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.
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
After Ry(-ß), µ lies in the y-z plane
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
cos µ = a/ ||u||
sinµ = u2 / ||u||
Rigid body transformations
Do not distort shapes – line lengths and angles are preserved
Rotations, Translations, and combinations of bothTransformation Types
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 theseAffine Transformation Properties