Transformations computer graphics
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1. The document discusses various 2D and 3D transformations including translation, scaling, rotation, reflection, and shearing. 2. It explains the basic transformation types including rigid-body, affine, and free-form and provides examples of each. 3. Key concepts covered include the homogeneous coordinate system, composition of transformations using matrices, and expressing transformations like translation, scaling, and rotation using matrices.
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Transformations computer graphics
- 2. The most basic ones Translation Scaling Rotation Shear And others, e.g., perspective transform, projection, etc Basic types of transformations Rigid-body: preserves length and angle Affine: preserves parallel lines, not angles or lengths Free-form: anything goes
- 3. BasicTransformations Homogeneous coordinate system Composition of transformations
- 4. (4,5) (7,5) Y XBefore Translation 1 * 100 10 01 1 y x d d y x TPP d d T y x P y x P y x y x FormsHomogeniou x’ = x + dx y’ = y + dy (7,1) (10,1) X Y Translation by (3,-4)
- 5. (4,5) (7,5) Y X (2,5/4) (7/2,5/4) X Y Before Scaling Scaling by (1/2, 1/4) y x y x y x sy sx y x s s PPS ysy xsx * * * 0 0 * * * Types of Scaling: Differential ( sx != sy ) Uniform ( sx = sy ) 1 * 100 00 00 1 FormsHomogeniou y x s s y x y x
- 6. sin cos r r v cossinsincos sinsincoscos rry rrx expand cossin sincos sin cos but yxy yxx ry rx sin cos r r v
- 7. (5,2) (9,2) Y X (2.1,4.9) (4.9,7.8) X Y Before Rotation Rotation of 45 deg. w.r.t. origin 1 * 100 0cossin 0sincos 1 FormsHomogeniou y x y x cos*sin* sin*cos* * cossin sincos * yx yx y x PPR yyx xyx cos*sin* sin*cos*
- 9. 100 01 01 100 01 001 100 010 01 b a SHbSH a SH xyyx unit cube Sheared in X direction Sheared in Y direction Sheared in both X and Y direction
- 11. Translation, scaling and rotation are expressed (non-homogeneously) as: translation: P = P +T Scale: P = S · P Rotate: P = R · P Composition is difficult to express, since translation not expressed as a matrix multiplication Homogeneous coordinates allow all three to be expressed homogeneously, using multiplication by 3 3 matrices W is 1 for affine transformations in graphics
- 12. P2d is a projection of Ph onto the w = 1 plane So an infinite number of points correspond to : they constitute the whole line (tx, ty, tw) x y w Ph(x,y,w) P2d(x,y,1) w=1
- 13. 1. Rigid-body Transformation Preserves parallelism of lines Preserves angle and length e.g. any sequence of R() and T(dx,dy) 2. Affine Transformation Preserves parallelism of lines Doesn’t preserve angle and length e.g. any sequence of R(), S(sx,sy) and T(dx,dy) unit cube 45 deg rotaton Scale in X not in Y
- 14. 100 2221 1211 y x trr trr The following Matrix is Orthogonal if the upper left 2X2 matrix has the following properties 1.A) Each row are unit vector. sqrt(r11* r11 + r12* r12) = 1 B) Each column are unit vector. sqrt(c11* c11 + c12* c12) = 1 2.A) Rows will be perpendicular to each other (r11 , r12 ) . ( r21 , r22) = 0 B) Columns will be perpendicular to each other (c11 , c12 ) . (c21 ,c22) = 0 e.g. Rotation matrix is orthogonal 100 0cossin 0sincos • Orthogonal Transformation Rigid-Body Transformation • For any orthogonal matrix B B-1 = BT
- 15. • In general matrix multiplication is not commutative • For the following special cases commutativity holds i.e. M1.M2 = M2.M1 M1 M2 Translate Translate Scale Scale Rotate Rotate Uniform Scale Rotate • Some non-commutative Compositions: Non-uniform scale, Rotate Translate, Scale Rotate, Translate Original Transitional Final
- 16. Create new affine transformations by multiplying sequences of the above basic transformations. q = CBAp q = ( (CB) A) p = (C (B A))p = C (B (Ap) ) etc. matrix multiplication is associative. But to transform many points, best to do M = CBA then do q = Mp for any point p to be rendered. To transform just a point, better to do q = C(B(Ap)) For geometric pipeline transformation, define M and set it up with the model-view matrix and apply it to any vertex subsequently defined to its setting.
- 17. R,P= Q(x,y) P(h,k) Step 1: Translate P(h,k) to origin T(-h ,-k) Q1(x’,y’) P1 (0,0) Step 2: Rotate w.r.t to origin R* Q2(x’,y’) P2 (0,0) Step 3: Translate (0,0) to P(h,k0) T(h ,k) * P3(h,k) Q3(x’+h, y’ +k)
- 18. T(-h ,-k)S(sx,sy)*T(h ,k) *Ssx,sy,P= (4,3) (1,1) (4,1) S3/2,1/2,(1,1) Step 1: Translate P(h,k) to origin (4,2) (0,0) (4,0) T(-1,-1) Step 2: Scale S(sx,sy) w.r.t origin (6,1) (6,0)(0,0) S(3/2,1/2) (7,1) Step 3: Translate (0,0) to P(h,k) (7,2) (1,1) T(1,1)
- 19. Step 1: Translate (0,b) to origin T(0 ,-b)ML = Step 2: Rotate - degrees Step 3: Mirror reflect about X-axis R(-) *T(0 ,b) * Step 4: Rotate degrees Step 5: Translate origin to (0,b) M x*R() * (0,b) Y X t O Y XO Y XO Y XO Y XO (0,b) Y X t O
- 20. Schaum’s outline series: Problems: 4.1 4.2 4.3, 4.4, 4.5 => R,P 4.6, 4.7, 4.8 => Ssx,sy,P 4.9, 4.10, 4.11, 4.21 => ML 4.12 => Shearing
- 22. Basics of 3D geometry Basic 3DTransformations CompositeTransformations
- 23. Thumb points to +ve Z-axis Fingers show +ve rotation from X toY axis Y X Z (out of page) Y X Z (larger z are away from viewer) Right-handed orentation Left-handed orentation
- 24. Transformation – is a function that takes a point (or vector) and maps that point (or vector) into another point (or vector). A coordinate transformation of the form: x’ = axx x + axy y + axz z + bx , y’ = ayx x + ayy y + ayz z + by , z’ = azx x + azy y + azz z + bz , is called a 3D affine transformation. 11000 ' ' ' z y x baaa baaa baaa w z y x zzzzyzx yyzyyyx xxzxyxx The 4th row for affine transformation is always [0 0 0 1]. Properties of affine transformation: – translation, scaling, shearing, rotation (or any combination of them) are examples affine transformations. – Lines and planes are preserved. – parallelism of lines and planes are also preserved, but not angles and length.
- 26. 1 * * * 1 * 1000 000 000 000 *),,( z y x z y x zyx sz sy sx z y x s s s PPsssS Original scale all axes scaleY axis zsz ysy xsx z y x * * *
- 27. 1 cos*sin* sin*cos* 1 * 1000 0100 00cossin 00sincos *, z yx yx z y x PPR k For 3D-Rotation 2 parameters are needed Angle of rotation Axis of rotation Rotation about z-axis:
- 30. Some of the composite transformations to be studied are: AV,N = aligning a vector V with a vector N R,L = rotation about an axis L( V, P ) Ssx,sy,P= scaling w.r.t. point P
- 31. Av = R,i V = aI + bJ + cK x y z b a c k 22 λ λ cos λ sin byaxis-about xRotate:1Step cb c b b |V| x y z b a k ( 0, b,c) b |V| x y z a k |V| ( a, 0,) ( 0, 0,) ( 0, b,c)
- 32. Av = R,iR-,j * 22 λ λ cos λ sin byaxis-about xRotate:1Step cb c b 222 |V| |V| λ )cos( |V| )sin( -byaxis-yaboutVRotate:2Step cba a P( a, b, c) b x y z b a c k |V| ( a, 0,) ( 0, b,c)b x y z b a c |V| ( 0, 0,|V|) ( 0, b,c) a
- 33. AV -1 = AV T AV,N = AN -1 * AV 1000 0 00 0 λλ λ - λ -λ V c V b V a bc V ac V ab V VA
- 34. Let the axis L be represented by vectorV and passing through point P 1. Translate P to the origin 2. AlignV with vector k 3. Rotate about k 4. Reverse step 2 5. Reverse step 1 R,L = T-PAV *R,k *AV -1 *T-P -1 * V P Q Q' L z x y k
- 35. Let the plane be represented by plane normal N and a point P in that plane x y z
- 36. Let the plane be represented by plane normal N and a point P in that plane 1. Translate P to the origin MN,P = T-P x y z
- 37. Let the plane be represented by plane normal N and a point P in that plane 1. Translate P to the origin 2. Align N with vector k MN,P = T-PAN * x y z
- 38. Let the plane be represented by plane normal N and a point P in that plane 1. Translate P to the origin 2. Align N with vector k 3. Reflect w.r.t xy-plane MN,P = T-PAN *S1,1,-1 * x y z
- 39. x y z Let the plane be represented by plane normal N and a point P in that plane 1. Translate P to the origin 2. Align N with vector k 3. Reflect w.r.t xy-plane 4. Reverse step 2 MN,P = T-PAN *S1,1,-1 *AN -1 *
- 40. Let the plane be represented by plane normal N and a point P in that plane 1. Translate P to the origin 2. Align N with vector k 3. Reflect w.r.t xy-plane 4. Reverse step 2 5. Reverse step 1 MN,P = T-PAN *S1,1,-1 *AN -1 *T-P -1 * x y z
- 41. The CompositeTransform must have – Translation of P1 to Origin T z x y 3P 1P 2PT – Some Combination of Rotations R R x y z 2P 3P 1P z x y 3P 1P 2P Fig. 1 Fig. 2 Translate points in fig. 1 into points in fig 2 such that: – P1 is at Origin – P1P2 is along positive z-axis – P1P3 lies in positive y-axis half of yz plane
- 43. z z z z zyx zyx zyx TT R zR yR xR zRzRzR yRyRyR xRxRxR RRkR 21 21 21 21 21 21 PP PP PP PP PP PP . . . 1 0 0 ... ... ... ˆ 1 R z x y 3P 1P 2P x y z 2P 3P 1P Rz kˆ k PP PP R axis-zalongPPalignsR 21 21 21 ˆ
- 44. R x y z 2P 3P 1P z x y 3P 1P 2P x 2131 2131 x x x 2131 2131 zyx zyx zyx T 2131 2131T R PPPP PPPP zR yR xR PPPP PPPP zRzRzR yRyRyR xRxRxR RR PPPP PPPP iR . . . 0 0 1 ... ... ... ˆ 1 Rx iˆ iR R ˆ 2131 2131 2131 PPPP PPPP axis-xalongPPPPaligns Rz kˆ
- 45. yxz y y y xz zyx zyx zyx T xz T RRR zR yR xR RR zRzRzR yRyRyR xRxRxR RRRRjR . . . 0 1 0 ... ... ... ˆ 1 jR R ˆ xz xz RR axis-yalongRRaligns R x y z 2P 3P 1P z x y 3P 1P 2P Rx iˆ Rz kˆ Ry jˆ
- 46. Schaum’s outline series: Problems: 6.1 6.2, 6.5, 6.9, 6.10, 6.11, 6.12 Av 6.3, 6.4 R,L 6.6, 6.7, 6.8 MN,P
- 47. OpenGL transformation commands Transformation Order Hierarchical Modeling
- 48. OpenGL uses 3 stacks to maintain transformation matrices: Model &View transformation matrix stack Projection matrix stack Texture matrix stack You can load, push and pop the stack The top most matrix from each stack is applied to all graphics primitive until it is changed M N Model-View Matrix Stack Projection Matrix Stack Graphics Primitives (P) Output N•M•P
- 49. Specify current matrix (stack) : void glMatrixMode(GLenum mode) ▪ Mode : GL_MODELVIEW, GL_PROJECTION, GL_TEXTURE Initialize current matrix. void glLoadIdentity(void) ▪ Sets the current matrix to 4X4 identity matirx void glLoadMatrix{f|d}(cost TYPE *M) ▪ Sets the current matrix to 4X4 matrix specified by M Note: current matrix Top most matrix of the current matrix stack A B C A B I A B M glLoadMatrix(M)
- 50. ConcatenateCurrent Matrix: void glMultMatrix(constTYPE *M) ▪ Multiplies current matrix C, by M. i.e. C = C*M Caveat: OpenGL matrices are stored in column major order. Best use utility function glTranslate, glRotate, glScale for common transformation tasks. 161284 151173 141062 13951 mmmm mmmm mmmm mmmm
- 51. Each time an OpenGL transformation M is called the current MODELVIEW matrix C is altered: Cvv CMvv glTranslatef(1.5, 0.0, 0.0); glRotatef(45.0, 0.0, 0.0, 1.0); CTRvv Note: v is any vertex placed in rendering pipeline v’ is the transformed vertex from v.
- 53. As a Global System Objects moves but coordinates stay the same Think of transformation in reverse order as they appear in code As a Local System Objects moves and coordinates move with it Think of transformation in same order as they appear in code There is aWorld Coordinate System where: All objects are defined Transformations are inWorld Coordinate space Two Different Views
- 54. Local View Translate Object Then Rotate glLoadIdentity(); glMultiMatrixf( T); glMultiMatrixf( R); draw_ the_ object( v); v’ = ITRv Global View Rotate Object Then Translate Effect is same, but perception is different
- 55. glLoadIdentity(); glMultiMatrixf( R); glMultiMatrixf( T); draw_ the_ object( v); v’ = ITRv Local View Rotate Object Then Translate Global View Translate Object Then Rotate Effect is same, but perception is different
- 56. Many graphical objects are structured Exploit structure for Efficient rendering Concise specification of model parameters Physical realism Often we need several instances of an object Wheels of a car Arms or legs of a figure Chess pieces Encapsulate basic object in a function Object instances are created in “standard” form Apply transformations to different instances Typical order: scaling, rotation, translation
- 57. A B C C – void glPushMatrix(void); A B – void glPopMatrix(void); A B C C m glGetFloatv – void glGetFloatv(GL_MODELVIEW_MATRIX, *m); A B C Some of the OpenGL functions helpful for hierarchical modeling are:
- 58. A scene graph is a hierarchical representation of a scene We will use trees for representing hierarchical objects such that: Nodes represent parts of an object Topology is maintained using parent-child relationship Edges represent transformations that applies to a part and all the subparts connected to that part typedef struct treenode { GLfloat m[16]; // Transformation void (*f) ( ); // Draw function struct treenode *sibling; struct treenode *child; } treenode; Scene Sun Star X Earth Venus Saturn Moon Ring
- 59. Initializing transformation matrix for node treenode torso, head, ...; /* in init function */ glLoadIdentity(); glRotatef(...); glGetFloatv(GL_MODELVIEW_MATRIX, torso.m); Initializing pointers torso.f = drawTorso; torso.sibling = NULL; torso.child = &head;
- 60. To render the hierarchy: Traverse the scene graph depth-first Going down an edge: ▪ push the top matrix onto the stack ▪ apply the edge's transformation(s) At each node, render with the top matrix Going up an edge: ▪ pop the top matrix off the stack
- 61. Recursive definition void traverse (treenode *root) { if (root == NULL) return; glPushMatrix(); glMultMatrixf(root->m); root->f(); if (root->child != NULL) traverse(root->child); glPopMatrix(); if (root->sibling != NULL) traverse(root->sibling); } C is really not the right language for this !!