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3-D Mathematical Preliminaries

3-D Mathematical Preliminaries. Y. Y. Z. X. X. Z. Coordinate Systems. Left-handed. Right-handed. coordinate. coordinate. system. system. •Translation •Scale •Rotation. Basic Transformations. TP = (x + t x , y + t y , z + t z ). Translation in Homogeneous Coordinates. T. P.

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3-D Mathematical Preliminaries

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  1. 3-D Mathematical Preliminaries

  2. Y Y Z X X Z Coordinate Systems Left-handed Right-handed coordinate coordinate system system

  3. •Translation •Scale •Rotation Basic Transformations

  4. TP = (x + tx, y + ty, z + tz) Translation in Homogeneous Coordinates T P

  5. SP = (sxx, syy, szz) Scale Note: A scale may also translate an object!

  6. Positive Rotations are defined as follows Axis of rotation is Direction of positive rotation is y to z z to x x to y Rotations

  7. About the z axis Rz(ß) P = About the x axis Rx(ß) P = About the y axis Ry(ß) P = Rotations

  8. xy Shear SHxyP = Shears Y Y X X Z Z

  9. If Tr is any transformation, then it is possible that the 4th component of the point will not be 1. TrP = Tr(x,y,z,1) = (x', y', z',w) Pplotted = (x'/w, y'/w, z'/w) Transformations may be appended together via matrix multiplication. Mapping a 4D point into R3

  10. a =  (u12 + u32) b =  (u12 + u22) c =  (u22 + u32) cosß = u3/a sinß = u1/a Rotation About An Arbitrary Axis P2 1. Translate one end of the axis to the origin [P2-P1] = [ u1, u2, u3] Z U P1 c u3 Y a ß u2 b u1 X

  11. 2. Rotate the coordinate axes about the y-axis an angle -ß Z Z a c u3 a Y ß µ Y u2 u2 b u1 X X After Ry(-ß), µ lies in the y-z plane

  12. 4. When u is aligned with the z-axis, apply the original rotation, R, about the z-axis. 5. Apply the inverses of the transformations in reverse order. 3. Rotate the coordinate axes about the x-axis through an angle µ to align the z-axis with U U Z Rx(µ) cos µ = a/ ||u|| sinµ = u2 / ||u|| µ Y X

  13. T-1 Ry(ß)Rx(-µ)RRx(µ)Ry(-ß)T P Rotation About an Arbitrary Axis

  14. A transformation maps points in one coordinate system to points in another coordinate system. Rigid body transformations Do not distort shapes – line lengths and angles are preserved Rotations, Translations, and combinations of both Transformation Types

  15. Affine transformations Keep parallel lines and planes parallel. Parallelograms map into oter parallelograms. but do not necessarily preserve line lengths or angles Preserve collinearity and “flatness” so the image of a plane or line is another plane or line. Rotations, Translations, Scales, Shears, and combinations of these Affine Transformation Properties

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