parallel. In perspective projection, lines converge to a single point called the center of projection, creating the illusion of depth. In parallel projection, lines remain parallel as they are projected onto the view plane. Perspective projection is more realistic but parallel projection preserves proportions. Perspective projections can be one-point, two-point, or three-point depending on the number of principal vanishing points. Orthographic projections use perpendicular lines while oblique projections are at an angle. Common parallel projections include isometric, dimetric, trimetric, cavalier and cabinet views.
The document discusses composite transformations, which involve performing two or more transformations in sequence. It provides examples that two successive translations can be represented as a single translation, and two successive rotations can be represented as a single rotation. It also explains that scaling an object with respect to a fixed point can be achieved through a sequence of translations, scaling around the origin, and inverse translations, as represented by a composite matrix.
Projection is the transformation of a 3D object into a 2D plane by mapping points from the 3D object to the projection plane. There are two main types of projection: perspective projection and parallel projection. Perspective projection uses lines that converge to a single point, while parallel projection uses parallel lines. Perspective projection includes one-point, two-point, and three-point perspectives. Parallel projection includes orthographic projection, which projects lines perpendicular to the plane, and oblique projection, where lines are parallel but not perpendicular to the plane.
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 document discusses the 3D viewing pipeline which transforms 3D world coordinates to 2D viewport coordinates through a series of steps. It then describes parallel and perspective projections. Parallel projection preserves object scale and shape while perspective projection does not due to foreshortening effects. Perspective projection involves projecting 3D points along projection rays to a view plane based on a center of projection. Other topics covered include vanishing points, different types of perspective projections, and how viewing parameters affect the view volume and object positioning in the view plane coordinates.
This document discusses various attributes that can be used to modify the appearance of graphical primitives like lines and curves when displaying them, including line type (solid, dashed, dotted), width, color, fill style (hollow, solid, patterned), and fill color/pattern. It describes how these attributes are specified in applications and how different rendering techniques like rasterization can be used to display primitives with various attribute settings.
The document discusses two algorithms for filling polygons: boundary fill and flood fill. Boundary fill starts at a point inside the polygon and fills pixels until it reaches the boundary color. Flood fill replaces all pixels of a specified interior color with a fill color. Both can be implemented with 4-connected or 8-connected pixels. Flood fill colors the entire area but uses more memory, while boundary fill stops at the boundary and is more efficient.
Three key points about advanced computer graphics and 3D viewing:
1. 3D viewing involves establishing a viewing coordinate system and transforming 3D world coordinates to 2D viewing coordinates using translations and rotations. Projections like parallel and perspective then project the viewing coordinates onto a 2D view plane.
2. Common projections used in 3D viewing are parallel projections, which project lines parallel to the view plane, and perspective projections, which simulate how the human eye sees and cause objects to appear smaller with distance.
3. Viewing pipelines involve modeling, transformations between coordinate systems, projections, clipping to a view volume, and normalization before rendering the 2D image. Technologies like OpenGL help specify common operations like projections, view
The document discusses different techniques for filling polygons, including boundary fill, flood fill, and scan-line fill methods. It provides details on how each technique works, such as using a seed point and filling neighboring pixels for boundary fill, replacing all pixels of a selected color for flood fill, and drawing pixels between edge intersections for each scan line for scan-line fill. Examples are given to illustrate the filling process for each method.
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.
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.
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.
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 discusses 2D and 3D geometric transformations. It describes two types of transformations: geometric transformations that alter the object itself, and coordinate transformations that alter the coordinate system. Several common 2D geometric transformations are covered, including translation, rotation, scaling, reflection and shear. Matrix representations are introduced to combine multiple transformations into a single operation. The concept of homogeneous coordinates is explained for representing 2D transformations with 3x3 matrices. Finally, a general method for 2D rotation around a pivot point is described.
This document discusses various 2D geometric transformations including translation, rotation, scaling, and more complex transformations. Translation moves an object by adding offsets to x and y coordinates. Rotation repositions an object along a circular path defined by a rotation angle and pivot point. Scaling changes an object's size by multiplying x and y coordinates by scaling factors. More advanced topics covered include reflection, shear transformations, and performing multiple transformations sequentially as composites.
This document discusses 2D transformations in computer graphics, including translation, rotation, and scaling. Translation moves an object by adding offsets to x and y coordinates. Rotation uses trigonometry and rotation matrices to reposition objects around a central point. Scaling enlarges or shrinks objects by multiplying their coordinates by scaling factors. Homogeneous coordinates allow representing these transformations with matrix multiplications.
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.
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.
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 types of 2D transformations like scaling, reflection, rotation and translations as well as their corresponding matrix representations. It also discusses representing transformations in homogeneous coordinates and the concept of screen and world coordinates in the context of mapping between them.
The document discusses 2D geometric transformations including translation, scaling, and homogeneous coordinates. Translation moves an object along a straight path by adding distances to the x- and y-coordinates. Scaling changes the size of an object by multiplying the x- and y-coordinates by scaling factors. Homogeneous coordinates represent translation using a transformation matrix that adds translation amounts to the x- and y-values.
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.
1) 2D geometric transformations include translations, scaling, and rotations. They can be represented by transformation matrices.
2) Translation moves an object by adding offsets to x and y coordinates. It can be represented by a 3x3 matrix with 1s on the diagonal and offsets as the last column.
3) Scaling enlarges or shrinks an object by multiplying x and y coordinates by scaling factors. It can be represented by a 2x2 diagonal matrix with scaling factors.
4) Rotation rotates an object by applying a trigonometric transformation to x and y coordinates. It can be represented by a 2x2 rotation matrix containing cosine and sine of the rotation angle.
The document discusses 2D transformations in computer graphics, including translation, rotation, scaling, and shearing. Translation moves an object by adding offsets to x and y coordinates. Rotation rotates objects around the origin by applying trigonometric functions to x and y. Scaling enlarges or shrinks objects along the x- and y-axes. Shearing distorts objects along an axis based on their position on the other axis. Homogeneous coordinates allow transformations like translation, rotation, and scaling to be expressed using matrix multiplication.
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.
1. The document discusses various 2D transformations including translation, rotation, scaling, reflection, shearing, and their representation using homogeneous coordinates and homogeneous transformations. All transformations can be represented as matrix multiplication using homogeneous coordinates.
2. Homogeneous coordinates allow geometric transformations to be expressed as matrix multiplications, enabling efficient concatenation of multiple transformations. Any 2D point (x,y) can be represented as a 3D homogeneous coordinate (x,y,1).
3. Common transformations like translation, rotation, scaling, etc. that were previously represented using vector addition can now be uniformly represented using matrix multiplications in homogeneous coordinates. This allows multiple transformations to be applied sequentially with a single matrix multiplication.
This document discusses 2D transformations in computer graphics, including rotation, reflection, and shearing. It explains rotation using trigonometric equations to express transformed coordinates in terms of an angle, and represents rotation using a rotation matrix. Reflection is described as rotating an object 180 degrees about an axis, and reflection about the x-axis is represented using a matrix. Shearing is defined as a transformation that changes an object's shape by sliding its layers, and shearing matrices for the x and y directions are provided.
This document discusses different types of 2D transformations including translation, rotation, and scaling. Translation moves an object by adding a translation vector to the original coordinates. Rotation rotates an object around an origin by applying a rotation matrix to the original coordinates. Scaling resizes an object by multiplying the original coordinates by scaling factors. These transformations can be represented using matrix algebra and are important for manipulating 2D graphics.
This document discusses different types of coordinate transformations, including translation, rotation, scaling, and reflection. Translation moves all points the same distance in the same direction. Rotation turns the coordinate system around a fixed point. Scaling changes the units of measurement along the axes. Reflection mirrors the coordinate system across an axis. Each transformation has a corresponding inverse that undoes the original transformation.
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.
Introduction of Transformations (1).pptxChumchumKumar
This document discusses different types of geometric transformations, focusing on translation. It defines translation as the movement of an object from one position to another without deformation. Translation is represented by adding offsets (tx, ty) to the original x and y coordinates. The offsets form the translation vector or shift vector (Tx, Ty). Translation of multiple points can be represented using matrix operations, where the translated point P' is equal to the original point P plus the translation vector T. Examples are provided to demonstrate translating single points and geometric shapes.
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We have designed & manufacture the Lubi Valves LBF series type of Butterfly Valves for General Utility Water applications as well as for HVAC applications.
Covid Management System Project Report.pdfKamal Acharya
CoVID-19 sprang up in Wuhan China in November 2019 and was declared a pandemic by the in January 2020 World Health Organization (WHO). Like the Spanish flu of 1918 that claimed millions of lives, the COVID-19 has caused the demise of thousands with China, Italy, Spain, USA and India having the highest statistics on infection and mortality rates. Regardless of existing sophisticated technologies and medical science, the spread has continued to surge high. With this COVID-19 Management System, organizations can respond virtually to the COVID-19 pandemic and protect, educate and care for citizens in the community in a quick and effective manner. This comprehensive solution not only helps in containing the virus but also proactively empowers both citizens and care providers to minimize the spread of the virus through targeted strategies and education.
Online train ticket booking system project.pdfKamal Acharya
Rail transport is one of the important modes of transport in India. Now a days we
see that there are railways that are present for the long as well as short distance
travelling which makes the life of the people easier. When compared to other
means of transport, a railway is the cheapest means of transport. The maintenance
of the railway database also plays a major role in the smooth running of this
system. The Online Train Ticket Management System will help in reserving the
tickets of the railways to travel from a particular source to the destination.
An In-Depth Exploration of Natural Language Processing: Evolution, Applicatio...DharmaBanothu
Natural language processing (NLP) has
recently garnered significant interest for the
computational representation and analysis of human
language. Its applications span multiple domains such
as machine translation, email spam detection,
information extraction, summarization, healthcare,
and question answering. This paper first delineates
four phases by examining various levels of NLP and
components of Natural Language Generation,
followed by a review of the history and progression of
NLP. Subsequently, we delve into the current state of
the art by presenting diverse NLP applications,
contemporary trends, and challenges. Finally, we
discuss some available datasets, models, and
evaluation metrics in NLP.
2. 2D Transformations
“Transformations are the operations applied to
geometrical description of an object to change its
position, orientation, or size are called geometric
transformations”.
3. Translation
Translation is a process of changing the position
of an object in a straight-line path from one co-
ordinate location to another.
We can translate a two dimensional point by
adding translation distances, tx and ty.
Suppose the original position is (x ,y) then new
position is (x’, y’).
Here x’=x + tx and y’=y + ty.
6. Matrix form of the equations:
X’ = X + tx and Y’ = Y + ty is
P = x P’ = x’ T= tx
y y’ ty
we can write it,
P’= P + T
7. Translate a polygon with co-ordinates A(2,5) B(7,10) and C(10,2) by 3
units in X direction and 4 units in Y direction.
A’ = A +T
= 2 + 3 = 5
5 4 9
B’ = B + T
= 7 + 3 = 10
10 4 14
C’ = C + T
= 10 + 3 = 13
2 4 7
8. Rotation
A two dimensional rotation is applied to an object by
repositioning it along a circular path in the xy plane.
Using standard trigonometric equations , we can express
the transformed co-ordinates in terms of
x’ = r cos( coscosr sinsin
y’ = r sin( cosr sin
The original co-ordinates of the point is
x = r cos
y = r sin
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After substituting equation 2 in equation 1 we get
x’=x cos
Y’=x sin + y cos
11. That equation can be represented in matrix form
x’ y’ = x y cos
-sin cos
we can write this equation as,
P’ = P . R
Where R is a rotation matrix and it is given as
R = cos
-sin cos
12. A point (4,3) is rotated counterclockwise by angle of 45.
find the rotation matrix and the resultant point.
R = cos = cos45 sin45
- cos -sin45 cos45
= 1/√2 1/√2
- 1/√2 1/√2
P’ = 4 3 1/√2 1/√2
- 1/√2 1/√2
= 4/√2 – 3/√2 4/√2 + 3/√2
= 1/√2 7/√2
13. Scaling
A scaling transformation changes the size of an object.
This operation can be carried out for polygons by
multiplying the co-ordinates values (x , y) of each vertex
by scaling factors Sx and Sy to produce the transformed
co-ordinates (x’ , y’).
x’ = x . Sx
y’ = y . Sy
In the matrix form
x’ y’ = x y Sx 0
0 Sy
= P . S
15. Homogeneous co-ordinates for Translation
The homogeneous co-ordinates for translation are given as
T = 1 0 0
0 1 0
tx ty 1
Therefore , we have
x’ y’ 1 = x y 1 1 0 0
0 1 0
tx ty 1
= x + tx y + ty 1
16. Homogeneous co-ordinates for rotation
The homogeneous co-ordinates for rotation are given as
R = cos sin
-sin cos
0 0 1
Therefore , we have
x’ y’ 1 = x y 1 cos sin
-sin cos
0 0 1
= x cos - y sin x sin + y cos 1
17. Homogeneous co-ordinates for scaling
The homogeneous co-ordinate for scaling are given as
S = Sx 0 0
0 Sy 0
0 0 1
Therefore , we have
x’ y’ 1 = x y 1 Sx 0 0
0 Sy 0
0 0 1
= x . Sx y . Sy 1
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28. CONCLUSION
To manipulate the initially created
object and to display the
modified object without having to
redraw it, we use
Transformations.
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