Chemistry

Linear transformations


General affine mappings

The transformation P.'=T(P.)represented by the system of linear equations

x'=ax+byy'=cx+dy

is referred to as affine mapping with a fixed coordinate origin in two dimensions. The coefficients a,b,c,d are elements of an invertible matrix

T=abcd.

If one also takes into account a translation (shift) around a constant vectort=ef, the general affine mapping in two dimensions reads:

x'y'=abcdxy+ef

or in vector form:

r'=Tr+t.

In 3D space, an affine mapping is expressed by:

x'=a11x+a12y+a13z+b1y'=a21x+a22y+a23z+b2z'=a31x+a32y+a33z+b3.

Affine mapping is characterized in that collinearity is preserved, i.e.

  • all points lying on a segment remain on the segment after the transformation, and
  • the ratio of the lengths of lines is retained after the transformation, e.g. the midpoint of a straight line remains the midpoint after the transformation.

In contrast, i. In general, the length of a segment and the angle between two straight lines are not obtained.

An isometric mapping or isometry is subject to the restriction that the matrix Tis orthogonal, i.e.TTT=TTT=I.. Lengths (and angles) are preserved.

Put in the matrix T b=c=0 , the following transformation types are possible:

  • Unit transformation (isometry)
  • Scaling
  • Reflection (isometry)

Removal of the restriction b=c=0 results in:

  • Shear b0 and c=0 or b=0 and c0
  • Rotation (isometry) b0 and c0

In the animation, two-dimensional affine images can be visualized on a rectangle. The Elements a,b,c,d the transformation matrix can be freely selected and with the help of the parameters apply to the figure:

T(s=0)=1001andT(s=1)=abcd.

Here are some cases:

Tab. 1
Affine images
matrixRemarks
-100-1Orthogonal matrix. Isometry. Reflection in y=x and then in y=-x. Equivalent to turn around π.
1-101No isometry. x-Shear.
0100Not invertible. No affine mapping.

Note: When producing the map, the surface of the globe is mapped onto a plane. But this map is not an isometry, which Gauss proved (known as Theorema Egregium). In an isometric illustration, the Gaussian curvature is retained, that for a sphere (radius r) 1r2 and one level is equal to 0.


Video: Robotics: Transformation Matrices - Part 1 (January 2022).