Abstract
Geometric elements are categorized as points, lines, surfaces, and solids. Surfaces and solids also have many subcategories. Points, lines, circles, and arbitrary curves are the basic 2D geometric primitives, or generators, from which other, more complex geometric objects can be derived or algorithmically produced. For example, by taking a straight line and moving it through a circular path lying in a plane normal to the line, while keeping the line normal to the plane, one can create a cylinder. This chapter defines, illustrates, and describes how to create points, lines, circles, polygons, polygonals, and arbitrary curves in the plane. The concept of rigid body in the plane and its properties is given due attention.
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Notes
- 1.
Sometimes, an open polygon is referred to as a polygonal.
- 2.
- 3.
Its eigenvalues (http://mathworld.wolfram.com/Eigenvalue.html) are positive.
- 4.
The proofmass is the triangular plate in the same figure.
- 5.
In a right circular cone, the intersections of all planes normal to the axis \(\mathcal {A}\) are circles.
- 6.
On a practical note, an ellipse can be quickly constructed using a pencil whose tip is used to keep one string taut, whose two ends are, in turn, attached to the two foci of the desired ellipse.
- 7.
A right circular cone is defined as the surface generated by a line \(\mathcal {L}\) that turns around a fixed line \(\mathcal {A}\), the cone axis, while pivoting about a point P of \(\mathcal {A}\) and making a constant angle with \(\mathcal {A}\)—See Fig. 2.12 and Sect. 3.3.2. A cone thus has two nappes.
- 8.
This trajectory is not exactly a parabola because of the drag effect of the air. Golf balls on the moon would describe exact parabolas.
- 9.
The qualifier is intended to denote emitting sources located a distance from the focus much larger (several orders of magnitude) than the antenna focal distance, \(\overline{VF}\) in Fig. 2.9.
- 10.
Points at which the value of a function becomes unbounded are termed singular.
- 11.
See: Teng et al. (2008) and the references therein.
- 12.
Only the top part is shown in Fig. 2.18.
- 13.
Figliolini et al. (2019).
- 14.
Mortenson (1985).
- 15.
Belzile et al. (2020).
References
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Figliolini G, Stachel H, Angeles J (2019) Kinematic properties of planar and spherical logarithmic spirals: applications to the synthesis of involute tooth profiles. Mech Mach Theory 136:14–26
Mortenson ME (1985) Geometric Modeling. Wiley, New York
Teng CP, Bai S, Angeles J (2008) Shape synthesis in mechanical design. Acta Polytech 47(6):56–62
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Angeles, J., Pasini, D. (2020). 2D Objects. In: Fundamentals of Geometry Construction. Springer Tracts in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-43131-0_2
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DOI: https://doi.org/10.1007/978-3-030-43131-0_2
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