3D Transformations

1
3D Transformations
2
3D homogeneous coordinates

• In 2D, we added a dimension so translations could
  be computed by matrix multiplication, just like
  all the other transformations.
• In 3D, well do the same trick, projecting up
  into a 4D space to do transformations, then
  projecting back down after.

3
Translation
(x,y,z)
T(tx,ty,tz)
(x,y,z)
4
Scaling
5
Rotation

• How should we specify rotations?
• In 2D, it was always counterclockwise in the
  xy-plane.
• In 3D, we have more choices.
• xz-plane, yz-plane, an arbitrary plane.
• We could specify these in terms of the vector
  perpendicular to the plane of rotation.
• z axis, y-axis, x-axis, arbitrary axis

6
Rotation about a major axis
y
y
z
x
z
y
x
z
x
7
Rotation about an arbitrary axis

• How do we specify this?
• R(rx, ry, rz, ?)
• So we need a vector and an angle
• The axis of rotation is from the origin and
  through the point r
• The rotation is counterclockwise about the axis

y
z
x
8
What are the steps?

• Rotate the axis of rotation so it lies on some
  major axis.
• Apply specified rotation about major axis.
• Apply inverse rotation to return axis or rotation
  to original orientation.

y
y
x
z
x
z
9
Math!

• We need to figure out the rotations to align the
  axes.
• First, compute a unit vector pointing in the same
  direction as the axis of rotation.
• Now we can compute the rotation directly from the
  unit vector.

10
More math!

• First, rotate u into the xz-plane rotation
  around the x-axis.
• We can temporarily ignore the x component of u to
  do this

y
u lt0,b,cgt
ulta,b,cgt
?
uzlt0,0,1gt
z
x
ulta,0,dgt
11
Picture this
Second, rotate u onto the z-axis rotation
around the y-axis.
?
uzlt0,0,1gt
ulta,0,dgt
12
Putting it together

• Dont forget the rotation about the z-axis!
• P R(ux,?) R(uy,?) R(uz,?) R(uy,?)
  R(ux,?) P

13
Other rotations

• What if the axis of rotation does not pass
  through the origin?
• Similar process as in 2D, translate to the
  origin, rotate as normal, translate back.
• We just need to know a point on the axis that we
  can translate to the origin.
• Only way to specify such a rotation is to give
  two points on the line or one point and a
  direction, so the requirement is easily satisfied.

14
Another way to get the rotation matrix

• We can compute it as a coordinate system
  transformation

y
uz
uy
ux
z
x