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Compliances of Basic Flexible-Hinge Segments
Published in Nicolae Lobontiu, Compliant Mechanisms, 2020
The circle and the ellipse are not the only planar curves resulting in right corner-filleted flexure segments. Rational Bézier curves can also be used to generate right corner-filleted flexible hinges, and these curves are actually conic-section curves, as shown here. Conic-section curves include the parabola, hyperbola and ellipse (or circle, as a limit case of an ellipse with equal semi-axes). They result by intersecting a conic surface with a plane that is in one of the following positions: at an angle with respect to the cone axis and intersecting both the lateral surface and the circular base of the cone – this results in a parabola, parallel to the cone axis – producing a hyperbola, and at an angle with respect to the cone axis and intersecting the lateral cone surface but not the circular base of the cone–this results in an ellipse, perpendicular to the cone axis – producing a circle; the parabola, hyperbola and ellipse are illustrated in Figure 2.18.
Ellipses, Spheres, Spiral Forms, and Random Curves
Published in Craig Attebery, The Complete Guide To Perspective Drawing, 2018
A look at conic sections can further explain this phenomenon. A conic section is the intersection of a plane and a cone. The silhouette of a sphere is circular. When that round shape is projected to the eye, the visual pyramid is conical. A cone intersecting a flat plane at an oblique angle creates an ellipse (Figure 8.18). Thus, a sphere plotted in perspective is not round unless its center is aligned with the focal point. A sphere plotted in perspective is an ellipse.
Orbit Dynamics and Properties
Published in Yaguang Yang, Spacecraft Modeling, Attitude Determination, and Control Quaternion-based Approach, 2019
Spacecraft orbits are closely related to conic sections. A conic section is the intersection of a plane and a right circular cone. Different intersections result in different orbital shapes: circle, ellipse, parabola, and hyperbola (see Figure 2.2). Since parabolic orbit is of no importance in the context of spacecrafts, therefore, only the circle, ellipse, and hyperbola orbits have been discussed here.
Non-equivalent notions of the eccentricity of the conics: an exploratory study with high school teachers
Published in International Journal of Mathematical Education in Science and Technology, 2023
Antonio Rivera-Figueroa, Ernesto Bravo-Díaz
Given the diversity of approaches to conics, and their corresponding definitions of eccentricity, that appear in textbooks and mathematical literature, we propose as objectives of our research to find out what is the experience of teachers about the eccentricity of conics, and whether the non-equivalence of the eccentricity definitions causes them some conflict when teaching these curves. The result is that it does not cause any conflict because, in general, the teachers base the teaching of the conics on what we have called a bifocal approach. The focus-directrix approach is generally unknown to teachers, and they only mention in the courses the cone-plane approach to introduce and interpret the curves as the intersection of a cone with a variable plane. Salinas and Pulido (2015) comment that in the teaching of the conics: ‘the attribute of being conic is illustrated with figures, which show the intersection of a plane with a cone’ (p. 148). We would add that in the teaching of the ellipse, parabola, and hyperbola, their illustration with the intersection of a cone and a plane only plays the role of justifying the name of conic sections. In this sense, the mere illustration in three-dimensional space cannot be considered an approach to those curves. A cone-plane approach to conics should include the location of foci and the characterization of the ellipse and the hyperbola by their focal properties (see, e.g. Apostol, 1966, p. 497; and Berttand, 2015). Also, it should include the determination of the directrices and calculating the eccentricities (e.g. Brannan et al., 2012). However, this approach to conics is not appropriate for high school. So, it is sufficient that the teachers mention the cone and the plane to interpret the conics.