3-D Mathematical Preliminaries

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# 3-D Mathematical Preliminaries - PowerPoint PPT Presentation

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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Y

Y

Z

X

X

Z

Coordinate Systems

Left-handed

Right-handed

coordinate

coordinate

system

system

SP = (sxx, syy, szz)Scale

Note: A scale may also translate an object!

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
About the z axis Rz(ß) P =

About the x axis Rx(ß) P =

About the y axis Ry(ß) P =

Rotations
xy Shear

SHxyP =

Shears

Y

Y

X

X

Z

Z

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
a =  (u12 + u32)

b =  (u12 + u22)

c =  (u22 + u32)

cosß = u3/a

sinß = u1/a

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 -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

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

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 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