Transforms
Each frame f defines a linear transform of 3D space: a point with
global 3D coordinates (x, y, z) is mapped to point with same coordinates
relative to f. If iFrame is
IFrame_DG, implemented by f,
the global coordinates of the transformed point are obtained by
iFrame.ToGlobal(x,y,z)
This correspondence establishes mapping between
frames and invertible transforms which preserve distances and
angles
Similarly, a vector (vx,vy,vz) in global coordinates is transformed into vector
with same coordinates relative to f and its global
coordinates are obtained with iFrame.ToGlobalVector(vx,vy,vz)
A 3D axis can be transformed in the same way, and as the transformation preserves
angles this also defines transformation of sets of axes of coordinates in 3D
Frame, which coincides with the global frame, is mapped to identity transformation
Constructing basic transforms
Call iFrame.Reset(); to make f an identity transformation
To obtain transformation of a translation by (vx,vy,vz) execute:
iFrame.Reset();
iFrame.Translate(vx,vy,vz);
To obtain transformation of a rotation:
iFrame.Reset();
iFrame.Rotate(angle, xAxis, yAxis, zAxis, vxDirAxis, vyDirAxis, vzDirAxis);
where (xAxis, yAxis, zAxis) is a point where the axis passes through and
(vxDirAxis, vyDirAxis, vzDirAxis) is the direction of the axis
Multiplication of transforms
Transforms have a multiplication (composition) operation: f*g(point)
= f(g(point)). The identical transform is a unit relative to this
operation. Inverse transform is a transform g, such that
f*g=identity.
Using the above correspondence, this operation can be extended to frames. So
frames can be multiplied, inverted and divided.
The product of two frames f = f1*f0 can be obtained with:
iFrame0.ToGlobalFrame(iFrame1);
after which, iFrame1 becomes product of the two.
To keep iFrame0 unchanged, clone it using
IFrame_DG.Clone():
Transforming 3D objects
There are two significantly different ways of applying transforms to entities:
a) Moving the entity as whole without modifying the geometry b) Transforming internal
surface of the entity. The second method means that surface of the entity
has to be moved relatively to the local frame
Moving the object as whole
This is the preferred way of moving entities around when the surface does not
need to be modified in any way, only position and orientation has to be
adjusted. This method is very fast and does not depend on geometry size
as only the local frame is modified.
This type of transformation can be executed with:
IFrame_DG iFrameEntity = iEntity.GetLocation();
iFrame.ToGlobalFrame(iFrameEntity);
Transforming surface
To apply a generic transform IFrame_DG iFrame, use:
iEntity > IGeometricObject_DG > iTransf
iTransf.Transform(iFrame);
IGeometricObject_DG.Translate(),
IGeometricObject_DG.Rotate(), etc.,
are shortcuts for basic transformations.
Transforming surface means operations on all elements of the geometry,
which can be computationaly expensive, for example in case of a dense mesh. When there are several
consequtive surface transformation required it is much faster to calculate the
total transform first by multuplying them in sequence as described above and
then apply the result to the surface
Transforming surface is
inevitable when there is need to modify position/orientation of the local frame relative to the
surface (for more convenient rotations etc.). For example, translating local frame
by a vector v means translating the whole object by v and translating the
surface by v (opposite direction).
See also: Transform sample
