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CS 4731: Computer Graphics Lecture 12: More 3D Viewing

CS 4731: Computer Graphics Lecture 12: More 3D Viewing. Emmanuel Agu. 3D viewing under the hood. Modeling Viewing Projection Transformation Transformation Transformation

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CS 4731: Computer Graphics Lecture 12: More 3D Viewing

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  1. CS 4731: Computer GraphicsLecture 12: More 3D Viewing Emmanuel Agu

  2. 3D viewing under the hood Modeling Viewing Projection Transformation Transformation Transformation Viewport Transformation Display

  3. 3D viewing under the hood • Topics of Interest: • Viewing transformation • Projection transformation

  4. Viewing Transformation • Transform the object from world to eye space • Construct eye coordinate frame • Construct matrix to perform coordinate transformation • Flexible Camera Control

  5. Viewing Transformation • Recall OpenGL way to set camera: • gluLookAt (Ex, Ey, Ez, cx, cy, cz, Up_x, Up_y, Up_z) • The view up vector is usually (0,1,0) • Remember to set the OpenGL matrix mode to GL_MODELVIEW first • Modelview matrix: • combination of modeling matrix M and Camera transforms V • gluLookAt fills V part of modelview matrix • What does gluLookAt do with parameters (eye, COI, up vector) you provide?

  6. center of interest (COI) eye Eye Coordinate Frame • Known: eye position, center of interest, view-up vector • To find out: new origin and three basis vectors Assumption: direction of view is orthogonal to view plane (plane that objects will be projected onto) o 90

  7. v n is pointing away from the world because we use left hand coordinate system N = eye – COI n = N / | N | u Center of interest (COI) eye n Remember u,v,n should be all unit vectors Eye Coordinate Frame • Origin: eye position (that was easy) • Three basis vectors: • one is the normal vector (n) of the viewing plane, • other two (u and v) span the viewing plane world origin (u,v,n should all be orthogonal)

  8. v V_up u COI eye n Eye Coordinate Frame • How about u and v? • We can get u first - • u is a vector that is perp • to the plane spanned by • N and view up vector (V_up) U = V_up x n u = U / | U |

  9. v V_up u COI eye n Eye Coordinate Frame • How about v? Knowing n and u, getting v is easy v = n x u v is already normalized

  10. v V_up u COI eye n Eye Coordinate Frame Eye space origin: (Eye.x , Eye.y, Eye.z) Basis vectors: n = (eye – COI) / | eye – COI| u = (V_up x n) / | V_up x n | v = n x u • Put it all together

  11. v u y n world x z World to Eye Transformation • Transformation matrix (Mw2e) ? P’ = Mw2e x P 1. Come up with the transformation sequence to move eye coordinate frame to the world 2. And then apply this sequence to the point P in a reverse order P

  12. v u y n world x z World to Eye Transformation • Rotate the eye frame to “align” it with the world frame • Translate (-ex, -ey, -ez) Rotation: ux uy uz 0 vx vy vz 0 nx ny nz 0 0 0 0 1 Translation: 1 0 0 -ex 0 1 0 -ey 0 0 1 -ez 0 0 0 1 (ex,ey,ez)

  13. v u y n (ex,ey,ez) world x z World to Eye Transformation • Transformation order: apply the transformation to the object in a reverse order - translation first, and then rotate Mw2e = ux uy ux 0 1 0 0 -ex vx vy vz 0 0 1 0 -ey nx ny nz 0 0 0 1 -ez 0 0 0 1 0 0 0 1

  14. v u y n world x z World to Eye Transformation • Head tilt: Rotate your head by d • Just rotate the object about the eye space z axis - d • Mw2e= cos(-d) -sin(-d) 0 0 ux uy ux 0 1 0 0 -ex sin(-d) cos(-d) 0 0 vx vy vz 0 0 1 0 -ey 0 0 1 0 nx ny nz 0 0 0 1 -ez 0 0 0 1 0 0 0 1 0 0 0 1 Why -d? When you rotate your head by d, it is like rotate the object by –d

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