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Transformations

Transformations. CS 445/645 Introduction to Computer Graphics David Luebke, Spring 2003. Admin. Call roll Assn 1 News flash: Assn 1 not necessarily a gimme Turn-in instructions by email soon Problems with lab?. Recap: OpenGL: Modeling Transforms.

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Transformations

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  1. Transformations CS 445/645Introduction to Computer Graphics David Luebke, Spring 2003

  2. Admin • Call roll • Assn 1 • News flash: Assn 1 not necessarily a gimme • Turn-in instructions by email soon • Problems with lab? David Luebke 210/15/2014

  3. Recap: OpenGL: Modeling Transforms • OpenGL provides several commands for performing modeling transforms: • glTranslate{fd}(x, y, z) • Creates a matrix T that transforms an object by translating (moving) it by the specified x, y, and z values • glRotate{fd}(angle, x, y, z) • Creates a matrix R that transforms an object by rotating it counterclockwise angle degrees about the vector {x, y, z} • glScale{fd}(x, y, z) • Creates a matrix S that scales an object by the specified factors in the x, y, and z directions David Luebke 310/15/2014

  4. Recap:OpenGL: Matrix Manipulation • Each of these postmultiplies the current matrix • E.g., if current matrix is C, then C=CS • The current matrix is either the modelview matrix or the projection matrix (also a texture matrix, won’t discuss) • Set these with glMatrixMode(), e.g.: glMatrixMode(GL_MODELVIEW); glMatrixMode(GL_PROJECTION); • WARNING: common mistake ahead! • Be sure that you are in GL_MODELVIEW mode before making modeling or viewing calls! • Ugly mistake because it can appear to work, at least for a while… David Luebke 410/15/2014

  5. Recap:OpenGL: Matrix Manipulation • More matrix manipulation calls • To replace the current matrix with an identity matrix: glLoadIdentity() • Postmultiply the current matrix with an arbitrary matrix: glMultMatrix{fd}(float/double m[16]) • Copy the current matrix and push it onto a stack: glPushMatrix() • Discard the current matrix and replace it with whatever’s on top of the stack: glPopMatrix() • Note that there are matrix stacks for both modelview and projection modes David Luebke 510/15/2014

  6. OpenGL: Specifying Color • Can specify other properties such as color • To produce a single aqua-colored triangle: glColor3f(0.1, 0.5, 1.0); glVertex3fv(v0); glVertex3fv(v1); glVertex3fv(v2); • To produce a Gouraud-shaded triangle: glColor3f(1, 0, 0); glVertex3fv(v0); glColor3f(0, 1, 0); glVertex3fv(v1); glColor3f(0, 0, 1); glVertex3fv(v2); • In OpenGL, colors can also have a fourth component  (opacity) • Generally want  = 1.0 (opaque); David Luebke 610/15/2014

  7. OpenGL: Specifying Normals • Calling glColor() sets the color for all vertices following, until the next call to glColor() • Calling glNormal() sets the normal vector for the following vertices, till next glNormal() • So flat-shaded lighting requires: glNormal3f(Nx, Ny, Nz); glVertex3fv(v0);glVertex3fv(v1);glVertex3fv(v2); • While smooth shading requires: glNormal3f(N0x, N0y, N0z); glVertex3fv(v0); glNormal3f(N1x, N1y, N1z); glVertex3fv(v1); glNormal3f(N2x, N2y, N2z); glVertex3fv(v2); • (Of course, lighting requires additional setup…) David Luebke 710/15/2014

  8. More OpenGL • Other things you’ll need to know: • To clear the screen: glClearColor(r, g, b, a); glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT); • Nate Robins has an excellent set of OpenGL tutorials that help illustrate many of these concepts & functions: http://www.cs.utah.edu/~narobins/opengl.html Also http://xmission.com/~nate/tutors.html • Next: the math and concepts underlying the transformation calls David Luebke 810/15/2014

  9. Translations • For convenience we usually describe objects in relation to their own coordinate system • We can translate or move points to a new position by adding offsets to their coordinates: • Note that this translates all points uniformly David Luebke 910/15/2014

  10. Scaling • Scaling a coordinate means multiplying each of its components by a scalar • Uniform scaling means this scalar is the same for all components:  2 David Luebke 1010/15/2014

  11. Scaling • Non-uniform scaling: different scalars per component: • How can we represent this in matrix form? X  2,Y  0.5 David Luebke 1110/15/2014

  12. Scaling • Scaling operation: • Or, in matrix form: scaling matrix David Luebke 1210/15/2014

  13. (x’, y’) (x, y)  2-D Rotation x’ = x cos() - y sin() y’ = x sin() + y cos() (Draw it) David Luebke 1310/15/2014

  14. 2-D Rotation • This is easy to capture in matrix form: • 3-D is more complicated • Need to specify an axis of rotation • Simple cases: rotation about X, Y, Z axes David Luebke 1410/15/2014

  15. 3-D Rotation • What does the 3-D rotation matrix look like for a rotation about the Z-axis? • Build it coordinate-by-coordinate David Luebke 1510/15/2014

  16. 3-D Rotation • What does the 3-D rotation matrix look like for a rotation about the Y-axis? • Build it coordinate-by-coordinate David Luebke 1610/15/2014

  17. 3-D Rotation • What does the 3-D rotation matrix look like for a rotation about the X-axis? • Build it coordinate-by-coordinate David Luebke 1710/15/2014

  18. 3-D Rotation • General rotations in 3-D require rotating about an arbitrary axis of rotation • Deriving the rotation matrix for such a rotation directly is a good exercise in linear algebra • Standard approach: express general rotation as composition of canonical rotations • Rotations about X, Y, Z David Luebke 1810/15/2014

  19. Composing Canonical Rotations • Goal: rotate about arbitrary vector A by  • Idea: we know how to rotate about X,Y,Z So, rotate about Y by  until A lies in the YZ plane Then rotate about X by  until A coincides with +Z Then rotate about Z by  Then reverse the rotation about X (by -) Then reverse the rotation about Y (by -) David Luebke 1910/15/2014

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