Chapter 8 - Transformations Part 1

What do geometric transformations do?

-Move points in space from one location to another

-Change the description of a point from one coord system to another (like Working Coord system to Model Coord system)

-Will always involve motion

What is a rigid-body transformation?

A transformation that changes an object’s position or orientation w/o deforming its shape

Types of geometric transformations

-Translation

-Rotation

-Scaling

-Reflection

-Shearing

-Orthogonal projection

What are two important characteristics of geometric transformations?

-Uniqueness (only one new point for an old point)

-Concatenation or combination of transformations

(two transformations can be combined to give a single transformation which gives the same effect as the sequential application of the original two)

What is the difference between transformation and mapping?

Transformation moves a point from one location in a coordinate system to another. Mapping moves a point from one coordinate system to another, but does not change the point’s location in the modeling space.

Where is mapping used?

-Implicitly during CAD construction

-Useful in assembly modeling

Types of mapping

-Translational mapping

-Rotational mapping

-General mapping (a combination of the above two)

What does a 2D rotation do to a point?

Rotates the point about the coordinate origin by an angle theta, without changing its distance from the origin or the object’s shape / size

How can two successive rotations be combined?

Multiply their rotation matrices together.

How do you reverse a rotation?

Apply the rotation in the opposite direction using the inverse rotation matrix.

Why are translation and rotation not commutative?

Performing a translation followed by a rotation generally produces a different result from rotating first and then translating.

What do S_x, S_y, and S_z represent?

Scaling factors in the X, Y, and Z directions

What happens when a scaling factor is less than or greater than 1?

A factor between 0 and 1 compresses the object, a factor greater then 1 stretches it

What is reflection in geometric modeling?

A transformation that creates a mirror image of a model across a plane, axis, or point.

How do coordinates change when reflecting across the XY, YZ, or XZ plane?

-XY-plane: (x,y,z) → (x,y,-z)

-YZ-plane: (x,y,z) → (-x,y,z)

-XZ-plane: (x,y,z) → (x,-y,z)

How do coordinates change when reflecting across the X-, Y-, or Z-axis in 3D?

-X-axis: (x,y,z) → (x,-y,-z)

-Y-axis: (x,y,z) → (-x,y,-z)

-Z-axis: (x,y,z) → (-x,-y,z)

How do you reflect a 3D point through the origin?

(x, y, z) → (-x, -y, -z)

What is shearing in geometric modeling?

A transformation that shifts coordinates in one direction by an amount proportional to another coordinate.

What happens when a model is sheared parallel to the X-axis by a factor m?

The X coordinate changes in proportion to another coordinate, while the other coordinates remain unchanged.

What is orthogonal projection?

Projecting points onto a plane along directions perpendicular to that plane. Aka projection removes the coordinate perpendicular to the selected plane.

How do coordinates change when projecting onto the XY-, XZ-, or YZ-plane?

-XY-plane: (x,y,z) → (x,y,0)

-XZ-plane: (x,y,z) → (x,0,z)

-YZ-plane: (x,y,z) → (0,y,z)