I plan to add this transformation to my upcoming book on computational geometry. This mapping provides a one-to-one correspondence between RGB color and locations on a dance floor for a positioning correction in a project on robotic square dancers. Fun!

# Tag Archives: mapping

# Tutorial: multi-linear regression

You’ve probably seen classical equations for linear regression, which is a procedure that finds the straight line that best fits a set of discrete points . You might also be aware that similar formulas exist to find a straight line that is a best (least squares) fit to a continuous function .

The bottom of this post provides a link to a tutorial on how to generalize the concept of linear regression to fit a function that takes a vector as input and produces a vector as output. In mechanics, the most common example of this type of function is a mapping function that describes material deformation: the input vector is the initial location of a point on a body, and the output vector is the deformed location of the same point. The image shows a collection of input vectors (initial positions, as grey dots) and a collection of output vectors (deformed locations as blue dots). The affine fit to these descrete data is the pink parallelogram. Continue reading

# Tutorial: Slideshow introduction to mappings in continuum mechanics

Each time you generate output from input, you are using a mapping. The mappings in continuum mechanics have similarities with simple functions y=f(x) that you already know. This slideshow (which apparently renders properly only when viewed from PowerPoint on a PC rather than Mac) provides a step-by-step introduction to mappings of the type used in Continuum Mechanics.

You may download the rest of the document here.

# Tutorial: Functional and Structured Tensor Analysis for Engineers

A step-by-step introduction to tensor analysis that assumes you know nothing but basic calculus. Considerable emphasis is placed on a notation style that works well for applications in materials modeling, but other notation styles are also reviewed to help you better decipher the literature. Topics include: matrix and vector analysis, properties of tensors (such as “orthogonal”, “diagonalizable”, etc.), dyads and outer products, axial vectors, axial tensors, scalar invariants and spectral analysis (eigenvalues/eigenvectors), geometry (e.g., the equations for planes, ellipsoids, etc.), material symmetry such as transverse isotropy, polar decomposition, and vector/tensor calculus theorems such as the divergence theorem and Stokes theorem. (A draft of this document was last released publically on Aug. 3, 2003. The non-public version is significantly expanded in anticipation of formal publication.)

You may download the rest of the document here.