1. 1 Chapter 4 Geometric Objects and Transformation

2. 2 Objectives Introduce the elements of geometry Scalars Vectors Points Develop mathematical operations among them in a coordinate-free manner Define basic primitives Line segments Polygons

3. 3 Basic Elements Geometry is the study of the relationships among objects in an n-dimensional space In computer graphics, we are interested in objects that exist in three dimensions Want a minimum set of primitives from which we can build more sophisticated objects We will need three basic elements Scalars Vectors Points

4. 4 Scalars Scalars can be defined as members of sets which can be combined by two operations (addition and multiplication) obeying some fundamental axioms (associativity, commutivity, inverses) Examples include the real and complex number under the ordinary rules with which we are familiar Scalars alone have no geometric properties

5. 5 Vectors Physical definition: a vector is a quantity with two attributes Direction Magnitude Directed line segments Examples include Force Velocity Directed Line segment

6. 6 Vector Operations Every vector has an inverse Same magnitude but points in opposite direction Every vector can be multiplied by a scalar There is a zero vector Zero magnitude, undefined orientation The sum of any two vectors is a vector Use head-to-tail axiom v -v vv v=u+wv=u+w u w

7. 7 Vectors Lack Position These vectors are identical Same length and magnitude Vectors spaces insufficient for geometry Need points

8. 8 Points Location in space Operations allowed between points and vectors Point-point subtraction yields a vector Equivalent to point-vector addition P=v+Q v=P-Q

9. 9 Coordinate-Free Geometry Objects and coordinate system Objects without coordinate system

10. 10 Linear Vector and Euclidean Spaces Mathematical system for manipulating vectors Operations Scalar-vector multiplication u=  v Vector-vector addition: w=u+v 1 P = P 0 P = 0 (zero vector) Expressions such as v=u+2w-3r Euclidean space is an extension of vector space that adds a measure of size of distance

11. 11 Affine Spaces Point + a vector space Operations Vector-vector addition Scalar-vector multiplication Point-vector addition Scalar-scalar operations

12. 12 The Computer-Science View Abstract data types(ADTs) vector u, v; point p, q; scalar a, b; In C++, by using classes and overloading operator, we could write: q = p + a*v;

13. 13 Geometric ADTs Textbook notations: , ,  denote scalars P, Q, R define points u, v, w denote vectors |  v| = |  ||v|, v = P – Q P = v + Q

14. 14 Lines Consider all points of the form P(  )=P 0 +  d Set of all points that pass through P 0 in the direction of the vector d

15. 15 Parametric Form This form is known as the parametric form of the line More robust and general than other forms Extends to curves and surfaces Two-dimensional forms Explicit: y = mx +h Implicit: ax + by +c =0 Parametric: x(  ) =  x 0 + (1-  )x 1 y(  ) =  y 0 + (1-  )y 1

16. 16 Rays and Line Segments If  >= 0, then P(  ) is the ray leaving P 0 in the direction d If we use two points to define v, then P(  ) = Q +  (R-Q)=Q+  v =  R + (1-  )Q For 0<=  <=1 we get all the points on the line segment joining R and Q

17. 17 Space Partitioning u v A B CD E p=u+vp=u+v E:  >=0,  >=0,  +  =1 F:  >=0,  >=0,  +  >1 A:  >=0,  >=0,  +  <=1 B:  =0 C:  <0,  <0 D:  >=0,  <0 F

18. 18 Convexity An object is convex iff for any two points in the object all points on the line segment between these points are also in the object P Q Q P

19. 19 Affine Sums Consider the “ sum ” P=  1 P 1 +  2 P 2 +…..+  n P n Can show by induction that this sum makes sense iff  1 +  2 + …..  n =1 in which case we have the affine sum of the points P 1  P 2,…..P n If, in addition,  i >=0, we have the convex hull of P 1  P 2,…..P n

20. 20 Convex Hull Smallest convex object containing P 1  P 2,…..P n Formed by “ shrink wrapping ” points

21. 21 Dot and Cross: Products

22. 22 Linear Independence A set of vectors v 1, v 2, …, v n is linearly independent if  1 v 1 +  2 v 2 +..  n v n =0 iff  1 =  2 = … =0 If a set of vectors is linearly independent, we cannot represent one in terms of the others If a set of vectors is linearly dependent, at least one can be written in terms of the others

23. 23 Dimension In a vector space, the maximum number of linearly independent vectors is fixed and is called the dimension of the space In an n-dimensional space, any set of n linearly independent vectors form a basis for the space Given a basis v 1, v 2,…., v n, any vector v can be written as v=  1 v 1 +  2 v 2 +….+  n v n where the {  i } are unique

24. 24 Planes and Normals Every plane has a vector n normal (perpendicular, orthogonal) to it From point-two vector form P( ,  )=R+  u+  v, we know we can use the cross product to find n = u  v and the equivalent form (P(   )-P)  n=0 Assume P=(x 0, y 0, z 0 ) and n=(n x, n y, n z ), then the plane equation= n x x+n y y+n z z=nx 0 +ny 0 +nz 0

25. 25 Three-Dimensional Primitives Hollow objects Objects can be specified by vertices Simple and flat polygons (triangles) Constructive Solid Geometry (CSG) 3D curves 3D surfacesVolumetric Objects

26. 26 Constructive Solid Geometry

27. 27 Representation Until now we have been able to work with geometric entities without using any frame of reference, such a coordinate system Need a frame of reference to relate points and objects to our physical world. For example, where is a point? Can ’ t answer without a reference system World coordinates Camera coordinates

28. 28 Coordinate Systems Consider a basis v 1, v 2,…., v n A vector is written v=  1 v 1 +  2 v 2 +….+  n v n The list of scalars {  1,  2, ….  n }is the representation of v with respect to the given basis We can write the representation as a row or column array of scalars a=[  1  2 ….  n ] T =

29. 29 Example v=2v 1 +3v 2 -4v 3 a=[2 3 – 4] Note that this representation is with respect to a particular basis For example, in OpenGL we start by representing vectors using the world basis but later the system needs a representation in terms of the camera or eye basis

30. 30 Problem in Coordinate Systems Which is correct? Both are because vectors have no fixed location v v

31. 31 Frames – 1/2 Coordinate System is insufficient to present points If we work in an affine space we can add a single point, the origin, to the basis vectors to form a frame

32. 32 Frames – 2/2 Frame determined by (P 0, v 1, v 2, v 3 ) Within this frame, every vector can be written as v=  1 v 1 +  2 v 2 +….+  n v n Every point can be written as P = P 0 +  1 v 1 +  2 v 2 +….+  n v n

33. 33 Representations and N-tuples

34. 34 Change of Coordinate Systems Consider two representations of a the same vector with respect to two different bases. The representations are v=  1 v 1 +  2 v 2 +  3 v 3 = [  1  2  3 ] [v 1 v 2 v 3 ] T =  1 u 1 +  2 u 2 +  3 u 3 = [  1  2  3 ] [u 1 u 2 u 3 ] T a=[  1  2  3 ] b=[  1  2  3 ] where

35. 35 Representing Second Basis in terms of the First Each of the basis vectors, u1,u2, u3, are vectors that can be represented in terms of the first basis u 1 =  11 v 1 +  12 v 2 +  13 v 3 u 2 =  21 v 1 +  22 v 2 +  23 v 3 u 3 =  31 v 1 +  32 v 2 +  33 v 3 v

36. 36 Matrix Form The coefficients define a 3 x 3 matrix and the basis can be related by see text for numerical examples M = a=M T b  b=(M T ) -1 a

37. 37 Confusing Points and Vectors Consider the point and the vector P = P 0 +  1 v 1 +  2 v 2 +….+  n v n v=  1 v 1 +  2 v 2 +….+  n v n They appear to have the similar representations P=[  1  2  3 ] v=[  1  2  3 ] which confuse the point with the vector A vector has no position v p v can place anywhere fixed

38. 38 A Single Representation If we define 0P = 0 and 1P =P then we can write v=  1 v 1 +  2 v 2 +  3 v 3 = [  1  2  3 0 ] [v 1 v 2 v 3 P 0 ] T P = P 0 +  1 v 1 +  2 v 2 +  3 v 3 = [  1  2  3 1 ] [v 1 v 2 v 3 P 0 ] T Thus we obtain the four-dimensional homogeneous coordinate representation v = [  1  2  3 0 ] T P = [  1  2  3 1 ] T

39. 39 Homogeneous Coordinates The general form of four dimensional homogeneous coordinates is p=[x y x w] T We return to a three dimensional point (for w  0) by x  x/w y  y/w z  z/w If w=0, the representation is that of a vector Note that homogeneous coordinates replaces points in three dimensions by lines through the origin in four dimensions

40. 40 Homogeneous Coordinates and Computer Graphics Homogeneous coordinates are key to all computer graphics systems All standard transformations (rotation, translation, scaling) can be implemented by matrix multiplications with 4 x 4 matrices Hardware pipeline works with 4 dimensional representations For orthographic viewing, we can maintain w=0 for vectors and w=1 for points For perspective we need a perspective division

41. 41 Change of Frames We can apply a similar process in homogeneous coordinates to the representations of both points and vectors Consider two frames Any point or vector can be represented in each P0P0 v1v1 v2v2 v3v3 Q0Q0 u1u1 u2u2 u3u3

42. 42 Representing One Frame in Terms of the Other Extending what we did with change of bases u 1 =  11 v 1 +  12 v 2 +  13 v 3 u 2 =  21 v 1 +  22 v 2 +  23 v 3 u 3 =  31 v 1 +  32 v 2 +  33 v 3 Q 0 =  41 v 1 +  42 v 2 +  43 v 3 +  44 P 0 using a 4 x 4 matrix: M =

43. 43 Working with Representations Within the two frames any point or vector has a representation of the same form a=[  1  2  3  4 ] in the first frame b=[  1  2  3  4 ] in the second frame where  4   4  for points and  4   4  for vectors and The matrix M is 4 x 4 and specifies an affine transformation in homogeneous coordinates a=M T b

44. 44 Modeling of a Colored Cube Modeling Converting to the camera frame Clipping Projecting Removing hidden surfaces Rasterizing Demo

45. 45 Representing a Mesh Consider a mesh There are 8 nodes and 12 edges 5 interior polygons 6 interior (shared) edges Each vertex has a location v i = (x i y i z i ) v1v1 v2v2 v7v7 v6v6 v8v8 v5v5 v4v4 v3v3 e1e1 e8e8 e3e3 e2e2 e 11 e6e6 e7e7 e 10 e5e5 e4e4 e9e9 e 12

46. 46 Simple Representation List all polygons by their geometric locations Inefficient and unstructured Consider moving a vertex to a new locations

47. 47 Inward and Outward Facing Polygons The order {v 0, v 3, v 2, v 1 } and {v 1, v 0, v 3, v 2 } are equivalent in that the same polygon will be rendered by OpenGL but the order {{v 0, v 1, v 2, v 3 } is different The first two describe outwardly facing polygons Use the right-hand rule = counter-clockwise encirclement of outward-pointing normal OpenGL treats inward and outward facing polygons differently

48. 48 Geometry versus Topology Generally it is a good idea to look for data structures that separate the geometry from the topology Geometry: locations of the vertices Topology: organization of the vertices and edges Example: a polygon is an ordered list of vertices with an edge connecting successive pairs of vertices and the last to the first Topology holds even if geometry changes

49. 49 Vertex Lists Put the geometry in an array Use pointers from the vertices into this array Introduce a polygon list Each location appears only once!,z 0

50. 50 The Color Cube void colorcube( ) { polygon(0,3,2,1); polygon(2,3,7,6); polygon(0,4,7,3); polygon(1,2,6,5); polygon(4,5,6,7); polygon(0,1,5,4); } Note that vertices are ordered so that we obtain correct outward facing normals 0 56 2 4 7 1 3

51. 51 Bilinear Interpolation Assuming a linear variation, then we can make use of the same interpolation coefficients in coordinates for the interpolation of other attributes.

52. 52 Scan-line Interpolation A polygon is filled only when it is displayed It is filled scan line by scan line Can be used for other associated attributes with each vertex

53. 53 General Transformations A transformation maps points to other points and/or vectors to other vectors

54. 54 Linear Function (Transformation) Transformation matrix for homogeneous coordinate system:

55. 55 Affine Transformations – 1/2 Line preserving Characteristic of many physically important transformations Rigid body transformations: rotation, translation Scaling, shear Importance in graphics is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints

56. 56 Affine Transformations – 2/2 Every linear transformation (if the corresponding matrix is nonsingular) is equivalent to a change in frames However, an affine transformation has only 12 degrees of freedom because 4 of the elements in the matrix are fixed and are a subset of all possible 4 x 4 linear transformations

57. 57 Translation Move (translate, displace) a point to a new location Displacement determined by a vector d Three degrees of freedom P ’ =P+d P P´P´ d

58. 58 How Many Ways? Although we can move a point to a new location in infinite ways, when we move many points there is usually only one way objecttranslation: every point displaced by same vector

59. 59 Rotation (2D) – 1/2 Consider rotation about the origin by  degrees radius stays the same, angle increases by  x´ =x cos  –y sin  y´ = x sin  + y cos  x = r cos  y = r sin  x´ = r cos (  r  cos  cos  r  sin  sin  y´ = r sin (  r  cos  sin  r  sin  cos 

60. 60 Rotation (2D) – 2/2 Using the matrix form: There is a fixed point Could be extended to 3D Positive direction of rotation is counterclockwise 2D rotation is equivalent to 3D rotation about the z-axis

61. 61 (Non-)Rigid-Body Transformation Translation and rotation are rigid-body transformation Non-rigid-body transformations

62. 62 Scaling Expand or contract along each axis (fixed point of origin) x´=s x x y´=s y x z´=s z x Uniform and non-uniform scaling

63. 63 Reflection corresponds to negative scale factors original s x = -1 s y = 1 s x = -1 s y = -1s x = 1 s y = -1

64. 64 Transformation in Homogeneous Coordinates With a frame, each affine transformation is represented by a 4  4 matrix of the form:

65. 65 Translation Using the homogeneous coordinate representation in some frame p=[ x y z 1] T p´=[x´ y´ z´ 1] T d=[dx dy dz 0] T Hence p´= p + d or x´=x+d x y´=y+d y z´=z+d z note that this expression is in four dimensions and expresses that point = vector + point

66. 66 Translation Matrix We can also express translation using a 4 x 4 matrix T in homogeneous coordinates p´=Tp where This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together

67. 67 Rotation about the Z axis Rotation about z axis in three dimensions leaves all points with the same z Equivalent to rotation in two dimensions in planes of constant z or in homogeneous coordinates p ’ =R z (  )p x´=x cos  –y sin  y´= x sin  + y cos  z´=z

68. 68 Rotation Matrix

69. 69 Rotation about X and Y axes Same argument as for rotation about z axis For rotation about x axis, x is unchanged For rotation about y axis, y is unchanged

70. 70 Scaling Matrix x´=s x x y´=s y x z´=s z x p´=Sp

71. 71 Shear Helpful to add one more basic transformation Equivalent to pulling faces in opposite directions

72. 72 Shear Matrix Consider simple shear along x axis x’ = x + y cot  y’ = y z’ = z

73. 73 Inverses Although we could compute inverse matrices by general formulas, we can use simple geometric observations Translation: T -1 (d x, d y, d z ) = T(-d x, -d y, -d z ) Rotation: R -1 (  ) = R(-  ) Holds for any rotation matrix Note that since cos(-  ) = cos(  ) and sin(-  )=-sin(  ) R -1 (  ) = R T (  ) Scaling: S -1 (s x, s y, s z ) = S(1/s x, 1/s y, 1/s z )

74. 74 Concatenation We can form arbitrary affine transformation matrices by multiplying together rotation, translation, and scaling matrices Because the same transformation is applied to many vertices, the cost of forming a matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p The difficult part is how to form a desired transformation from the specifications in the application

75. 75 Order of Transformations Note that matrix on the right is the first applied Mathematically, the following are equivalent p´ = ABCp = A(B(Cp)) Note many references use column matrices to present points. In terms of column matrices p´ T = p T C T B T A T

76. 76 Rotation about a Fixed Point and about the Z axis  f

77. 77 Sequence of Transformations

78. 78 General Rotation about the Origin A rotation by q about an arbitrary axis can be decomposed into the concatenation of rotations about the x, y, and z axes R(  ) = R x (  ) R y (  ) R z (  )   are called the Euler angles Note that rotations do not commute We can use rotations in another order but with different angles  x z y v

79. 79 Decomposition of General Rotation   

80. 80 The Instance Transformation In modeling, we often start with a simple object centered at the origin, oriented with the axis, and at a standard size We apply an instance transformation to its vertices to Scale Orient Locate Display lists

81. 81 Rotation about an Arbitrary Axis – 1/3 1. Move the fixed point to the origin 2. Rotate through a sequence of rotations 

82. 82 Rotation about an Arbitrary Axis – 2/3 Normalize u:     Final rotation matrix:

83. 83 Rotation about an Arbitrary Axis – 3/3         Rotate the line segment to the plane of y=0, and the line segment is foreshortened to: Rotate clockwise about the y-axis, so: Final transformation matrix:

84. 84 Matrix Stacks In many situations we want to save transformation matrices for use later Traversing hierarchical data structures (Chapter 9) Avoiding state changes when executing display lists OpenGL maintains stacks for each type of matrix Access present type by glPushMatrix(); glPopMatrix();

85. 85 Interfaces to 3D Applications One of the major problems in interactive computer graphics is how to use two-dimensional devices such as a mouse to interface with three dimensional objects Example: how to form an instance matrix? Some alternatives Virtual trackball 3D input devices such as the spaceball Use areas of the screen Distance from center controls angle, position, scale depending on mouse button depressed

86. 86 Using Areas of the Screen Each button controls rotation, scaling and translation, separately Example: Left button: closer to the center, no rotation, moving up or down, rotate about x-axis, moving left or right, rotate about y-axis, moving to the corner, rotate about x-y-axes Right button: translation Middle button: scaling (zoom-in or zoom-out)

87. 87 Physical Trackball The trackball is an “ upside down ” mouse If there is little friction between the ball and the rollers, we can give the ball a push and it will keep rolling yielding continuous changes Two possible modes of operation Continuous pushing or tracking hand motion Spinning

88. 88 A Trackball from a Mouse Problem: we want to get the two behavior modes from a mouse We would also like the mouse to emulate a frictionless (ideal) trackball Solve in two steps Map trackball position to mouse position Use GLUT to obtain the proper modes

89. 89 Trackball Frame origin at center of ball

90. 90 Projection of Trackball Position We can relate position on trackball to position on a normalized mouse pad by projecting orthogonally onto pad

91. 91 Reversing Projection Because both the pad and the upper hemisphere of the ball are two-dimensional surfaces, we can reverse the projection A point (x,z) on the mouse pad corresponds to the point (x,y,z) on the upper hemisphere where y = if r  |x|  0, r  |z|  0

92. 92 Computing Rotatoins Suppose that we have two points that were obtained from the mouse. We can project them up to the hemisphere to points p 1 and p 2 These points determine a great circle on the sphere We can rotate from p 1 to p by finding the proper axis of rotation and the angle between the points

93. 93 Using the Cross Product The axis of rotation is given by the normal to the plane determined by the origin, p 1, and p 2 n = p 1  p 1

94. 94 Obtaining the Angle The angle between p 1 and p 2 is given by If we move the mouse slowly or sample its position frequently, then  will be small and we can use the approximation | sin  | = sin 

95. 95 Rotation Matrix