graphicsLecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '01Lecture 7 --- 6.837 Fall '016.837 LECTURE 71.Geometric Image Transformations2.Two-Dimensional Geometric Transforms3.Translations4.Groups and Composition5.Rotations6.Euclidean Transforms7.Problems with this Form8.Choose a Subspace9.Playing with Euclidean Transforms10.Similitude Transforms11.Playing with Similitude Transforms12.Affine Transformations13.Affine Properties13a.Examples of Affine Transformations14.Playing with Affine Transforms15.Determining Affine Transforms16.Solution Method17.Projective Transformations18.Projection19.Projective Transforms20.Degrees of Freedom21.Specifying a projective transform22.Projective Example23.Next Time Lecture 7 Outline 6.837 Fall '01Lecture 7 --- 6.837 Fall '01http://graphics/classes/6.837/F01/Lecture07/index.html [10/2/2001 5:26:14 PM]Geometric Image TransformationsAlgebraic Groups● Euclidean● Affine● Projective● Bovine● Lecture 7 Slide 1 6.837 Fall '01Lecture 7 --- 6.837 Fall '01http://graphics/classes/6.837/F01/Lecture07/Slide01.html [10/2/2001 5:20:39 PM]Two-Dimensional Geometric TransformsGeometric transforms are functions that map points from one place to anotherGeometric transformscan be applyed todrawing primitives(lines, conics, triangles)● pixel coordinates of an image (or sprites)● We'll begin with simple transforms and generalize them.Lecture 7 Slide 2 6.837 Fall '01Lecture 7 --- 6.837 Fall '01http://graphics/classes/6.837/F01/Lecture07/Slide02.html [10/2/2001 5:20:41 PM]TranslationsTranslations are a simple family of two-dimensional transforms. Translations were at theheart of our Sprite implementations in Project #1.Translations have the following formx' = x + txy' = y + tyFor every translation there exists an inverse function which undoes the translation. In ourcase the inverse looks like:x = x' - txy = y' - tyThere also exists a special translation, called the identity, that leaves every point unchanged.x' = x + 0y' = y + 0Lecture 7 Slide 3 6.837 Fall '01Lecture 7 --- 6.837 Fall '01http://graphics/classes/6.837/F01/Lecture07/Slide03.html [10/2/2001 5:20:45 PM]Groups and CompositionFor Translations:There exists an inverse mapping for each function1. There exists an identity mapping2. The composition operation is associative3. The functions are "closed under composition"4. These properties might seem trivial at first glance, but they are actually veryimportant, because when these conditions are shown for any class of functions andtheir two-argument composition operation, then they form an algebraic group. Oneof the consequences is that any series of translations can be composed to a singletranslation. Another consequence is that the inverse is unique.Lecture 7 Slide 4 6.837 Fall '01Lecture 7 --- 6.837 Fall '01http://graphics/classes/6.837/F01/Lecture07/Slide04.html [10/2/2001 5:20:47 PM]RotationsAnother group of 2-transforms are the rotations about the origin.Lecture 7 Slide 5 6.837 Fall '01Lecture 7 --- 6.837 Fall '01http://graphics/classes/6.837/F01/Lecture07/Slide05.html [10/2/2001 5:20:49 PM]Euclidean TransformsThe union of translations and rotation functions defines the Euclidean SetProperties of Euclidean Transformations:They preserve distances● They preserve angles● How do you represent these functions?Lecture 7 Slide 6 6.837 Fall '01Lecture 7 --- 6.837 Fall '01http://graphics/classes/6.837/F01/Lecture07/Slide06.html [10/2/2001 5:21:35 PM]Problems with this FormMust consider Translation and Rotation separately● Computing the inverse transform involves multiple steps● Order matters between the R and T partsThese problem can be remedied by considering our 2 dimensional image plane as a 2Dsubspace within 3D.● Lecture 7 Slide 7 6.837 Fall '01Lecture 7 --- 6.837 Fall '01http://graphics/classes/6.837/F01/Lecture07/Slide07.html [10/2/2001 5:21:38 PM]Choose a SubspaceWe can use any planar subspace as long as it does not contain the originWLOG assume the our 2D space of points lies on the 3D plane z = 1Now we can express all Euclidean Transforms in matrix form:This gives us a three parameter group of Transformations.Lecture 7 Slide 8 6.837 Fall '01Lecture 7 --- 6.837 Fall '01http://graphics/classes/6.837/F01/Lecture07/Slide08.html [10/2/2001 5:21:40 PM]Playing with Euclidean TransformsIn what order are the translation and rotation performed?● Will this family of transforms always generate points on our choosen3-D plane?Why?● Lecture 7 Slide 9 6.837 Fall '01Lecture 7 --- 6.837 Fall '01http://graphics/classes/6.837/F01/Lecture07/Slide09.html [10/2/2001 5:21:42 PM]Similitude TransformsWe can define a 4-parameter superset of Euclidean Transforms with additional capabilitiesProperties of Similitudes:Distance between any 2 pointsare changed by a fixed ratio● Angles are preserved● Maintains "similar" shape(similar triangles, circles map to circles, etc)● Lecture 7 Slide 10 6.837 Fall '01Lecture 7 --- 6.837 Fall '01http://graphics/classes/6.837/F01/Lecture07/Slide10.html [10/2/2001 5:21:44 PM]Playing with Similitude TransformsAdds reflections● Scales in x and y must be the same. Why?● Order?● Will this family of transforms always generate points on our choosen3-D plane?Why?● Lecture 7 Slide 11 6.837 Fall '01Lecture 7 --- 6.837 Fall '01http://graphics/classes/6.837/F01/Lecture07/Slide11.html [10/2/2001 5:21:45 PM]Affine TransformationsA 6-parameter group of transformsLecture 7 Slide 12 6.837 Fall '01Lecture 7 --- 6.837 Fall '01http://graphics/classes/6.837/F01/Lecture07/Slide12.html [10/2/2001 5:21:48 PM]Affine PropertiesTo the right is a simple illustration of how wecan map our parameters into the
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