Shape Transformations
(MEMS Fabrication Techniques)
Examples of shape transformations
The fundamental process we will examine is the transformation from one shape to another.
Such changes occur at all length scales, a few macro scale examples are listed below. These different examples all share the same mathematics.
• Painting:
Consider a two dimensional curve defining the path of a paint brush. As the paint brush is swept along this curve the initial path is mapped to two output curves, an inner and an outer path. By changing the shape of the brush we can model anisotropic transformations. Note that some corners are rounded and others are sharp. This dichotomy of corner behavior is a fundamental part of shape transformation.
• Fonts:
If the painting path is the outline of a letter, typographic fonts may be developed with a compact representation. By parametrically defining a brush which changes along the path, a wide variety of fonts may be generated.
• Milling:
The ideal cutting path of a milling machine is enlarged by the finite size of the milling cutter. The output shapes are called offset surfaces. Offsets surfaces are difficult to calculate (because they often contain cusps) and are the subject of much research. They are commercially important because NC (numerically controlled) milling machines must be able to predict the result of particular milling actions.
• Robotic Path Planning:
Consider a circular two-dimensional robot in an obstacle filled room. Path planning algorithms attempt to move the robot from one location to another. If the obstacles are painted with a brush the same size and shape as the robot, the robot may be
transformation examples: (A) painting, (B) profilers, (C) cams
replaced with a point and the path planing done on this new system. This method is valid for non-circular robots as long as rotation is not permitted.
• Profilers:
One way to measure the surface profile of a sample is to drag a stylus across the sample. One of the limitations of such a system is that the measurable height changes are limited by the shape of the stylus. In fact the output is a convolution of the actual shape and the stylus shape. This is a serious limitation at smaller scales.
• Optics:
In most materials, the propagation of light is isotropic; a circular wave front remains circular. Birefringent materials have indices of refraction that depend on orientation.
The evolution of the wave front is anisotropic in this case and the wave front changes shape.
• Cams:
When a rocker arm is in tangential contact with a cam the output motion of the arm depends on the shape of the cam. If the cam is defined by R(µ) and rotates at an angular rate !, then the output as a function of time is
©. This output can be expressed in Cartesian coordinates (© vs. time) or polar coordinates (radius: ©, angle: time). In polar coordinates the motion of the cam is the transformation of the shape defined by R(µ) to a new shape ©(time).
Minkowski transforms
Minkowski transformations are a mathematical method of modeling shape transformations, some examples of which were given above. Minkowski algebra allows us to add two shapes represented by a list of vectors from one vertex of a polygon to the next vertex.
If the polygons are both convex, then the Minkowski addition of two polygons is found by combining the two vector lists and slope sorting them (see Figure 2.13). Slope sorting means that the combined list of vectors is rearranged so that the vectors with the smallest slopes (with respect to the x axis) are the first vectors in the sorted list. The asterisks in
Example of Minkowski Addition (convex/convex). The slope diagrams are shown below
When one shape is nonconvex then the algorithm is more involved. Between two consecutive vertices of the nonconvex polygon perform the following steps:
i) if the rotation from one vertex to the next is counterclockwise (positive angle), insert into the nonconvex list the portion of the convex vector list that is between the two nonconvex vertices (this is the same as the convex/convex case).
ii) if the rotation from one vertex to the next is clockwise (negative angle), insert into the nonconvex list the negative of the portion of the convex vector list that is between the two nonconvex vertices. Note that the output shape is non-simple and has self intersections. The final shape in the outer perimeter of the shape with the set of intersecting regions trimmed off. The finding of such self intersections is a non-trivial part of Minkowski transformations.
Minkowski subtraction is performed in the same fashion as addition but the negative of the vectors are used. Minkowski transformations will be revisited later.
Cesar Augusto Suarez
CI 17394384
CAF
No hay comentarios:
Publicar un comentario