Transformation: about

Principle of use

The transformations are geometric functions that permit the creation of new objects, starting from objects already created.

Various functions

The various available functions are:

  • translation
  • rotation
  • affinity
  • helix
  • composed

Translation

A translation is defined by a direction and a distance.

2D / 3D domain
Translation vector Translation defined by 2 points and a ratio
  • the direction and the distance are defined by:
    • a working coordinate system
    • a vector (its components DX and DY define the direction and the amplitude of the vector)
  • the direction is defined by two points (vector tail and vector head)
  • the distance is equal to the distance between the two points (vector tail and vector head) multiplied by the ratio (proportionality coefficient)

Rotation

A rotation is defined by a rotation axis and an angle.

2D / 3D domain 3D domain
Rotation defined by angles and a pivot point (its coordinates or reference number)

Rotation defined by 3 points and 1 angle
  • rotation axis is defined by:
    • a working coordinate system
    • and a pivot point
  • rotation angle is defined about Z-axis
  • rotation axis is defined by direction and position:
    • a head point and a tail point give the direction
    • and a pivot point defines the position
  • angle is defined in the plane perpendicular to the axis
Note: The positive value of the angle corresponds to the anti-clockwise rotation.

Affinity

Affinity is defined with respect to a point, to a straight line or to a plane (for 3D domain).

The result of this transformation application depends on the affinity ratio (see the table below).

Ratio Result
k = -1 symmetry
k = 1 identity
k = 0 projection
k > 1 increasing (increasing homothety)
0 < k < 1 reducing (reducing homothety)
k < -1 increasing (increasing negative homothety)
-1 < k < 0 reducing (reducing negative homothety)
2D / 3D domain
Affine transformation with respect to a point Affine transformation with respect to a line
3D domain
Affine transformation with respect to a plane
CAUTION: The affinity ratio = 0 causes an error in the application of affinity transformation with respect to a point because the line is degenerated and reduced to a point.

Helix

The helix transformation is used only for 3D study domain.

A helix is defined by a coordinate system, an axis, a height and an angle.

Domain 3D
Helix
  • axis of helix is defined by:
    • a pivot point
    • and a directing vector
  • length of displacement is defined by a helix height
  • angle of helix α is defined in the plane perpendicular to the axis (-90° ≤ α ≤ 90°)

Composed transformation

It is possible to create composed geometric functions.

Domain 2D / 3D
Transformation combining two transformations
  • a translation following OY axis (defined by DX=0, and DY)
  • a rotation about Z-axis (defined by the pivot point and an angle α)
CAUTION: You do not obtain the same result by using different order of two transformations.

Parameter setting

The characteristics of transformation are parametrized expressions. The vector components, the coordinates of pivot point, the rotation angles and the ratios of affinity can be defined using algebraic expression.

The algebraic expression can contain:

  • constants
  • geometric parameters (created beforehand)
  • basic mathematical functions using operators: +, -, *, /, ( )
  • usual mathematical functions admitted by FORTRAN.

The mathematical functions are described in section Functions.