Conic Sections
Conic sections are the curves formed by the intersection of a plane with a cone. The four main types of conic sections are the circle, ellipse, parabola, and hyperbola. They are fundamental in geometry and have applications in various fields such as physics, engineering, and astronomy. Each type of conic section has distinct properties and equations that describe its shape.
7 Key excerpts on "Conic Sections"
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...In this case, the two descriptor sides are [ AB ] and [ BC ], corresponding to straight lines defined by Eq. (13.26) with (x 1, y 1) replaced by (x 0, y 0) and (x 2, y 2) replaced by (x 1, y 1), and Eq. (13.26) as such, respectively; the former degenerates to (13.30) or else (13.31) after solving for y – whereas the latter becomes (13.32) which eventually breaks down to (13.33) upon isolation of y, followed by isolation of the term in x and elimination of the pending parenthesis – or simply (13.34) As emphasized before, the triangle will not be defined in full until Eqs. (13.31) and (13.34) are coupled with the horizontal axis, with x 0 = 0 and x 2 = h 2 serving as initial and final abscissae, respectively. 13.3 Conical Curves A conic section (or conical curve) is obtained as the intersection of the surface of a cone with a plane – as exemplified in Fig. 13.4 ; remember that a cone is, in turn, the solid produced by a π rad -rotation of an isosceles triangle about its height (which thus becomes its symmetry axis). A circle or an ellipse arises when the intersection of the cone with the plane is a closed curve – as emphasized in Fig. 13.4 ; the latter results from an intersection plane with inclination angle of amplitude below that of a generating line – whereas a circle will be at stake when the cutting plane is parallel to the plane of the generating circle of the cone, and thus perpendicular to its symmetry axis. If the (inclined) cutting plane is parallel to exactly one generating line of the cone, then the conic becomes unbounded and is termed parabola. In the remaining case, the outcome is a hyperbola – with the (inclined) plane intersecting both halves (or nappes) of the cone, and thus producing two separate unbounded curves...
eBook - ePubIbn al-Haytham's Theory of Conics, Geometrical Constructions and Practical Geometry
A History of Arabic Sciences and Mathematics Volume 3
- Roshdi Rashed(Author)
- 2013(Publication Date)
- Routledge(Publisher)
...INTRODUCTION Conic Sections AND GEOMETRICAL CONSTRUCTIONS Greek geometers soon noticed that ‘geometrical’ constructions were not applicable only to plane problems and that ‘constructible’ problems were not simply those that could be solved by means of straightedge and compasses. This important discovery led some mathematicians to investigate the properties of curves other than the circle, in particular Conic Sections. The story of these constructions has been told so many times that it need not detain us here. 1 Let us merely remind ourselves that as early as the fourth century BC conics are called upon in the solution of a three-dimensional problem. Menæchmus made use of a parabola and a hyperbola to solve the problem of the duplication of the cube. Was this something unusual or an established practice? Is this a procedure that lays the foundations for the theory of conics? The answers to questions like these are hidden in the mists surrounding the origins of the theory. What concerns us here is to note that from a very early date Conic Sections were used to construct solutions to three-dimensional problems in geometry. A little later, towards the mid third century BC, Conon of Samos — according to the account given by Apollonius — takes up the whole problem once again. Apollonius in fact informs us, in the preamble to the fourth book of his Conics, that Conon of Samos had investigated the intersection of Conic Sections, and had tried to find the number of their points of intersection, ‘the greatest number of points in which Conic Sections that do not completely coincide can meet one another’. 2 This is the only echo, distant as it is, to tell us about the form and purpose of Conon’s work...
eBook - ePub- Michael Harrison, Patrick Waldron(Authors)
- 2011(Publication Date)
- Routledge(Publisher)
...Conic Sections, quadratic forms and definite matrices DOI: 10.4324/9780203829998-6 f 4.1 Introduction So far, we have concentrated on linear equations, which represent lines in the plane, planes in three-dimensional space or, as will be seen in Section 7.4.1, hyperplanes in higher dimensions. In this chapter, we consider equations that also include second-order or squared terms, and that represent the simplest types of nonlinear curves and surfaces. The concepts of matrix quadratic form and definite matrix are of considerable importance in economics and finance, as will be seen in the detailed study of our several applied examples later. Quadratic forms relate importantly to the algebraic representation of Conic Sections. Also, in Theorem 10.2.5, it will be seen that the definiteness of a matrix is an essential idea in the theory of convex functions. This chapter gives definitions and simple illustrations of the concepts of quadratic form and definiteness before going on to establish a number of theorems relating to them. We will return to quadratic forms in Chapter 14, where the important general problem of maximization or minimization of a quadratic form subject to linear inequality constraints is studied. 4.2 Conic Sections In this section, we consider equations representing Conic Sections in two dimensions. There are a number of equivalent ways of describing and classifying Conic Sections. We begin with a geometric approach. 1 Consider the curve traced out in the coordinate plane ℝ 2 by a point P = (x, y), which moves so that its distance from a fixed point (the focus S) is always in a constant ratio (the eccentricity ϵ ≥ 0) to its perpendicular distance from a fixed straight line (the directrix L). This curve is called: an ellipse when 0 < ϵ < 1; a parabola when ϵ = 1; a hyperbola when ϵ > 1; and a circle as ϵ → 0, as we shall see later. 4.2.1 Parabola Consider first the case of ϵ = 1, i.e. the parabola...
eBook - ePubManual of Engineering Drawing
Technical Product Specification and Documentation to British and International Standards
- Colin H. Simmons, Dennis E. Maguire(Authors)
- 2012(Publication Date)
- Butterworth-Heinemann(Publisher)
...Chapter 14 Conic Sections and Interpenetration of Solids Consider a right circular cone, i.e. a cone whose base is a circle and whose apex is above the centre of the base (Fig. 14.1). • The true face of a section through the apex of the cone will be a triangle. • The true face of a section drawn parallel to the base will be a circle. • The true face of any other section which passes through two opposite generators will be an ellipse. • The true face of a section drawn parallel to the generator will be a parabola. FIGURE 14.1 Conic Sections: section AA – triangle; section BB – circle; section CC – parabola; section DD – hyperbola; section EE – rectangular hyperbola; section FF – ellipse. If a plane cuts the cone through the generator and the base on the same side of the cone axis, then a view on the true face of the section will be a hyperbola. The special case of a section at right-angles to the base gives a rectangular hyperbola. To Draw an Ellipse from Part of a Cone Figure 14.2 shows the method of drawing the ellipse, which is a true view on the surface marked AB of the frustum of the given cone. 1. Draw a centre line parallel to line AB as part of an auxiliary view. 2. Project points A and B onto this line and onto the centre lines of the plan and end elevation. 3. Take any horizontal section XX between A and B and draw a circle in the plan view of diameter D. 4. Project the line of section plane XX onto the end elevation. 5. Project the point of intersection of line AB and plane XX onto the plan view. 6. Mark the chord-width W in the plan, in the auxiliary view and the end elevation...
eBook - ePubManual of Engineering Drawing
Technical Product Specification and Documentation to British and International Standards
- Colin H. Simmons, Dennis E. Maguire(Authors)
- 2009(Publication Date)
- Butterworth-Heinemann(Publisher)
...Chapter Conic Sections and interpenetration of solids Consider a right circular cone, i.e. a cone whose base is a circle and whose apex is above the centre of the base (Fig. 12.1).If a plane cuts the cone through the generator and the base on the same side of the cone axis, then a view on the true face of the section will be a hyperbola. The special case of a section at right-angles to the base gives a rectangular hyperbola. The true face of a section through the apex of the cone will be a triangle. The true face of a section drawn parallel to the base will be a circle. The true face of any other section which passes through two opposite generators will be an ellipse. The true face of a section drawn parallel to the generator will be a parabola. Fig. 12.1. Conic Sections: section AA – triangle; section BB – circle; section CC – parabola; section DD – hyperbola; section EE – rectangular hyperbola; section FF – ellipse. To draw an ellipse from part of a cone Figure 12.2 shows the method of drawing the ellipse, which is a true view on the surface marked AB of the frustum of the given cone. Draw a centre line parallel to line AB as part of an auxiliary view. Project points A and B onto this line and onto the centre lines of the plan and end elevation. Take any horizontal section XX between A and B and draw a circle in the plan view of diameter D. Project the line of section plane XX onto the end elevation. Project the point of intersection of line AB and plane XX onto the plan view. Mark the chord-width W on the plan, in the auxiliary view and the end elevation. These points in the auxiliary view form part of the ellipse. Repeat with further horizontal sections between A and B, to complete the views as shown. Fig...
eBook - ePubManual of Engineering Drawing
British and International Standards
- Colin H. Simmons, Dennis E. Maguire, Neil Phelps(Authors)
- 2020(Publication Date)
- Butterworth-Heinemann(Publisher)
...14 Conic Sections and interpenetration of solids Abstract There is often difficulty in handling problems involving two- and three-dimensional geometrical constructions. The examples in this chapter are included in order to provide a background in solving engineering problems connected with lines, planes, and space. Copying a selection of these examples on the drawing board or on CAD equipment will certainly enable the reader to visualize and position the lines in space which form each part of a view, or the boundary, of a three-dimensional object. It is a necessary part of draughting to be able to justify every line and dimension which appears on a drawing correctly. In this chapter examples of constructing an ellipse, a parabola and a rectangular hyperbola from a part of a cone. Many objects are formed by a collection of geometrical shapes such as cubes, cones, spheres, cylinders, prisms, and pyramids, and where any two of these shapes meet, some sort of curve of intersection or interpenetration results. It is necessary to be able to construct these curves to complete drawings in orthographic projection or to draw patterns and developments. Some of the most commonly found examples of interpenetration are also included such as a cones, cylinders and square prisms. Keywords Ellipse; Conical; Parabola; Hyperbola; Interpenetration Consider a right circular cone, i.e. a cone whose base is a circle and whose apex is above the center of the base (Fig. 14.1). • The true face of a section through the apex of the cone will be a triangle. • The true face of a section drawn parallel to the base will be a circle. • The true face of any other section which passes through two opposite generators will be an ellipse. • The true face of a section drawn parallel to the generator will be a parabola. If a plane cuts the cone through the generator and the base on the same side of the cone axis, then a view on the true face of the section will be a hyperbola...
eBook - ePub- J. F. Scott(Author)
- 2016(Publication Date)
- Routledge(Publisher)
...Forthen the quantityxwould not appear at all in the equation; there would therefore be no purpose in giving known values toy; the equation would have to be solved fory, and since it is now of the third degree, its value can only be determined by finding the root of a cubic equation, and this, in general, cannot be effected without the aid of the Conic Sections. Further, if there are not more than nine lines given, so long as they are not all parallel, the equation can be expressed so as to be of no higher degree than the fourth, and such an equation can always be solved by means of the Conic Sections, as will presently appear. Similarly, if there are not more than thirteen lines an equation can be derived not higher than the sixth degree; it will be shown that such can be solved by means of a curve one degree higher than the Conic Sections and so on*.Before enlarging upon the solutions he has outlined, Descartes considers it necessary to make some general statements upon the nature of curves. It is with these that the second book opens.It is, however, pertinent at this point to indicate the difference between a plane locus and a plane problem, a solid locus and a solid problem, a supersolid locus and a supersolid problem. A plane locus is a straight line or the circumference of a circle ; a problem is considered plane if it can be solved by finding the intersection of straight lines or the circumferences of circles, i.e. if it can be solved by means of a straight edge and a pair of compasses. A solid problem is one which can be solved by one of the three Conic Sections; although these are plane curves they were called solid because they were first approached by three dimensional considerations. A supersolid locus is a curve more complex than the Conic Sections; the supersolid problem is one the solution of which involves the use of a curve more complex than the Conic Sections, whatever the line which cuts it†.Notes*Lettres, III, No. 69, p...
