1Computer Graphics (Fall 2005)Computer Graphics (Fall 2005)COMS 4160, Lecture 2: Review of Basic Mathhttp://www.cs.columbia.edu/~cs4160To DoTo Do Complete Assignment 0; e-mail by tomorrow Download and compile skeleton for assignment 1 Read instructions re setting up your system Ask TA if any problems, need visual C++ etc. We won’t answer compilation issues after next lecture Are there logistical problems with getting textbooks, programming (using MRL lab etc.?), office hours? About first few lectures Somewhat technical: core mathematical ideas in graphics HW1 is simple (only few lines of code): Lets you see how to use some ideas discussed in lecture, create imagesMotivation and OutlineMotivation and Outline Many graphics concepts need basic math like linear algebra Vectors (dot products, cross products, …) Matrices (matrix-matrix, matrix-vector mult., …) E.g: a point is a vector, and an operation like translating or rotating points on an object can be a matrix-vector multiply Much more, but beyond scope of this course (e.g. 4162) Chapters 2.4 (vectors) and 5.2.1,5.2.2 (matrices) Worthwhile to read all of chapters 2 and 5 Should be refresher on very basic material for most of you If not understand, talk to me (review in office hours)VectorsVectors Length and direction. Absolute position not important Use to store offsets, displacements, locations  But strictly speaking, positions are not vectors and cannot be added: a location implicitly involves an origin, while an offset does not.=Usually written as or in bold. Magnitude written as aaVector AdditionVector Addition Geometrically: Parallelogram rule In cartesian coordinates (next), simply add coordsaba+b= b+aCartesian CoordinatesCartesian Coordinates X and Y can be any (usually orthogonal unit) vectorsXA = 4 X +3Y()22 TxAAxyAxyy===+2Vector MultiplicationVector Multiplication Dot product (2.4.3) Cross product (2.4.4) Orthonormal bases and coordinate frames (2.4.5,6) Note: book talks about right and left-handed coordinate systems. We always use right-handedDot (scalar) productDot (scalar) productabφ?ab ba==ii1coscosab a bababφφ−==ii()() () ( )ab c ab acka b a kb k a b+=+==iiiii iDot product: some applications in CGDot product: some applications in CG Find angle between two vectors (e.g. cosine of angle between light source and surface for shading) Finding projection of one vector on another (e.g. coordinates of point in arbitrary coordinate system) Advantage: can be computed easily in cartesian componentsProjections (of b on a)Projections (of b on a)abφ??baba→=→=cosabba baφ→= =i2aabbaba aaa→= → =iDot product in Cartesian componentsDot product in Cartesian components?ababxxabyy•= • =abab ababxxab xx yyyy•= • = +Vector MultiplicationVector Multiplication Dot product (2.4.3) Cross product (2.4.4) Orthonormal bases and coordinate frames (2.4.5,6) Note: book talks about right and left-handed coordinate systems. We always use right-handed3Cross (vector) productCross (vector) product Cross product orthogonal to two initial vectors Direction determined by right-hand rule Useful in constructing coordinate systems (later)abφab ba×=−×sinab abφ×=Cross product: PropertiesCross product: Properties0()() ( )ab baaaabc abacakb kab×=−××=×+=×+××=×xyzyx zyz xzy xzx yxzy×=+×=−×=+×=−×=+×=−Cross product: Cartesian formula?Cross product: Cartesian formula?ab baaaa ababbbb ababxyz yzyzab x y z zx xzxyz xyyx−×= = −−*000aa baabaa bzy xab Ab z x yyx z−×= = −−Dual matrix of vector aVector MultiplicationVector Multiplication Dot product (2.4.3) Cross product (2.4.4) Orthonormal bases and coordinate frames (2.4.5,6) Note: book talks about right and left-handed coordinate systems. We always use right-handedOrthonormalOrthonormalbases/coordinate framesbases/coordinate frames Important for representing points, positions, locations Often, many sets of coordinate systems (not just X, Y, Z) Global, local, world, model, parts of model (head, hands, …) Critical issue is transforming between these systems/bases Topic of next 3 lecturesCoordinate FramesCoordinate Frames Any set of 3 vectors (in 3D) so that 10()()( )uvwuv vw uwwuvppuu pvv pww== =====×=++ii iiii4Constructing a coordinate frameConstructing a coordinate frame Often, given a vector a (viewing direction in HW1), want to construct an orthonormal basis Need a second vector b (up direction of camera in HW1) Construct an orthonormal basis (for instance, camera coordinate frame to transform world objects into in HW1)Constructing a coordinate frame?Constructing a coordinate frame?awa=We want to associate w with a, and v with b But a and b are neither orthogonal nor unit norm And we also need to find ubwubw×=×vwu=×MatricesMatrices Can be used to transform points (vectors) Translation, rotation, shear, scale (more detail next lecture) Section 5.2.1 and 5.2.2 of text Instructive to read all of 5 but not that relevant to courseWhat is a matrixWhat is a matrix Array of numbers (m×n = m rows, n columns) Addition, multiplication by a scalar simple: element by element135204MatrixMatrix--matrix multiplicationmatrix multiplication Number of columns in second must = rows in first Element (i,j) in product is dot product of row i of first matrix and column j of second matrix13369452278304MatrixMatrix--matrix multiplicationmatrix multiplication Number of columns in second must = rows in first Element (i,j) in product is dot product of row i of first matrix and column j of second matrix9 213 7 352041339618331928 242636441728 2    =      5MatrixMatrix--matrix multiplicationmatrix multiplication Number of columns in second must = rows in first Element (i,j) in product is dot product of row i of first matrix and column j of second matrix9 213 7 352041339618331928 242636441728 2    =      MatrixMatrix--matrix