Abstract
Affine transformations allow the production of complex shapes using much simpler shapes. For example, an ellipse (ellipsoid) with axes offset from the origin of the given coordinate frame and oriented arbitrarily with respect to the axes of this frame can be produced as an affine transformation of a circle (sphere) of unit radius centered at the origin of the given frame.
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Notes
- 1.
Faute-de-mieux, we term bump the point of a Lamé curve where its curvature is a maximum.
- 2.
These are the points at which the curvature of a continuous and smooth curve changes sign. In the case at hand, these points are \(A^\prime \) and \(B^\prime \).
- 3.
The inverse of \({\varvec{P}}\) was obtained with computer algebra.
- 4.
The factoring is based on the polar decomposition (PD) of \({\varvec{M}}\) and the eigenvalue decomposition of the symmetric, positive-definite factor of the PD. These two decompositions lying outside of the scope of the book, they are not discussed here. This factoring is related to, but simpler than, the most popular singular-value decomposition of matrix \({\varvec{M}}\): Strang (1986).
- 5.
This is the circle traced by the center of the circle playing the role of the generatrix of the torus, which is produced by a rotation of the circle about the axis of the torus.
- 6.
Leonhard Euler proved this in 1776.
- 7.
The bisector plane of two given points is the locus of points equidistant from the two points.
- 8.
The qualifier must come from the plane containing the arrow upon being thrown by the archer.
- 9.
See Exercise 4.1.
- 10.
In the same way as the coordinate axes X and Y divide the plane into four quadrants, in 3D space the coordinate axes divide the space into eight octants.
- 11.
Maplesoft’s Maple 10.
- 12.
Notice that \({\varvec{t}}=(s/(2p)){\varvec{e}}\).
- 13.
Bevel gears were introduced in Sect. 3.3.2.
- 14.
Chaudhary et al. (2016).
- 15.
Koenderink (1990).
References
Chaudhary M, Angeles J, Morozov A (2016) Design and kinematic analysis of a spherical cam-roller mechanism for an automotive differential. CSME Trans 40:243–252
Koenderink JJ (1990) Solid shape. The MIT Press, Cambridge
Singh S (2015) Design synthesis of custom-molded earphone sleeve. M.Eng. Project, Department of Mechanical Engineering, McGill University, Montreal
Strang G (1986) Introduction to applied mathematics. Wellesley-Cambridge Press, Cambridge
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Angeles, J., Pasini, D. (2020). Affine Transformations. In: Fundamentals of Geometry Construction. Springer Tracts in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-43131-0_4
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