XPM 2D Transformations Week 2, Lecture 3

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1 CS 430/585 Computer Graphics I XPM 2D Transformations Week 2, Lecture 3 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel University Overview XPM file format Mathematical preliminaries 2D Affine transformations Homogeneous coordinates Discussion of homework #1 Lecture Credits: Most pictures are from Foley/VanDam; Additional and extensive thanks also goes to those credited on individual slides XPM Format XPM Basics Encoded pixels C code ASCII Text file Viewable on Unix w/ xv and other tools On Windows with IrfanVIew Translate w/ XPixelMap (XPM) Native file format in X Windows Color cursor and icon bitmaps Files are actually C source code Read by compiler instead of viewer Successor of X BitMap (XBM) B-W format convert 3 4

2 XPM Supports Color XPM: Defining Grayscales and Colors Each pixel specified by an ASCII char key describes the context this color should be used within. You can always use c for color. Colors can be specified: color name # followed by the RGB code in hexadecimal RGB 24 bits (2 characters 0 - f ) for each color. 5 6 XPM: Specifying Color XPM Example Color Name black white red green blue RGB # # ff ff ff # # ff # 00 ff 00 # ff Color Array of C strings The XPM format assumes the origin (0,0) is in the upper lefthand corner. First string is width height ncolors cpp Then you have "ncolors" strings associating characters with colors. And last you have "height" strings of "width" * "chars_per_pixel" characters 7 8

3 Geometric Preliminaries Affine Geometry Scalars + Points + Vectors and their ops Euclidian Geometry Affine Geometry lacks angles, distance New op: Inner/Dot product, which gives Length, distance, normalization Angle, Orthogonality, Orthogonal projection Projective Geometry Affine Geometry Affine Operations: Affine Combinations: where are points and Example: 9 10 Mathematical Preliminaries Vector: an n-tuple of real numbers Vector Operations Vector addition: u + v = w Commutative, associative, identity element (0) Scalar multiplication: cv Note: Vectors and Points are different Can not add points Can find the vector between two points 11 Linear Combinations & Dot Products A linear combination of the vectors v 1, v 2, v n is any vector of the form α 1 v 1 + α 2 v α n v n where α i is a real number (i.e. a scalar) Dot Product: a real value u 1 v 1 + u 2 v u n v n written as u v 12

4 Fun with Dot Products Euclidian Distance from (x,y) to (0,0) 2 2 in general: 2 2 x + y x 1 + x xn which is just: r x x r This is also the length of vector v: v or v Normalization of a vector: Orthogonal vectors: 2 Projections & Angles Angle between vectors, Projection of vectors Pics/Math courtesy of Dave UMD-CP Matrices and Matrix Operators Matrix Multiplication A n-dimensional vector: Matrix Operations: Addition/Subtraction Identity Multiplication Scalar Matrix Multiplication Implementation issue: Where does the index start? (0 or 1, it s up to you ) 15 Sum over rows & columns Recall: multiplication is not commutative Identity Matrix: 1s on diagonal 0s everywhere else 16

5 Matrix Determinants A single real number Computed recursively Example: a det b c = ad bc d Uses: Find vector ortho to two other vectors Determine the plane of a polygon Matrix Transpose & Inverse Matrix Transpose: Swap rows and cols: Facts about the transpose: Matrix Inverse: Given A, find B such that AB = BA = I (only defined for square matrices) D Affine Transformations 2D Affine Transformations All represented as matrix operations on vectors! Parallel lines preserved, angles/lengths not Scale Rotate Translate Reflect Shear 19 Pics/Math courtesy of Dave UMD-CP Example 1: rotation and non uniform scale on unit cube Example 2: shear first in x, then in y Note: Preserves parallels Does not preserve lengths and angles 20

6 2D Transforms: Translation 2D Transforms: Scale Rigid motion of points to new locations Stretching of points along axes: Defined with column vectors: In matrix form: as 21 or just: 22 2D Transforms: Rotation Rotation of points about the origin 2D Transforms: Rotation Substitute the 1 st two equations into the 2 nd two to get the general equation Positive Angle: CCW Negative Angle: CW Matrix form: or just: 23 24

7 Homogeneous Coordinates Observe: translation is treated differently from scaling and rotation Homogeneous coordinates: allows all transformations to be treated as matrix multiplications Example: A 2D point (x,y) is the line (x,y,w), where w is any real #, in 3D homogenous coordinates. To get the point, homogenize by dividing by w (i.e. w=1) 25 Recall our Affine Transformations 26 Pics/Math courtesy of Dave UMD-CP Matrix Representation of 2D Affine Transformations Translation: Scale: Composition of 2D Transforms Rotate about a point P1 Translate P1 to origin Rotate Translate back to P1 Rotation: Shear: Reflection: F y = 27 28

8 Composition of 2D Transforms Scale object around point P1 P1 to origin Scale Translate back to P1 Composition of 2D Transforms Scale + rotate object around point P1 and move to P2 P1 to origin Scale Rotate Translate to P Programming assignment 1 Write XPM output procedures. You may assume B/W images for now This is the only assignment that deals with raster graphics Create data structure to hold points in memory (so called world model) Implement translation, rotation and scaling (of the world model, not bitmap) Implement World to view-port transformation Programming assignment 1 Understand XPM image format Implement XPM image writer Implement 2D transformations 31 32