3D Cartesian Coordinate Systems

3D Cartesian Coordinate Systems

3D Cartesian Coordinate Systems

3D Cartesian Coordinate Systems

Translation

2D Translation

x' = x + t_x
y' = y + t_y

3D Translation

x' = x + t_x
y' = y + t_y
z' = z + t_z

Translation

2D Translation

x' = x + t_x
y' = y + t_y

3D Translation

x' = x + t_x
y' = y + t_y
z' = z + t_z

Scaling

2D Scaling

x' = x ^. s_x
y' = y ^. s_y

3D Scaling

x' = x ^. s_x
y' = y ^. s_y
z' = z ^. s_z

Scaling

2D Scaling

x' = x ^. s_x
y' = y ^. s_y

3D Scaling

x' = x ^. s_x
y' = y ^. s_y
z' = z ^. s_z

Scale with respect to Fixed Point

Scale with respect to Fixed Point

Translate fixed point to origin

Scale with respect to Fixed Point

Scale

Scale with respect to Fixed Point

Translate fixed point back

Positive Rotation about the Coordinate Axis

Z-axis

Positive Rotation about the Coordinate Axis

X-axis

Positive Rotation about the Coordinate Axis

Y-axis

Rotation about Z-Axis

Rotation about Z-Axis

x' = x cos [theta] - y sin [theta]
y' = y sin [theta] + y cos [theta]
z' = z

Rotation about Z-Axis

x' = x cos [theta] - y sin [theta]
y' = y sin [theta] + y cos [theta]
z' = z

Rotation About Z Axis

Rotation About Z Axis

Rotation About X Axis

Rotation About X Axis

Rotation About Y Axis

Rotation About Y Axis

Rotation about an Axis that is parallel to X Axis

Original position of Object

Rotation about an Axis that is parallel to X Axis

Translate rotation axis onto x axis

Rotation about an Axis that is parallel to X Axis

Rotate object through angle u

Rotation about an Axis that is parallel to X Axis

Translate rotation axis to original position

Rotation About Arbitrary Axis

To rotate about axis through P1P2

P

Rotation About Arbitrary Axis

(1) Translate P1 to origin

T

Rotation About Arbitrary Axis

(2) Rotate until P2 lies on z axis

R₁

Rotation About Arbitrary Axis

(3) Perform desired rotation

R₁

Rotation About Arbitrary Axis

Rotate axis back

R₁^-1

Rotation About Arbitrary Axis

Translate axis back

T^-1

Rotation Axis Specification

1. Point and 2 rotation angles

Rotation Axis Specification

2. 2 points

Rotation Axis Specification

2. 2 points

V = P₂ - P₁

U = V / | V | = (a, b, c)

Rotation Axis Specification

2. 2 points

V = P₂ - P₁

U = V / | V | = (a, b, c)

a = (x₂ - x₁)/ | V |
b = (y₂ - y₁)/ | V |
c = (z₂ - z₁)/ | V |

Rotating Rotation Axis onto Coordinate Axis

(1) Rotate about x-axis into xz plane

Rotating Rotation Axis onto Coordinate Axis

(1) Rotate about x-axis into xz plane

same as rotation of yz projection onto z
u' = (0, b, c)
u₂ = (0, 0, d)
cos [alpha] = (u' ^. u₂) / ( | u' |^. | u₂ | ) = cd/dd = c/d

where d = sqrt(b² + c²)

sin [alpha] = b/d

Rotating Rotation Axis onto Coordinate Axis

(1) Rotate about x-axis into xz plane

same as rotation of yz projection onto z
u' = (0, b, c)
u₂ = (0, 0, d)
cos [alpha] = (u' ^. u₂) / ( | u' |^. | u₂ | ) = cd/dd = c/d

where d = sqrt(b² + c²)

sin [alpha] = b/d

u'' = (a, 0, d)

Rotating Rotation Axis onto Coordinate Axis

(2) Rotate about y onto z axis

cos [beta] = d
sin [beta] = -a

Rotating Rotation Axis onto Coordinate Axis

(1) Rotate about x-axis into xz plane

(2) Rotate about y onto z axis

- composite rotation R₁ = R_y([beta]) R_x([alpha])