The document discusses different types of 3D reference systems and coordinate axes, including left-handed and right-handed systems. It also covers various 3D transformations like translation, scaling, and rotation around the X, Y, and Z axes. The document explains how to project 3D objects and scenes onto a 2D surface using different projection types, including parallel and perspective projections. It discusses one-point, two-point, and three-point perspective as well as orthographic and oblique projections.
This document discusses different types of 2D and 3D geometric transformations in computer graphics, including translation, rotation, scaling, shearing, and reflection. It provides examples of how to mathematically represent these transformations using matrix multiplication and homogeneous coordinates. Transformations are used to position and modify 3D objects, change viewing positions, and affect how something is viewed.
3D transformation in computer graphicsSHIVANI SONI
This document discusses different types of 2D and 3D transformations that are used in computer graphics, including translation, rotation, scaling, shearing, and reflection. It provides the mathematical equations and transformation matrices used to perform each type of transformation on 2D and 3D points and objects. Key types of rotations discussed are roll (around z-axis), pitch (around x-axis), and yaw (around y-axis). Homogeneous coordinates are introduced for representing 3D points.
This document discusses various 3D geometric transformations including translation, scaling, rotation, and coordinate transformations. It provides details on:
1) How translation, uniform scaling, and relative scaling transformations work in 3D space.
2) How rotations around the x, y, z axes as well as general 3D rotations around arbitrary axes are performed.
3) How quaternions can be used to represent rotations and how rotation matrices are derived from quaternions.
4) How reflections, shears, and different coordinate systems require coordinate transformations between systems.
This document discusses the Daroko blog, which provides real-world applications of various IT skills. It encourages readers to not just learn computer graphics and other topics but to apply them in business contexts. The blog covers topics like computer graphics, networking, programming, IT jobs, technology news, blogging, website building, and IT companies. It aims to help readers gain practical experience applying their IT knowledge. Readers are instructed to search "Daroko blog" online to access resources on various IT subjects and their business applications.
Here are the key steps to solve this problem:
1. Write the rotation matrix for 45 degree rotation about the x-axis:
R = [1 0 0]
[0 cos(45) -sin(45)]
[0 sin(45) cos(45)]
2. Write the scaling matrix with factors of 4 in each direction:
S = [4 0 0]
[0 4 0]
[0 0 4]
3. Pre-multiply the original coordinates by the rotation matrix, then scale the results by pre-multiplying the rotation matrix by the scaling matrix:
A' = SR(0,1,0)
= S(0,0.
This document discusses different types of 2D and 3D geometric transformations in computer graphics, including translation, rotation, scaling, shearing, and reflection. It provides examples of how to mathematically represent these transformations using matrix multiplication and homogeneous coordinates. Transformations are used to position and modify 3D objects, change viewing positions, and affect how something is viewed.
3D transformation in computer graphicsSHIVANI SONI
This document discusses different types of 2D and 3D transformations that are used in computer graphics, including translation, rotation, scaling, shearing, and reflection. It provides the mathematical equations and transformation matrices used to perform each type of transformation on 2D and 3D points and objects. Key types of rotations discussed are roll (around z-axis), pitch (around x-axis), and yaw (around y-axis). Homogeneous coordinates are introduced for representing 3D points.
This document discusses various 3D geometric transformations including translation, scaling, rotation, and coordinate transformations. It provides details on:
1) How translation, uniform scaling, and relative scaling transformations work in 3D space.
2) How rotations around the x, y, z axes as well as general 3D rotations around arbitrary axes are performed.
3) How quaternions can be used to represent rotations and how rotation matrices are derived from quaternions.
4) How reflections, shears, and different coordinate systems require coordinate transformations between systems.
This document discusses the Daroko blog, which provides real-world applications of various IT skills. It encourages readers to not just learn computer graphics and other topics but to apply them in business contexts. The blog covers topics like computer graphics, networking, programming, IT jobs, technology news, blogging, website building, and IT companies. It aims to help readers gain practical experience applying their IT knowledge. Readers are instructed to search "Daroko blog" online to access resources on various IT subjects and their business applications.
Here are the key steps to solve this problem:
1. Write the rotation matrix for 45 degree rotation about the x-axis:
R = [1 0 0]
[0 cos(45) -sin(45)]
[0 sin(45) cos(45)]
2. Write the scaling matrix with factors of 4 in each direction:
S = [4 0 0]
[0 4 0]
[0 0 4]
3. Pre-multiply the original coordinates by the rotation matrix, then scale the results by pre-multiplying the rotation matrix by the scaling matrix:
A' = SR(0,1,0)
= S(0,0.
This document provides an overview of 2D and 3D graphics transformations including:
1. 2D affine transformations like translation, rotation, scaling and shearing and their properties such as preserving lines and ratios.
2. Representing transformations with matrices and composing multiple transformations.
3. Drawing 3D wireframe models using projections including orthogonal and perspective projections.
4. 3D affine transformations and their elementary forms as well as composing rotations in 3D.
5. Non-affine transformations like fish-eye and false perspective distortions.
A transformation that slants the shape of an object is called the shear transformation. Like in 2D shear, we can shear an object along the X-axis, Y-axis, or Z-axis in 3D. This ppt is about the transformation
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
This document provides an overview of 3D transformations, including translation, rotation, scaling, reflection, and shearing. It explains that 3D transformations generalize 2D transformations by including a z-coordinate and using homogeneous coordinates and 4x4 transformation matrices. Each type of 3D transformation is defined using matrix representations and equations. Rotation is described for each coordinate axis, and reflection is explained for each axis plane. Shearing is introduced as a way to modify object shapes, especially for perspective projections.
The document discusses various geometric transformations in 3D modeling including translation, rotation, scaling, reflection, and shearing. Translation moves an object by adding amounts to the x, y, and z coordinates. Rotation requires specifying an axis and angle of rotation. Scaling changes the size of an object by multiplying the x, y, and z coordinates. Reflection mirrors objects across planes, while shearing slants objects along axes. All transformations can be represented using 4x4 matrices.
This document discusses 3D geometric transformations and 3D modelling. It describes basic 3D transformations like translation, rotation, scaling, and shear. It also discusses scaling an object by translating it to the origin, scaling it, and translating it back. Finally, it explains that 3D viewing involves generating a view of an object from any position, analogous to taking a photograph using a camera's position, orientation, and aperture size to project the scene onto a flat surface.
The document discusses various 2D and 3D transformations including translation, scaling, rotation, reflection, shearing, and homogeneous coordinates. It provides the mathematical definitions and matrix representations for each transformation type in 2D and 3D. It also covers topics like composition and inverse of transformations, classification of transformations, and properties of rigid body transformations.
3D transformations are represented by 4x4 matrices. They include translation, scaling, and rotation. Rotation can be about coordinate axes or arbitrary axes. For arbitrary axis rotation, the process involves translating the axis to the origin, rotating the axis to align with an axis, rotating about that axis, then applying the inverse transformations to return the axis to its original orientation.
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.
Slides from when I was teaching CS4052 Computer Graphics at Trinity College Dublin in Ireland.
These slides aren't used any more so they may as well be available to the public!
There are some mistakes in the slides, I'll try to comment below these.
This is the third lecture - on using linear algebra for transformations.
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.
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
This document discusses 3D transformations in computer graphics. It covers 3D translation, rotation, and scaling. Translation moves an object by adding offsets to the x, y, and z coordinates. Rotation in 3D can occur around any axis and is represented using 4x4 matrices. Scaling enlarges or shrinks an object along the x, y, and z axes by multiplying coordinates by scale factors. More complex transformations can be achieved by combining multiple simple transformations.
This document discusses 3D transformations in computer graphics. It describes how 3D transformations modify and reposition graphics in 3D space using translation, scaling, and rotation. Translation moves an object using vectors in the x, y, and z directions. Scaling changes an object's size using scaling factors for each axis. Rotation changes an object's angle by specifying a rotation axis, direction, and angle. Matrix representations of transformations are provided.
This document discusses 3D coordinate spaces and transformations, including translations, scaling, and rotations. It explains how 3D scenes are projected onto 2D planes for display through parallel or perspective projections. Parallel projections include orthographic and isometric views, while perspective projections can be one-point or two-point. Key elements of 3D viewing include the camera position, look vector, and up vector. Matrices are used to represent transformations in homogeneous coordinates.
This document discusses 3D transformations in computer graphics including translation, rotation, scaling, and shearing. Translation simply moves an object along the x, y, and z axes and can be represented by a 4x4 matrix. Rotation is more complex in 3D and involves rotating around the x, y, or z axes. Scaling enlarges or shrinks an object along the x, y, and z axes. Shearing skews an object by changing the coordinates of one axis based on the values of another axis. The document provides formulas and examples for performing each type of 3D transformation.
The document discusses 3D scaling, transformation, and rotation using homogeneous coordinates and 4x4 matrices. It explains that translation, scaling, and rotation can be represented by 4x4 matrices to allow combinations of transformations. Points in 3D space can be represented using homogeneous coordinates by adding a fourth w coordinate, and translations are represented by adding values to the fourth row of a transformation matrix. Using homogeneous coordinates and 4x4 matrices allows multiple transformations to be combined and represented with single matrices.
Comprehensive coverage of fundamentals of computer graphics.
3D Transformations
Reflections
3D Display methods
3D Object Representation
Polygon surfaces
Quadratic Surfaces
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 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 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.
The document discusses nanocatalysis, which involves using nanotechnology products as catalysts (called nanocatalysts). It describes the history and introduction of nanocatalysis, benefits of nanocatalysts, methods of synthesizing nanocatalysts both homogenously and heterogeneously, types of nanocatalysts, how catalytic activity depends on properties like composition and environment, applications in industries like petroleum refining and pharmaceuticals, and concludes that nanocatalysts offer opportunities to meet future demands through their high activity and selectivity.
This document provides an overview of 2D and 3D graphics transformations including:
1. 2D affine transformations like translation, rotation, scaling and shearing and their properties such as preserving lines and ratios.
2. Representing transformations with matrices and composing multiple transformations.
3. Drawing 3D wireframe models using projections including orthogonal and perspective projections.
4. 3D affine transformations and their elementary forms as well as composing rotations in 3D.
5. Non-affine transformations like fish-eye and false perspective distortions.
A transformation that slants the shape of an object is called the shear transformation. Like in 2D shear, we can shear an object along the X-axis, Y-axis, or Z-axis in 3D. This ppt is about the transformation
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
This document provides an overview of 3D transformations, including translation, rotation, scaling, reflection, and shearing. It explains that 3D transformations generalize 2D transformations by including a z-coordinate and using homogeneous coordinates and 4x4 transformation matrices. Each type of 3D transformation is defined using matrix representations and equations. Rotation is described for each coordinate axis, and reflection is explained for each axis plane. Shearing is introduced as a way to modify object shapes, especially for perspective projections.
The document discusses various geometric transformations in 3D modeling including translation, rotation, scaling, reflection, and shearing. Translation moves an object by adding amounts to the x, y, and z coordinates. Rotation requires specifying an axis and angle of rotation. Scaling changes the size of an object by multiplying the x, y, and z coordinates. Reflection mirrors objects across planes, while shearing slants objects along axes. All transformations can be represented using 4x4 matrices.
This document discusses 3D geometric transformations and 3D modelling. It describes basic 3D transformations like translation, rotation, scaling, and shear. It also discusses scaling an object by translating it to the origin, scaling it, and translating it back. Finally, it explains that 3D viewing involves generating a view of an object from any position, analogous to taking a photograph using a camera's position, orientation, and aperture size to project the scene onto a flat surface.
The document discusses various 2D and 3D transformations including translation, scaling, rotation, reflection, shearing, and homogeneous coordinates. It provides the mathematical definitions and matrix representations for each transformation type in 2D and 3D. It also covers topics like composition and inverse of transformations, classification of transformations, and properties of rigid body transformations.
3D transformations are represented by 4x4 matrices. They include translation, scaling, and rotation. Rotation can be about coordinate axes or arbitrary axes. For arbitrary axis rotation, the process involves translating the axis to the origin, rotating the axis to align with an axis, rotating about that axis, then applying the inverse transformations to return the axis to its original orientation.
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.
Slides from when I was teaching CS4052 Computer Graphics at Trinity College Dublin in Ireland.
These slides aren't used any more so they may as well be available to the public!
There are some mistakes in the slides, I'll try to comment below these.
This is the third lecture - on using linear algebra for transformations.
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.
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
This document discusses 3D transformations in computer graphics. It covers 3D translation, rotation, and scaling. Translation moves an object by adding offsets to the x, y, and z coordinates. Rotation in 3D can occur around any axis and is represented using 4x4 matrices. Scaling enlarges or shrinks an object along the x, y, and z axes by multiplying coordinates by scale factors. More complex transformations can be achieved by combining multiple simple transformations.
This document discusses 3D transformations in computer graphics. It describes how 3D transformations modify and reposition graphics in 3D space using translation, scaling, and rotation. Translation moves an object using vectors in the x, y, and z directions. Scaling changes an object's size using scaling factors for each axis. Rotation changes an object's angle by specifying a rotation axis, direction, and angle. Matrix representations of transformations are provided.
This document discusses 3D coordinate spaces and transformations, including translations, scaling, and rotations. It explains how 3D scenes are projected onto 2D planes for display through parallel or perspective projections. Parallel projections include orthographic and isometric views, while perspective projections can be one-point or two-point. Key elements of 3D viewing include the camera position, look vector, and up vector. Matrices are used to represent transformations in homogeneous coordinates.
This document discusses 3D transformations in computer graphics including translation, rotation, scaling, and shearing. Translation simply moves an object along the x, y, and z axes and can be represented by a 4x4 matrix. Rotation is more complex in 3D and involves rotating around the x, y, or z axes. Scaling enlarges or shrinks an object along the x, y, and z axes. Shearing skews an object by changing the coordinates of one axis based on the values of another axis. The document provides formulas and examples for performing each type of 3D transformation.
The document discusses 3D scaling, transformation, and rotation using homogeneous coordinates and 4x4 matrices. It explains that translation, scaling, and rotation can be represented by 4x4 matrices to allow combinations of transformations. Points in 3D space can be represented using homogeneous coordinates by adding a fourth w coordinate, and translations are represented by adding values to the fourth row of a transformation matrix. Using homogeneous coordinates and 4x4 matrices allows multiple transformations to be combined and represented with single matrices.
Comprehensive coverage of fundamentals of computer graphics.
3D Transformations
Reflections
3D Display methods
3D Object Representation
Polygon surfaces
Quadratic Surfaces
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 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 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.
The document discusses nanocatalysis, which involves using nanotechnology products as catalysts (called nanocatalysts). It describes the history and introduction of nanocatalysis, benefits of nanocatalysts, methods of synthesizing nanocatalysts both homogenously and heterogeneously, types of nanocatalysts, how catalytic activity depends on properties like composition and environment, applications in industries like petroleum refining and pharmaceuticals, and concludes that nanocatalysts offer opportunities to meet future demands through their high activity and selectivity.
This document discusses different ways to mathematically represent curves, including polynomial representations and parametric forms. It focuses on cubic polynomials and parametric representations, explaining that parametric form solves problems with explicit and implicit forms by allowing representation of curves with infinite slopes or multiple y-values for a given x-value. Parametric form also makes it easier to combine curve segments continuously. The document then discusses spline curves, which use piecewise cubic polynomial functions to fit smooth curves through points, and cubic splines specifically, providing the equations used to define cubic splines.
nano catalysis as a prospectus of green chemistry Ankit Grover
Nanocatalysis and green chemistry prospects.
Nanocatalysts have higher activity, selectivity, and efficiency than traditional catalysts due to their high surface area to volume ratio. They can be designed for sustainability by having properties like recyclability, durability, and cost-effectiveness. Examples discussed include gold nanoparticle catalysts for oxidation reactions and magnetically separable nanoparticle catalysts. Nanocatalyst applications highlighted are water splitting for hydrogen production and storage, and fuel cells.
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.
A polygon mesh is a 3D surface made of vertices, edges, and faces that defines the shape of a polyhedral object. It can be constructed using box modeling with subdivision and extrusion tools, inflation modeling by extruding a 2D shape, or connecting primitive 3D shapes. Polygon meshes are commonly represented through face-vertex or winged-edge structures and can be rendered with flat, Gouraud, or Phong shading models. However, polygons only approximate curved surfaces and lose geometric information.
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.
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.
Indifference curves show combinations of goods that provide the same level of satisfaction to a consumer. A consumer seeks to maximize utility by consuming the combination on their highest attainable indifference curve, which is tangent to their budget constraint. As more of a good is consumed, the marginal rate of substitution diminishes, following the law of diminishing marginal rate of substitution. When prices or income change, the budget constraint shifts, changing the optimal consumption bundle where the indifference curve is tangent to the new budget line.
Notes 2D-Transformation Unit 2 Computer graphicsNANDINI SHARMA
Notes of 2D Transformation including Translation, Rotation, Scaling, Reflection, Shearing with solved problem.
Clipping algorithm like cohen-sutherland-hodgeman, midpoint-subdivision with solved problem.
What is catalysis, its type and its applicationLovnish Thakur
This document will give you information about catalysis and type of catalysis like homogenious and heterogenious catalysis and its various application .
This document discusses computer graphics and how linear algebra is used to represent and manipulate 2D and 3D images on a computer screen. It explains that 2D graphics exist in the xy-plane and can undergo transformations like scaling, translation, and rotation represented by multiplication of matrices. 3D graphics add the z-axis and use homogeneous coordinates to represent perspective projections. Transformations in 3D include scaling along three axes and combining multiple transformations through composite matrices.
1) The document discusses various 3D geometric transformations including translation, scaling, rotation, and coordinate transformations.
2) It explains transformations like uniform scaling, translation of a point, and rotation about the X, Y, and Z axes.
3) Rotation about an arbitrary axis is described as translating the axis to the origin, rotating it onto an axis, rotating the object, and translating the axis back.
This document discusses 3D transforms in computer graphics. It introduces homogeneous coordinates which add a fourth coordinate (w) to represent 3D points. Common 3D transforms include scaling, reflection, and rotation around the x, y, and z axes. Scaling simply multiplies each coordinate by a factor. Reflection flips points across a plane. Rotation matrices contain trigonometric functions of the angle of rotation to transform the coordinates.
The document discusses representing points in 3D space using cylindrical and spherical coordinate systems. It explains that cylindrical coordinates extend the 2D polar coordinate system (r,θ) by adding a z-coordinate. Spherical coordinates represent points using three values - the radial distance ρ, the azimuthal angle θ, and the polar angle φ. The document also covers converting between rectangular and cylindrical/spherical coordinates, and discusses how the elements of integration change based on the coordinate system used.
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.
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.
7. Translation
3D
Transformation
Rotation
3D Transformation Supriya H. Madane Slide 7
8. Translation in 3D!
Remembering 2D transformations -> 3x3 matrices,
take a wild guess what happens to 3D transformations.
1 0 tx
x tx
T tx , t y 0 1 ty T=(tx, ty, tz)
y ty
0 0 1
1 0 0 tx
x tx
0 1 0 ty
T tx , t y , tz y ty
0 0 1 tz
z tz
0 0 0 1
3D Transformation Supriya H. Madane Slide 8
9. Scaling, 3D Style
sx 0 0 S=(sx, sy, sz)
sx 0 x
S sx , s y * 0 sy 0
0 sy y
0 0 1
sx 0 0 0
sx 0 0 x
0 sy 0 0
S sx , s y , sz 0 sy 0 * y
0 0 sz 0
0 0 sz z
0 0 0 1
3D Transformation Supriya H. Madane 9
10. Rotations about the Z axis
R=(0,0,1, )
cos sin 0
cos sin
R sin cos 0
sin cos
0 0 1
cos sin 0 0
sin cos 0 0
R (0,0,1, )
0 0 1 0
0 0 0 1
3D Transformation Supriya H. Madane 10
11. Rotations about the X axis
Let’s look at the other axis rotations
R=(1,0,0, )
1 0 0 0
0 cos sin 0
R (1,0,0, )
0 sin cos 0
0 0 0 1
3D Transformation Supriya H. Madane 11
12. Rotations about the Y axis
R=(0,1,0, )
cos 0 sin 0
0 1 0 0
R (0,1,0, )
sin 0 cos 0
0 0 0 1
3D Transformation Supriya H. Madane 12
13. Viewing in 3D
3D Transformation Supriya H. Madane Slide 13
14. Man-made objects often have “cube-like” shape.
These objects have 3 principal axes.
From www.loc.gov/ jefftour/cutaway.html
3D Transformation Supriya H. Madane Slide 14
15. Display device
(a screen) is • How do we map 3D objects to 2D space?
2D…
2D to 2D is • 2D window to world.. and a viewport on the 2D
surface.
straight • Clip what won't be shown in the 2D window, and
forward… map the remainder to the viewport.
3D to 2D is
• Solution : Transform 3D objects on
more to a 2D plane using projections
complicated…
3D Transformation Supriya H. Madane Slide 15
16. Rays converge on eye position Rays parallel to view plane
Perspective Parallel
Orthographic Oblique
Elevations Axonometric Cavalier Cabinet
Top Left Right Isometric Dimetric Trimetric
17. In 3D…
View volume in the world
Projection onto the 2D projection plane
A viewport to the view surface
Process…
1… clip against the view volume,
2… project to 2D plane, or window,
3… map to viewport.
3D Transformation Supriya H. Madane Slide 17
18. Conceptual Model of the 3D viewing process
3D Transformation Supriya H. Madane 18
19. 2 types of projections
perspective and parallel.
Key factor is the center of projection.
if distance to center of projection is finite : perspective
if infinite : parallel
3D Transformation Supriya H. Madane Slide 19
20. Perspective:
visual effect is similar to human visual system...
has 'perspective foreshortening'
size of object varies inversely with distance from the center of
projection.
angles only remain intact for faces parallel to projection
plane.
Parallel:
less realistic view because of no foreshortening
however, parallel lines remain parallel.
angles only remain intact for faces parallel to projection
plane.
3D Transformation Supriya H. Madane Slide
21. Any parallel lines not parallel to the projection
plane, converge at a vanishing point.
There are an infinite number of these, 1 for each of the
infinite amount of directions line can be oriented.
If a set of lines are parallel to one of the three
principle axes, the vanishing point is called an axis
vanishing point.
There are at most 3 such points, corresponding to the
number of axes cut by the projection plane.
3D Transformation Supriya H. Madane Slide 21
22. One point, two point, three point perspective
One point perspective: One principal axis intersects view plane
3D Transformation Supriya H. Madane Slide
23. One point, two point, three point perspective
Two point perspective: two principal axes intersect view plane
3D Transformation Supriya H. Madane 23
24. One point, two point, three point perspective
3D Transformation Supriya Three principal axes intersect view plane
Three point perspective: H. Madane 24
25. View Plane
Three point
Two point
3D Transformation Supriya H. Madane
One point 25
26. 2 principle types:
orthographic and oblique.
Orthographic :
direction of projection = normal to the projection plane.
Oblique :
direction of projection != normal to the projection
plane.
3D Transformation Supriya H. Madane Slide
27. Orthographic (or orthogonal) projections:
front elevation, top-elevation and side-elevation.
all have projection plane perpendicular to a principle axes.
Useful because angle and distance measurements can be
made...
However, As only one face of an object is shown, it can be
hard to create a mental image of the object, even when
several view are available.
3D Transformation Supriya H. Madane Slide