3.4 Linear transformation matrix

A linear transformation matrix is a mathematical tool for describing linear transformations

• Translate
• Scale
• Mirror
• Rotate
• Shear.

Any sequence of these operations in a 2-dimensional plane can be described by one linear transformation matrix of 6 parameters. It can do the following

• Transform a square into a square, a rotated square, a rectangle, a rotated rectangle, a rhomb, a rotated rhomb, a parallellogram or a rotated parallellogram
• Transform a circle into a circle, a rotated circle, an ellipse or a rotated ellipse
• Transform parallel lines into parallel lines.

Because parallel lines stay parallel, a linear transformation matrix can describe a 3D isometric projection, but not a 3D perspective projection.

The transformation of a triangle into another triangle uniquely identifies a linear transformation matrix (CADCalcMatrix).

Applying a linear transformation matrix with CADTransform is generally more efficient than a succession of singular operations (CADTranslate, CADScale, CADMirror, CADRotate and CADShear).

Objects of type Block insert and Raster image have a linear transformation matrix attached

• because linear transformation matrices are generally very efficient
• because linear transforms on those objects will not touch their original content.

The matrix that doesn’t change anything is called the identity matrix (CADIdentityMatrix). A matrix that describes one sequence of the operations scale, rotate and translate (in that order) is called a standard matrix (CADStandardMatrix). A matrix that undoes the effect of another matrix is called an inverse matrix (CADInvertMatrix).

Applying one matrix transformation on another is called a matrix multiplication (CADTransformMatrix). The term multiplication is somewhat misleading. When multiplying two numbers, the order of the two numbers is not important. But when multiplying two matrices the order is important.

Methods
CADIdentityMatrix, CADStandardMatrix, CADTranslateMatrix, CADRotateMatrix, CADScaleMatrix, CADMirrorMatrix, CADShearMatrix, CADTransformMatrix, CADFlipMatrix, CADStretchMatrix, CADInvertMatrix, CADCalcMatrix, CADGetMatrixPlane.