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Comprehensive Guide to 3D Transformations in OpenGL

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This guide provides a thorough overview of 3D transformations, including translation, rotation, scaling, and reflection in OpenGL. It covers the mathematical foundations and matrix operations necessary for manipulating 3D objects. You will learn how to apply transformations with various axes of rotation and the importance of using translation to align rotation axes. The document also features practical OpenGL commands for implementing transformations and managing the model-view matrix stack effectively.

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Comprehensive Guide to 3D Transformations in OpenGL

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  1. 3D Transformations

  2. y P’ P x z Translation x’ = x + tx y’ = y + ty z’ = z + tz P = P’ = T = P’ = T . P x y z 1 x’ y’ z’ 1 1 0 0 tx 0 1 0 ty 0 0 1 tz 0 0 0 1

  3. y x z Rotation z-axis rotation x’ = x.cosq – y.sinq y’ = x.sinq + y.cosq z’ = z R = P’ = R . P cos q -sin q 0 0 sin q cos q 0 0 0 0 1 0 0 0 0 1

  4. y x z y x z Rotation x-axis rotation replace x -> y -> z -> x in z-axis rotation y’ = y.cosq – z.sinq z’ = y.sinq + z.cosq x’ = x R = y-axis rotation replace x -> y -> z -> x in x-axis rotation z’ = z.cosq – x.sinq x’ = z.sinq + x.cosq y’ = y R = 1 0 0 0 0 cos q -sin q 0 0 sin q cos q 0 0 0 0 1 cos q 0 -sin q 0 0 1 0 0 sin q 0 cos q 0 0 0 0 1

  5. Rotation Rotation about an arbitrary axis parallel to a coordinate axis P’ = T-1 . Rx(q) . T. P y x z

  6. Rotation Rotation about an arbitrary axis NOT parallel to a coordinate axis y x z

  7. Rotation Rotation about an arbitrary axis NOT parallel to a coordinate axis 1. Translate so that rotation axis passes through the origin y x z

  8. Rotation Rotation about an arbitrary axis NOT parallel to a coordinate axis 1. Translate so that rotation axis passes through the origin 2. Rotate so that rotation axis coincides with a coordinate axis y x z

  9. Rotation Rotation about an arbitrary axis NOT parallel to a coordinate axis 1. Translate so that rotation axis passes through the origin 2. Rotate so that rotation axis coincides with a coordinate axis 3. Rotate y x z

  10. Rotation Rotation about an arbitrary axis NOT parallel to a coordinate axis 1. Translate so that rotation axis passes through the origin 2. Rotate so that rotation axis coincides with a coordinate axis 3. Rotate 4. Inverse rotation y x z

  11. Rotation Rotation about an arbitrary axis NOT parallel to a coordinate axis 1. Translate so that rotation axis passes through the origin 2. Rotate so that rotation axis coincides with a coordinate axis 3. Rotate 4. Inverse rotation 5. Inverse translation y x z

  12. Scaling S = P’ = S . P wrt a fixed point (xf, yf, zf) : T(xf, yf, zf) . S(sx, sy, sz) . T(-xf, -yf, -zf) sx 0 0 0 0 sy 0 0 0 0 sz 0 0 0 0 1 y x z

  13. x z Reflection 1800 rotation about x-axis a combination of translation and rotation y x z y

  14. y x z Shear 1 0 shx –shx.zref 0 1 shy –shy.zref 0 0 1 0 0 0 0 1 y x z

  15. OpenGL • glTranslate*(tx, ty, tz) • f (float) • d (double) • glRotate* (theta, vx, vy, vz) • (vx, vy, vz) vector defines the orientation of the rotation axis that passes through the coordinate origin • glScale*(sx, sy, sz)

  16. OpenGL • glMatrixMode(GL_MODELVIEW) • sets up the matrix for transformations (4x4 modelview matrix) • glLoadIdentity ( ) • assigns identity matrix to the current matrix • glLoadMatrix*(16-element array) • assigns a 16-element array (in column major order) to the current matrix • glMultMatrix*(16-element array) • postmultiplies a 16-element array (M’) with the current matrix (M) : M <- M.M’

  17. OpenGL glMatrixMode(GL_MODELVIEW); glLoadIdentity ( ); glMultMatrixf(M2); glMultMatrixf(M1); /* M = M2 . M1 */

  18. OpenGL Matrix Stack • Initially stack contains identity matrix • Maximum stack depth is 32 • glGetIntegerv (GL_MAX_MODELVIEW_STACK_DEPTH, stacksize) • returns the number of positions available in the modelview stack • glGetIntegerv (GL_MODELVIEW_STACK_DEPTH, nummats) • returns the number of matrices currently in the stack • glPushMatrix() • copies the current matrix at the top of the stack • glPopMatrix() • destroys the matrix at the top of the stack

  19. OpenGL glMatrixMode(GL_MODELVIEW); glColor3f(0.0, 0.0, 1.0); Recti(50, 100, 200, 150); glColor3f(1.0, 0.0, 0.0); glTranslatef(-200.0, -50.0, 0.0); Recti(50, 100, 200, 150); glLoadIdentity ( ); glRotatef(90.0, 0.0, 0.0, 1.0); Recti(50, 100, 200, 150); glLoadIdentity ( ); glScalef(-0.5, 1.0, 1.0); Recti(50, 100, 200, 150);

  20. OpenGL glMatrixMode(GL_MODELVIEW); glColor3f(0.0, 0.0, 1.0); Recti(50, 100, 200, 150); glPushMatrix(); glColor3f(1.0, 0.0, 0.0); glTranslatef(-200.0, -50.0, 0.0); Recti(50, 100, 200, 150); glPopMatrix(); glPushMatrix(); glRotatef(90.0, 0.0, 0.0, 1.0); Recti(50, 100, 200, 150); glPopMatrix(); glScalef(-0.5, 1.0, 1.0); Recti(50, 100, 200, 150);

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