CS488 - Introduction to Computer Graphics - Lecture 7

Affine Transformations in Practice

Rigid Transformations

Rotation

To specify a rotation you need

an axis of rotation, a
an angle of rotation, t
Ra, t( P )

Three basic rotations encompass all rotations geometrical demonstration

about the x-axis
about the y-axis
about the z-axis

You know the basic matrix for 2D rotation

/                \
|  cos(t)  sin(t) |
|                 |
| -sin(t)  cos(t) |
\                /

Generalize it into three dimensions

But only two are independent

E.g., Rj,90( P ) = Ri,-90( P ) Rk,90( P ) Ri,90( P )
Why does this work?
1. Ri,90( P ) rotates the y-axis into the z-axis
2. Rk,90( P ) rotates about the z-axis (formerly the y-axis)
3. Ri,-90( P ) rotates the z-axis (formerly the y-axis) back to the y-axis

Non-rigid transformations

Scaling

Three interpretations

Change of units
Change of frame
Movement of point

Others

Clipping

What is it?

Representations of Lines

Parametric: y = mx + b, z = nx + c
Implicit, L(t) = P + t v = P + t ( Q - P )
- n - unit vector normal to line
  - For L(t) = P + t ( Q - P )
  - n . ( Q - P ) = 0
- If P is any point on the line then d( R ) = ( R - P ) . n is the signed distance to the line
- Checks (for homework)
  1. If R is on the line then d( R ) = 0.
  2. If R is off the line then d( R ) != 0.

Clip a Point against a Half-space.

Representation of a Half-space.

Use the implicit representation of a line (P, n).
- P on the separating plane
- n perpendicular to the separating plane
Let the direction of the normal vector point to the inside.

Calculate d = ( R - P ) \dot n

If d > 0, inside.
If d < 0, outside.
If d = 0, on the line.

Clip a Line Segment to a Half-space.

The line segment

L(t) = R + t ( S - R ), 0 < t < 1.
R is at one end of the line segment
S is at the other end

Test if each of R and S are inside. Calculate

d(R) = ( R - P ) \dot n
d(S) = ( S - P ) \dot n

There are three cases

Both inside: keep the segment as is.
Both outside: discard the segment.
One inside, one outside: the segment crosses the boundary of the half-space.
- Calculate the intersection
  - ( L(t) - P ) \dot n = 0.
  - t = { ( P - R ) \dot n } / { ( S - R ) \dot n }
- Optimize (??)
  - t = { ( P - R ) \dot n } / { ( S - P ) \dot n - ( R - P )\dot. n }
- Replace one of the end points by L(t), (Which one?)

Clip a Line Segment to a Rectangular Parallellopiped

Straightforward, BUT what if the line segment crosses a corner?

Two Good Examination Questions

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