2.5.3 Compositions of Transformations
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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.
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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.
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This document provides an introduction to real-time rendering concepts. It discusses 3D mathematics including coordinate systems, primitives, and affine transformations. It then explains the graphics pipeline including programmable shaders and GPU architecture. Finally, it covers simulating light through reflection models and shading models. Key topics include primitive types and topologies, constructing polygon meshes, transforming objects, the view and projection transforms, and how shaders are used in the graphics pipeline to simulate lighting effects.
2.6.2 Rotations and Compositions
- The student can identify and draw rotations and compositions of transformations.
- Rotations involve turning a figure around a fixed point called the center of rotation. Specific rotations discussed are 90° and 180° about the origin.
- Compositions of transformations involve performing two or more transformations sequentially, such as a glide reflection which reflects a figure and then translates it.
Computer graphics
The document discusses 2D and 3D transformations used in computer graphics. It covers topics like scaling, rotation, translation and how they can be represented using matrices. Matrix representations allow multiple transformations to be combined through matrix multiplication. Both linear transformations like scaling and rotation, as well as affine transformations like translation, can be captured with matrices. The order of matrix multiplications is important, as transformations are not commutative. These concepts extend from 2D to 3D graphics using homogeneous coordinates.
04transformation2d
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.
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6.7 Exponential and Logarithmic Models
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* Use Newton's Law of Cooling
* Use logistic-growth models
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This document provides instruction on factoring polynomials and quadratic equations. It begins by reviewing factoring techniques like finding the greatest common factor and factoring trinomials and binomials. Examples are provided to demonstrate the factoring methods. The document then discusses solving quadratic equations by factoring, putting the equation in standard form, and setting each factor equal to zero. An example problem demonstrates solving a quadratic equation through factoring. The document concludes by assigning homework and an optional reading for the next class.
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* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
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* Use the vertical line test to determine a function
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* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
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* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
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This document discusses functions and their graphs. It defines increasing, decreasing and constant functions based on how the function values change as the input increases. Relative maxima and minima are points where a function changes from increasing to decreasing. Symmetry of functions is classified by the y-axis, x-axis and origin. Even functions are symmetric about the y-axis, odd functions are symmetric about the origin. Piecewise functions have different definitions over different intervals.
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1) Polynomial equations solved by factoring
2) Equations with radicals where radicals are eliminated by raising both sides to a power
3) Equations with rational exponents where both sides are raised to the reciprocal power
4) Equations quadratic in form where an algebraic substitution is made to transform into a quadratic equation
5) Absolute value equations where both positive and negative solutions must be considered.
1.5 Quadratic Equations (Review)
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- A function is a relation where each input has a single, unique output
- The domain is the set of inputs and the range is the set of outputs
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* Solve counting problems using permutations involving n distinct objects.
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* Calculate geometric probabilities
* Use geometric probability to predict results in real world situations.
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* Use summation notation.
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* Use the formula for the sum of the first n terms of a geometric series.
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Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
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Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
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1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
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Mysore WhatsApp group:- https://chat.whatsapp.com/EhqtHtCC75vCAX7gaO842N
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2.5.3 Compositions of Transformations
- 1. Compositions of Transformations The student is able to (I can): • Draw and identify compositions of transformations
- 2. composition of transformations Performing two or more transformations sequentially (one after another) to a figure. An example of a composition is a glideglideglideglide reflectionreflectionreflectionreflection: we reflect the figure and then translate it along a vector.
- 3. To describe a composite transformation using notation, state each of the transformations that make up the composite transformation and link them with the symbol . The transformations are performed in order from right to leftright to leftright to leftright to left. Example: To perform the transformation T R r rx-axis — Reflect across the x-axis R90° — Rotate 90° counterclockwise T2, 4 — Translate along the vector 〈2, 4〉 ° −2,4 90 x axisT R r
- 4. With most compositions, it is important to perform them in the order given. ° −x0 is9 axrR − °x axis 90Rr (Reflect across theReflect across theReflect across theReflect across the xxxx----axisaxisaxisaxis and then rotate 90rotate 90rotate 90rotate 90°°°°.) (Rotate 90Rotate 90Rotate 90Rotate 90°°°° and then reflect acrossreflect acrossreflect acrossreflect across the xthe xthe xthe x----axisaxisaxisaxis.)
- 5. Examples (a) Describe the composition (b) Graph the transformations 1. (1, 4), (—2, 1), (—4, 1): T—3, 1 ry-axis 2. (2, 1), (3, 5), (5, 2): R180° ry=2
- 6. Examples (a) Describe the composition (b) Graph the transformations 1. (1, 4), (—2, 1), (—4, 1): T—3, 1 ry-axis a) Reflect across the y-axis and translate along the vector 〈—3, 1〉translate along the vector 〈—3, 1〉 b)
- 7. Examples (a) Describe the composition (b) Graph the transformations 2. (2, 1), (3, 5), (5, 2): R180° ry=2 a) Reflect across the line y=2 and then rotate 180°.rotate 180°. b)