China
Africa
Explore chapters and articles related to this topic
Analytic Geometry
Published in Richard C. Dorf, Ronald J. Tallarida, Pocket Book of Electrical Engineering Formulas, 2018
Richard C. Dorf, Ronald J. Tallarida
A parabola is the set of all points (x, y) in the plane that are equidistant from a given line called the directrix and a given point called the focus. The parabola is symmetric about a line that contains the focus and is perpendicular to the directrix. The line of symmetry intersects the parabola at its vertex (Figure 4.4). The eccentricity e = 1.
P
Published in Splinter Robert, Illustrated Encyclopedia of Applied and Engineering Physics, 2017
[biomedical, computational] Symmetric graphical representation of a variety of mathematical expressions, but the most well known is the quadratic equation, for instance, mathematically expressed as y = ax2 + bx + c, or x = y2, respectively, in general notation: Ax2 + Bxy + cy2 + Dx + Ey + F = 0, where x and y are Cartesian coordinates and the remaining letters are constants. The parabola has an axis of symmetry that intersects with the graphical representation of the parabola in the vertex. The parabola is in the family of conic intersects: circle, ellipse, parabola, and hyperbola. In mechanics, the parabolic trajectory is well known and can be defined by the vector sum of the gravitational attraction and the propulsion force, where propulsion will be in vertical and horizontal directions. A horizontally released projectile (e.g., bullet from a gun) will have a vertical dependence on gravity (gravitational accelerationg) as a function of time (t), describing only one side of the parabolic track defined as y = −(1/2)gt2 (see Figure P.5).
Axially symmetric reflector antennas: Geometrical-optics models and efficient electrodynamic analysis of double-mirror structures
Published in Electromagnetics, 2020
Yuriy Sirenko, Paul Smith, Lyudmyla Velychko, Olena Velychko
The papers Moreira and Bergmann (2006) and Moreira, Prata, and Bergmann (2007) present the schemes for constructing well-directed reflector antennas with the funnel-shaped far-field pattern efficiently radiating symmetric - and -waves. These techniques are based on the well-known results of the geometrical theory of diffraction (James 1986) and on the following results of the theory of second-order curves (Korn and Korn 1961): (i) a tangent and a normal constructed at any point of an ellipse (hyperbola) bisect the angles between the straight lines connecting this point with the focuses; (ii) a tangent and a normal constructed at any point of a parabola bisect the angles between the parabola diameter parallel to the axis and the straight line connecting this point with the focus.
Incorporating the dynamic mathematics software GeoGebra into a history of mathematics course
Published in International Journal of Mathematical Education in Science and Technology, 2018
Another concept examined by the pre-service mathematics teachers in the history of mathematics course was Khayyam's solution of the cubic equation. In the geometric model, the x2 = ay parabola is drawn before the conic section is cut. Then a line is drawn as it will be tangent to the parabola. On this line a circle with a diameter [AG] = b/a2 is drawn which passing through tangent point and whose centre is over the line. Let the intersection points of the circle and the parabola, be A and C. If the foot of the vertical line descending from the point C to the line segment ([AG]) is D then the desired positive root of the cubic equation is the length of [AD] [18]. Discussions were held with the pre-service teachers about the geometric solution of a cubic equation in the form x3+a2x = b. Then, the pre-service teachers were given some time to move Khayyam's solution to a dynamic environment. The researcher asked different questions to the pre-service teachers about the independent variables a and b. The groups themselves had to be active so that they could explore the relationship between these variables and the geometric solution in the dynamic environment. Thus, the researcher constantly asked leading questions. At the end of the process, the pre-service teachers were asked to present their dynamic materials to the class. An example of a dynamic material presented by the pre-service teachers is given in Figure 2.
Note on Archimedes’ quadrature of the parabola
Published in International Journal of Mathematical Education in Science and Technology, 2022
Dusan Vallo, Jozef Fulier, Lucia Rumanova
If , then the point is a midpoint of the chord and it holds where is the tangent point of the tangent line and the parabola.