3 d mathematical preliminaries n.
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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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coordinate systems

Y

Y

Z

X

X

Z

Coordinate Systems

Left-handed

Right-handed

coordinate

coordinate

system

system

scale
SP = (sxx, syy, szz)Scale

Note: A scale may also translate an object!

rotations
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
rotations1
About the z axis Rz(ß) P =

About the x axis Rx(ß) P =

About the y axis Ry(ß) P =

Rotations
shears
xy Shear

SHxyP =

Shears

Y

Y

X

X

Z

Z

mapping a 4d point into r 3
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
rotation about an arbitrary axis
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

2 rotate the coordinate axes about the y axis an angle
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

3 rotate the coordinate axes about the x axis through an angle to align the z axis with u
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

transformation types
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
affine transformation properties
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