3D TransformationsGeometryScalar FieldExample Scalar FieldsVector SpacesExample Vector SpacesBasis VectorsVector CoordinatesInterpreting Vector CoordinatesLinear TransformsMatricesReading Matrix ExpressionsThe Basis is Important!Transformation ExamplePointsHow Vectors and Points DifferMaking Sense of PointsA Basis for PointsFramesPictures of FramesA Consistent ModelHomogeneous CoordinatesAffine CombinationsThe Points BetweenAffine TransformationsComposing TransformationsSame PointsSame Point in Different FramesMore Frame ChangesMore Frame ChangesNext Time1/26/0713D TransformationsComputer GraphicsCOMP 770 (236)Spring 2007Instructor: Brandon Lloyd1/26/072Geometry■ Geometric entities, such as points in space, exist without numbers.■ Coordinates are a naming scheme.° The same point can be described by different coordinates. ° Both vectors and points expressed by coordinates, but they are very different ■ Our plan1. understand the “things”2. THEN associate coordinates to them.TriangleRaleigh-Durham(50, 160)Go 7 miles southwest1/26/073Scalar Field■ Definition. A set S over which addition (+) and multiplication (.) are closed.■ These operators commute, associate, and distribute■ Both operators have a unique identity element■ Each element has a unique inverse under both operatorsSbaSbaSb,a ∈⋅∈+∈∀caba)cb(ac)ba()cb(ac)ba()cb(aabbaabbaSc,b,a⋅+⋅=+⋅⋅⋅=⋅⋅++=++⋅=⋅+=+∈∀01aaaa+=⋅=1()0 1aa aa−+−= ⋅ =1/26/074Example Scalar Fields■ Real Numbers■ Complex Numbers (given the standard definitions for addition and multiplication)■ Rational Functions (Ratios of polynomials)■ Notation: we will represent scalars by lower case letters. a, b, c, … are scalar variables.1/26/075Vector Spaces■ Vector space (V): scalars and vectors, denoted by .■ Two operations for vectors:° vector-vector addition° scalar-vector multiplication■ Vector-vector addition commutes and associates.■ There is also an additive identity, and an additive inverse for each vector■ Scalar-vector multiplication distributesxK,uv V u v V∀∈+∈GGKK()()uv vuu vw uv w+=+ + + = + +KK KK K KKKK Kvaua)vu(aubuau)ba(KKKKKKK+=++=+,uV aSauV∀∈∀∈ ∈GK0()0uuuu+=+−=KKKKK K1/26/076Example Vector Spaces■ Geometric Vectors (directed segments)■ N-tuples of scalars■ Not coincidentally, we can use N-tuples to represent vectorsuKvKwKuKvKwvuKKG=+u2KvK−(1,3,7) (3,5, 4)(2,2, 3) 2 (2,6,14)(3,5, 4) ( 2, 2,3)ttttttuuv wvuwv=+===− ==−=−−KKKKKKKK1/26/077Basis Vectors■ A vector basis is a subset of vectors from V that can be used to generate any other element in V, using just additions and scalar multiplications.■ A basis set, , is linearly dependentif: Otherwise, the basis set is linearly independent. A linearly independent basis set with ielementsis said to spanan i-dimensionalvector space.Basis vectors are physical things, not numbers.n21v,...,v,vKKK∑==≠∃n0iiin210vathat such0a,...,a,aK1/26/078Vector Coordinates■ A linearly independent basis set can be used to uniquely name or address a vector. This is the done by assigning the vector coordinates as follows:■ Note: we’ll use bold letters to indicate tuples of scalars that are interpreted as coordinates■ Our vectors are still abstract entities. So how do we interpret the equation above?13123 213tiiicxcvvvvcvc=⎡⎤⎢⎥== =⎡⎤⎣⎦⎢⎥⎢⎥⎣⎦∑cKKK K KK1/26/079Interpreting Vector Coordinates1vK2vK3vK33vcK11vcK22vcKctvKValid Interpretation11vcKctvK22vcK33vcKEqually Valid InterpretationRemember, vectors don’t have any notion of position1/26/0710Linear TransformsA linear transformation, L, is just a mapping from V to V which satisfies the following properties:Linearity implies:Expressing with a basis and coordinate vector gives:)u(a)ua(and)v()u()vu(KKKKKKLLLLL=+=+∑∑=⎟⎠⎞⎜⎝⎛=⇒iiiiii)v(cvc)x(xKKKKLLLxK11123 2 1 2 3 233() ( ) ( )ccvv v c v v v ccc⎡⎤⎡⎤⎢⎥⎢⎥⇒⎡⎤⎡ ⎤⎣⎦⎣ ⎦⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦KK K K K KLL LTransforms my basis vectors and leaves the coordinates unchanged1/26/0711MatricesLinear transformations are equivalent to those that can be expressed using matrices and matrix operations.We can interpret this expression in one of two wayschange of basis vectors change of coordinates111121311232 123212223233132333() ( ) ( )cmmmcvv vc vvvmmmccmmmc⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⇒⎡⎤⎡⎤⎣⎦⎣⎦⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦LL LKK K KKK11 12 13 1123 21 22 23 231 32 33 3mm m cvv v m m m cmm m c⎛⎞⎡⎤⎡⎤⎜⎟⎢⎥⎢⎥⎡⎤⎜⎣ ⎦ ⎟⎢⎥⎢⎥⎜⎟⎢⎥⎢⎥⎣⎦⎣⎦⎝⎠KK K11 12 13 1123 21 22 23 231 32 33 3mm m cvv v m m m cmm m c⎛⎞⎡⎤⎡ ⎤⎜⎟⎢⎥⎢ ⎥⎡⎤⎣⎦⎜ ⎟⎢⎥⎢ ⎥⎜⎟⎢⎥⎢ ⎥⎣⎦⎣ ⎦⎝⎠KK K1/26/0712Reading Matrix ExpressionsOften we desire to apply sequences of operations to vectors. Forinstance, we might want to rotate a particular vector, add it to some other vector, and then rotate the result back. In order to specify and interpret such sequences, you should become proficient at reading matrix expressions.Consider the following expression:()ttt tvv v m⇒= =cMcMccKK K K()dMcMccttttvvvvKKKK==⇒Think of this as changing from one space to another (i.e. world space to eye space)Think of this as moving an vector, changing its coordinates, within a common space. (i.e. rotate a normal around some axis)1/26/0713The Basis is Important!If you are given coordinates and told to transform them using a matrix, you have not been given enough information to determine the final mapping.Consider the matrix:If we apply this matrix to coordinates there mustbe some implied basis, because coordinates are not geometric entities (a basis is required to convert coordinates into a vector). Assume this implied basis is . Thus, our coordinates describe the vector . The resulting transform,, will stretch this vector by a factor of 2 in the direction of the first element of the basis set. Of course that direction depends entirely on .⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=100010002MctwvKK=twKMccttwwKK⇒twK1/26/0714Transformation Examplewt= [v1, v2, v3]nt= [v1, v2, v3]These vectors with identical initial and final coordinates are very different geometric entities1/26/0715PointsConceptually, Points and Vectors are very different.A point is a place in space. A vector describes a direction independent of