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Geometric Transformations

Geometric Transformations. Transformations. Local moving frame. Linear transformations Rigid transformations Affine transformations Projective transformations. T. Global reference frame. Linear Transformations. A linear transformation T is a mapping between vector spaces

hermione-mendez
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Geometric Transformations

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  1. Geometric Transformations

  2. Transformations Local moving frame • Linear transformations • Rigid transformations • Affine transformations • Projective transformations T Global reference frame

  3. Linear Transformations • A linear transformationT is a mapping between vector spaces • T maps vectors to vectors • linear combination is invariant under T • In 3D-spaces, T can be represented by a 3x3 matrix

  4. Affine Transformations • An affine transformationT is an mapping between affine spaces • T maps vectors to vectors, and points to points • T is a linear transformation on vectors • affine combination is invariant under T • In 3D-spaces, T can be represented by a 3x3 matrix together with a 3x1 translation vector

  5. Homogeneous Coordinates • Any affine transformation between 3D spaces can be represented by a 4x4 matrix • Affine transformation is linear in homogeneous coordinates

  6. Projective Spaces • Homogeneous coordinates • (x, y, z, w) = (x/w, y/w, z/w, 1) • Useful for handling perspective projection • But, it is algebraically inconsistent !!

  7. Examples of Affine Transformations • 2D rotation • 2D scaling

  8. Examples of Affine Transformations • 2D shear • 2D reflection

  9. Examples of Affine Transformations • 2D translation

  10. Examples of Affine Transformations • 2D transformation for vectors • Translation is simply ignored

  11. Examples of Affine Transformations • 3D rotation

  12. Pivot-Point Rotation • Rotation with respect to a pivot point (x,y)

  13. Fixed-Point Scaling • Scaling with respect to a fixed point (x,y)

  14. Scaling Direction • Scaling along an arbitrary axis

  15. Properties of Affine Transformations • Any affine transformation between 3D spaces can be represented as a combination of a linear transformation followed by translation • An affine transf. maps lines to lines • An affine transf. maps parallel lines to parallel lines • An affine transf. preserves ratios of distance along a line • An affine transf. does not preserve absolute distances and angles

  16. Review of Affine Frames • A frame is defined as a set of vectors {vi | i=1, …, N} and a point o • Set of vectors {vi} are bases of the associate vector space • o is an origin of the frame • N is the dimension of the affine space • Any point p can be written as • Any vector v can be written as

  17. Changing Frames • Affine transformations as a change of frame

  18. Changing Frames • Affine transformations as a change of frame

  19. Changing Frames • In case the xyz system has standard bases

  20. Rigid Transformations • A rigid transformationT is a mapping between affine spaces • T maps vectors to vectors, and points to points • T preserves distances between all points • T preserves cross product for all vectors (to avoid reflection) • In 3-spaces, T can be represented as

  21. Rigid Body Rotation • Rigid body transformations allow only rotation and translation • Rotation matrices form SO(3) • Special orthogonal group (Distance preserving) (No reflection)

  22. Rigid Body Rotation • R is normalized • The squares of the elements in any row or column sum to 1 • R is orthogonal • The dot product of any pair of rows or any pair columns is 0 • The rows (columns) of R correspond to the vectors of the principle axes of the rotated coordinate frame

  23. 3D Rotation About Arbitrary Axis

  24. 3D Rotation About Arbitrary Axis • Translation : rotation axis passes through the origin 2. Make the rotation axis on the z-axis 3. Do rotation 4. Rotation & translation

  25. 3D Rotation About Arbitrary Axis • Rotate u onto the z-axis

  26. 3D Rotation About Arbitrary Axis • Rotate u onto the z-axis • u’: Project u onto the yz-plane to compute angle a • u’’: Rotate u about the x-axis by angle a • Rotate u’’ onto the z-asis

  27. 3D Rotation About Arbitrary Axis • Rotate u’ about the x-axis onto the z-axis • Let u=(a,b,c) and thus u’=(0,b,c) • Let uz=(0,0,1)

  28. 3D Rotation About Arbitrary Axis • Rotate u’ about the x-axis onto the z-axis • Since we know both cos a and sin a, the rotation matrix can be obtained • Or, we can compute the signed angle a • Do not use acos() since its domain is limited to [-1,1]

  29. Euler angles • Arbitrary rotation can be represented by three rotation along x,y,z axis

  30. Gimble • Hardware implementation of Euler angles • Aircraft, Camera

  31. Euler Angles • Rotation about three orthogonal axes • 12 combinations • XYZ, XYX, XZY, XZX • YZX, YZY, YXZ, YXY • ZXY, ZXZ, ZYX, ZYZ • Gimble lock • Coincidence of inner most and outmost gimbles’ rotation axes • Loss of degree of freedom

  32. Euler Angles • Euler angles are ambiguous • Two different Euler angles can represent the same orientation • This ambiguity brings unexpected results of animation where frames are generated by interpolation.

  33. Taxonomy of Transformations • Linear transformations • 3x3 matrix • Rotation + scaling + shear • Rigid transformations • SO(3) for rotation • 3D vector for translation • Affine transformation • 3x3 matrix + 3D vector or 4x4 homogenous matrix • Linear transformation + translation • Projective transformation • 4x4 matrix • Affine transformation + perspective projection

  34. Rigid Taxonomy of Transformations Affine Projective

  35. Composite Transformations • Composite 2D Translation

  36. Composite Transformations • Composite 2D Scaling

  37. Composite Transformations • Composite 2D Rotation

  38. OpenGL Geometric Transformations • glMatrixMode(GL_MODELVIEW);

  39. OpenGL Geometric Transformations • Basic Transpormation: • glLoadIdentity(); • glTranslatef(tx, ty, tz); • glRotatef(theta, vx, vy, vz); angle-axis • (vx, vy, vz)is automatically normalized • glScalef(sx, sy, sz); • glLoadMatrixf(Glfloat elems[16]); • Multiplication • glMultMatrixf(Glfloat elems[16]); • The current matrix is postmultiplied by the matrix • Column major

  40. OpenGL Geometric Transformations • Getting the current matrix value: • glGetFloatv (GL_MODELVIEW_MATRIX, GLfloat elems[16]); • Column major GLfloat mat [16]; glGetFloatv (GL_MODELVIEW_MATRIX, mat);

  41. OpenGL Geometric Transformations • Matrix Direct Manipulation: • glLoadMatrixf(GLfloat elems[16]); • Column major • glMultMatrixf(GLfloat elems[16]); • The current matrix is postmultiplied by the matrix glLoadIdentity(); glMultMatrixf (M1); glMultMatrixf (M2); M = M1∙M2

  42. OpenGL GLUT Animation Function • GLUT Idle Callback fuction: • Idling: when there is nothing to do. • Redraw the scene: void MyIdle() { … // things to do … // when Idling } glutIdelFunc ( MyIdle ); glutPostRedisplay ( );

  43. Hierarchical Modeling • A hierarchical model is created by nesting the descriptions of subparts into one another to form a tree organization

  44. OpenGL Matrix Stacks • Four matrix modes • Modelview, projection, texture, and color • glGetIntegerv(GL_MAX_MODELVIEW_STACK_DEPTH, stackSize); • Stack processing • The top of the stack is the “current” matrix • glPushMatrix(); // Duplicate the current matrix at the top • glPopMatrix(); // Remove the matrix at the top

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