Conic Section

Showing posts with label Hyparabola. Show all posts

Saturday, 6 October 2012

Friday, 5 October 2012

Hyperbolas may be seen in many sundials. On any given day, the sun revolves in a circle on the celestial sphere, and its rays striking the point on a sundial traces out a cone of light. The intersection of this cone with the horizontal plane of the ground forms a conic section. At most populated latitudes and at most times of the year, this conic section is a hyperbola. In practical terms, the shadow of the tip of a pole traces out a hyperbola on the ground over the course of a day (this path is called the declination line). The shape of this hyperbola varies with the geographical latitude and with the time of the year, since those factors affect the cone of the sun's rays relative to the horizon. The collection of such hyperbolas for a whole year at a given location was called a pelekinon by the Greeks, since it resembles a double-bladed axe.

File:Akademia Ekonomiczna w Krakowie Pawilon C.JPG

Hyperbolas as declination lines on a sundial

2.Trilateration

A hyperbola is the basis for solving trilateration problems, the task of locating a point from the differences in its distances to given points — or, equivalently, the difference in arrival times of synchronized signals between the point and the given points. Such problems are important in navigation, particularly on water; a ship can locate its position from the difference in arrival times of signals from a LORAN or GPS transmitters. Conversely, a homing beacon or any transmitter can be located by comparing the arrival times of its signals at two separate receiving stations; such techniques may be used to track objects and people. In particular, the set of possible positions of a point that has a distance difference of 2a from two given points is a hyperbola of vertex separation 2a whose foci are the two given points.

3.Path followed by a particle

The paths followed by any particle in the classical Kepler problem is a conic section. In particular, if the total energy E of the particle is greater than zero (i.e., if the particle is unbound), the path of such a particle is a hyperbola. This property is useful in studying atomic and sub-atomic forces by scattering high-energy particles; for example, the Rutherford experiment demonstrated the existence of an atomic nucleus by examining the scattering of alpha particles from gold atoms. If the short-range nuclear interactions are ignored, the atomic nucleus and the alpha particle interact only by a repulsive Coulomb force, which satisfies the inverse square law requirement for a Kepler problem.

4.Korteweg-de Vries equation

The hyperbolic trig function appears as one solution to the Korteweg-de Vries equation which describes the motion of a soliton wave in a canal.

5.Angle trisection

As shown first by Apollonius of Perga, a hyperbola can be used to trisect any angle, an intensely studied problem of geometry. Given an angle, one first draws a circle centered on its middle point O, which intersects the legs of the angle at points A and B. One next draws the line through A and B and constructs a hyperbola of eccentricity ε=2 with that line as its transverse axis and B as one focus. The directrix of the hyperbola is the bisector of AB, and for any point P on the hyperbola, the angle ABP is twice as large as the angle BAP. Let P be a point on the circle. By the inscribed angle theorem, the corresponding center angles are likewise related by a factor of two, AOP = 2×POB. But AOP+POB equals the original angle AOB. Therefore, the angle has been trisected, since 3×POB = AOB.

6.Efficient portfolio frontier

In portfolio theory, the locus of mean-variance efficient portfolios (called the efficient frontier) is the upper half of the east-opening branch of a hyperbola drawn with the portfolio return's standard deviation plotted horizontally and its expected value plotted vertically; according to this theory, all rational investors would choose a portfolio characterized by some point on this locus.

7.Stones in a Lake

When two stones are thrown simultaneously into a pool of still water, ripples move outward in concentric circles. These circles intersect in points which form a curve known as the hyperbola.

8.Cooling Towers of Nuclear Reactors

The hyperboloid is the design standard for all nuclear
cooling towers. It is structurally sound and can be built with straight steel beams.

When designing these cooling towers, engineers are faced with two problems:
(1) the structutre must be able to withstand high winds and
(2) they should be built with as little material as possible.

The hyperbolic form solves both of these problems. For a given diameter and height of a tower and a given strength, this shape requires less material than any other form. A 500 foot tower can be made of a reinforced concrete shell only six or eight inches wide. See the pictures below (this nuclear power plant is located in Indiana).

9.Sonic Boom

In 1953, a pilot flew over an Air Force Base flying faster than the speed of sound. He damaged every building on the base.

As the plane moves faster than the speed of sound, you get a cone-like wave. Where the cone intersects the ground, it is an hyperbola.

The sonic boom hits every point on that curve at the same time. No sound is heard outside the curve. The

hyperbola is known as the "Sonic Boom Curve." In the picture below, the sonic boom is "visible" due to the humidity.

10.Gear transmission

Two hyperboloids of revolution can provide gear transmission between two skew axes. The cogs of each gear are a set of generating straight lines.

11.Lampshade

A household lamp casts hyperbolic shadows on a wall. just like..

12.Dulles Airport

Dulles Airport, designed by Eero Saarinen, is in the shape of a hyperbolic paraboloid. The hyperbolic paraboloid is a three-dimensional curve that is a hyperbola in one cross-section, and a parabola in another cross section.

Properties of hyperbolas

If a line intersects one branch of a hyperbola at M and N and intersects the asymptotes at P and Q, then MN has the same midpoint as PQ.

The following are concurrent: (1) a circle passing through the hyperbola's foci and centered at the hyperbola's center; (2) either of the lines that are tangent to the hyperbola at the vertices; and (3) either of the asymptotes of the hyperbola.

The following are also concurrent: (1) the circle that is centered at the hyperbola's center and that passes through the hyperbola's vertices; (2) either directrix; and (3) either of the asymptotes.

The product of the distances from a point P to one of the asymptotes along a line parallel to the other asymptote, and to the second asymptote along a line parallel to the first asymptote, is independent of the location of point P on the hyperbola.If the hyperbola is written in canonical form $\frac{x^2}{a^2} - \frac{y^2}{b^2}=1$ then this product is $\frac{a^2+b^2}{4}$ .

The product of the perpendicular distances from a point P on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2}=1$ or on its conjugate hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2}=-1$ to the asymptotes is a constant independent of the location of P: specifically, $\frac{a^2b^2}{a^2+b^2}$ , which also equals $(b/e)^2$ where e is the eccentricity of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2}=1$ .

The product of the slopes of lines from a point on the hyperbola to the two vertices is independent of the location of the point.

A line segment between the two asymptotes and tangent to the hyperbola is bisected by the tangency point.

The area of a triangle two of whose sides lie on the asymptotes, and whose third side is tangent to the hyperbola, is independent of the location of the tangency point. Specifically, the area is ab, where a is the semi-major axis and b is the semi-minor axis.

The distance from either focus to either asymptote is b, the semi-minor axis; the nearest point to a focus on an asymptote lies at a distance from the center equal to a, the semi-major axis. Then using the Pythagorean theorem on the right triangle with these two segments as legs shows that $a^2 + b^2 =c^2$ , where c is the semi-focal length (the distance from a focus to the hyperbola's center).

Rectangular hyperbola with horizontal/vertical asymptotes (Cartesian coordinates)

Rectangular hyperbolas with the coordinate axes parallel to their asymptotes have the equation

$(x-h)(y-k) = m \, \, \,$ .

These are equilateral hyperbolas (eccentricity $\varepsilon = \sqrt 2$ ) with semi-major axis and semi-minor axis given by $\text{[math]}$ .
The simplest example of rectangular hyperbolas occurs when the center (h, k) is at the origin:

$y=\frac{m}{x}\,$

describing quantities x and y that are inversely proportional. By rotating the coordinate axes counterclockwise by 45 degrees, with the new coordinate axes labelled $(x',y')$ the equation of the hyperbola is given by canonical form

$\frac{(x')^2}{(\sqrt{2m})^2}-\frac{(y')^2}{(\sqrt{2m})^2}=1$ .

Hyperbola in Elliptic coordinates

A family of confocal hyperbolas is the basis of the system of elliptic coordinates in two dimensions. These hyperbolas are described by the equation

$\left(\frac{x}{c \cos\theta}\right)^2 - \left(\frac{y}{c \sin\theta}\right)^2 = 1$

where the foci are located at a distance c from the origin on the x-axis, and where θ is the angle of the asymptotes with the x-axis. Every hyperbola in this family is orthogonal to every ellipse that shares the same foci. This orthogonality may be shown by a conformal map of the Cartesian coordinate system w = z + 1/z, where z= x + iy are the original Cartesian coordinates, and w=u + iv are those after the transformation.

Other orthogonal two-dimensional coordinate systems involving hyperbolas may be obtained by other conformal mappings. For example, the mapping w = z² transforms the Cartesian coordinate system into two families of orthogonal hyperbolas

Parametric equations of Hyperbola

East-west opening hyperbola:

$\begin{matrix} x = a\sec t + h \\ y = b\tan t + k \\ \end{matrix} \qquad \mathrm{or} \qquad\begin{matrix} x = \pm a\cosh t + h \\ y = b\sinh t + k \\ \end{matrix}$

North-south opening hyperbola:

$\begin{matrix} x = a\tan t + h \\ y = b\sec t + k \\ \end{matrix} \qquad \mathrm{or} \qquad\begin{matrix} x = a\sinh t + h \\ y = \pm b\cosh t + k \\ \end{matrix}$

In all formulae (h,k) are the center coordinates of the hyperbola, a is the length of the semi-major axis, and b is the length of the semi-minor axis.

Hyperbola in Polar coordinates

The polar coordinates used most commonly for the hyperbola are defined relative to the Cartesian coordinate system that has its origin in a focus and its x-axis pointing towards the origin of the "canonical coordinate system" as illustrated in the figure of the section "True anomaly".

Relative to this coordinate system one has that

$r = \frac{a(e^2-1)}{1+e\cos \theta}$

and the range of the true anomaly $\theta$ is:

$-\arccos {\left(-\frac{1}{e}\right)} < \theta < \arccos {\left(-\frac{1}{e}\right)}$

With polar coordinate relative to the "canonical coordinate system"

$x = R\, \cos t$

$y = R\, \sin t$

one has that

$R^2 =\frac{b^2}{e^2 \cos^2 t -1} \,$

For the right branch of the hyperbola the range of $t$ is:

$-\arccos {\left(\frac{1}{e}\right)} < t < \arccos {\left(\frac{1}{e}\right)}$

Hyperbola in Cartesian coordinates

An east-west opening hyperbola centered at (h,k) has the equation

$\frac{\left( x-h \right)^2}{a^2} - \frac{\left( y-k \right)^2}{b^2} = 1.$

The major axis runs through the center of the hyperbola and intersects both arms of the hyperbola at the vertices (bend points) of the arms. The foci lie on the extension of the major axis of the hyperbola.

The minor axis runs through the center of the hyperbola and is perpendicular to the major axis.

In both formulas a is the semi-major axis (half the distance between the two arms of the hyperbola measured along the major axis), and b is the semi-minor axis (half the distance between the asymptotes along a line tangent to the hyperbola at a vertex).

If one forms a rectangle with vertices on the asymptotes and two sides that are tangent to the hyperbola, the sides tangent to the hyperbola are 2b in length while the sides that run parallel to the line between the foci (the major axis) are 2a in length. Note that b may be larger than a despite the names minor and major.

If one calculates the distance from any point on the hyperbola to each focus, the absolute value of the difference of those two distances is always 2a.

The eccentricity is given by

$\varepsilon = \sqrt{1+\frac{b^2}{a^2}} = \sec\left(\arctan\left(\frac{b}{a}\right)\right) = \cosh\left(\operatorname{arsinh}\left(\frac{b}{a}\right)\right)$

If c equals the distance from the center to either focus, then

$\varepsilon = \frac{c}{a}$

where

$c = \sqrt{a^2 + b^2}$ .

The distance c is known as the linear eccentricity of the hyperbola. The distance between the foci is 2c or 2aε.

The foci for an east-west opening hyperbola are given by

$\left(h\pm c, k\right)$

and for a north-south opening hyperbola are given by

$\left( h, k\pm c\right)$ .

The directrices for an east-west opening hyperbola are given by

$x = h\pm a \; \cos\left(\arctan\left(\frac{b}{a}\right)\right)$

and for a north-south opening hyperbola are given by

$y = k\pm a \; \cos\left(\arctan\left(\frac{b}{a}\right)\right)$ .

Conic section analysis of the hyperbolic appearance of circles

Besides providing a uniform description of circles, ellipses, parabolas, and hyperbolas, conic sections can also be understood as a natural model of the geometry of perspective in the case where the scene being viewed consists of a circle, or more generally an ellipse. The viewer is typically a camera or the human eye. In the simplest case the viewer's lens is just a pinhole; the role of more complex lenses is merely to gather far more light while retaining as far as possible the simple pinhole geometry in which all rays of light from the scene pass through a single point. Once through the lens, the rays then spread out again, in air in the case of a camera, in the vitreous humor in the case of the eye, eventually distributing themselves over the film, imaging device, or retina, all of which come under the heading of image plane. The lens plane is a plane parallel to the image plane at the lens; all rays pass through a single point on the lens plane, namely the lens itself.

When the circle directly faces the viewer, the viewer's lens is on-axis, meaning on the line normal to the circle through its center (think of the axle of a wheel). The rays of light from the circle through the lens to the image plane then form a cone with circular cross section whose apex is the lens. The image plane concretely realizes the abstract cutting plane in the conic section model.

When in addition the viewer directly faces the circle, the circle is rendered faithfully on the image plane without perspective distortion, namely as a scaled-down circle. When the viewer turns attention or gaze away from the center of the circle the image plane then cuts the cone in an ellipse, parabola, or hyperbola depending on how far the viewer turns, corresponding exactly to what happens when the surface cutting the cone to form a conic section is rotated.

A parabola arises when the lens plane is tangent to (touches) the circle. A viewer with perfect 180-degree wide-angle vision will see the whole parabola; in practice this is impossible and only a finite portion of the parabola is captured on the film or retina.

When the viewer turns further so that the lens plane cuts the circle in two points, the shape on the image plane becomes that of a hyperbola. The viewer still sees only a finite curve, namely a portion of one branch of the hyperbola, and is unable to see the second branch at all, which corresponds to the portion of the circle behind the viewer, more precisely, on the same side of the lens plane as the viewer. In practice the finite extent of the image plane makes it impossible to see any portion of the circle near where it is cut by the lens plane. Further back however one could imagine rays from the portion of the circle well behind the viewer passing through the lens, were the viewer transparent. In this case the rays would pass through the image plane before the lens, yet another impracticality ensuring that no portion of the second branch could possibly be visible.

The tangents to the circle where it is cut by the lens plane constitute the asymptotes of the hyperbola. Were these tangents to be drawn in ink in the plane of the circle, the eye would perceive them as asymptotes to the visible branch. Whether they converge in front of or behind the viewer depends on whether the lens plane is in front of or behind the center of the circle respectively.

If the circle is drawn on the ground and the viewer gradually transfers gaze from straight down at the circle up towards the horizon, the lens plane eventually cuts the circle producing first a parabola then a hyperbola on the image plane. As the gaze continues to rise the asymptotes of the hyperbola, if realized concretely, appear coming in from left and right, swinging towards each other and converging at the horizon when the gaze is horizontal. Further elevation of the gaze into the sky then brings the point of convergence of the asymptotes towards the viewer.

By the same principle with which the back of the circle appears on the image plane were all the physical obstacles to its projection to be overcome, the portion of the two tangents behind the viewer appear on the image plane as an extension of the visible portion of the tangents in front of the viewer. Like the second branch this extension materializes in the sky rather than on the ground, with the horizon marking the boundary between the physically visible (scene in front) and invisible (scene behind), and the visible and invisible parts of the tangents combining in a single X shape. As the gaze is raised and lowered about the horizon, the X shape moves oppositely, lowering as the gaze is raised and vice versa but always with the visible portion being on the ground and stopping at the horizon, with the center of the X being on the horizon when the gaze is horizontal.

All of the above was for the case when the circle faces the viewer, with only the viewer's gaze varying. When the circle starts to face away from the viewer the viewer's lens is no longer on-axis. In this case the cross section of the cone is no longer a circle but an ellipse (never a parabola or hyperbola). However the principle of conic sections does not depend on the cross section of the cone being circular, and applies without modification to the case of eccentric cones.

It is not difficult to see that even in the off-axis case a circle can appear circular, namely when the image plane (and hence lens plane) is parallel to the plane of the circle. That is, to see a circle as a circle when viewing it obliquely, look not at the circle itself but at the plane in which it lies. From this it can be seen that when viewing a plane filled with many circles, all of them will appear circular simultaneously when the plane is looked at directly.

A common misperception about the hyperbola is that it is a mathematical curve rarely if ever encountered in daily life. The reality is that one sees a hyperbola whenever catching sight of portion of a circle cut by one's lens plane (and a parabola when the lens plane is tangent to, i.e. just touches, the circle). The inability to see very much of the arms of the visible branch, combined with the complete absence of the second branch, makes it virtually impossible for the human visual system to recognize the connection with hyperbolas such as y = 1/x where both branches are on display simultaneously.

Relation of Hyperbola to other conic sections

There are three major types of conic sections: hyperbolas, ellipses and parabolas. Since the parabola may be seen as a limiting case poised exactly between an ellipse and a hyperbola, there are effectively only two major types, ellipses and hyperbolas. These two types are related in that formulae for one type can often be applied to the other.

The canonical equation for a hyperbola is

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1.$

Any hyperbola can be rotated so that it is east-west opening and positioned with its center at the origin, so that the equation describing it is this canonical equation.
The canonical equation for the hyperbola may be seen as a version of the corresponding ellipse equation

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

in which the semi-minor axis length b is imaginary. That is, if in the ellipse equation b is replaced by ib where b is real, one obtains the hyperbola equation.

Similarly, the parametric equations for a hyperbola and an ellipse are expressed in terms of hyperbolic and trigonometric functions, respectively, which are again related by an imaginary number, e.g.,

$\cosh \mu = \cos i\mu$

Hence, many formulae for the ellipse can be extended to hyperbolas by adding the imaginary unit i in front of the semi-minor axis b and the angle. For example, the arc length of a segment of an ellipse can be determined using an incomplete elliptic integral of the second kind. The corresponding arclength of a hyperbola is given by the same function with imaginary parameters b and μ, namely, ib E(iμ, c).

Hyperbolic functions and equations

Just as the sine and cosine functions give a parametric equation for the ellipse, so the hyperbolic sine and hyperbolic cosine give a parametric equation for the hyperbola.

As
$\cosh^2 \mu - \sinh^2 \mu= 1$
one has for any value of $\mu$ that the point

$x = a\ \cosh\ \mu$

$y = b\ \sinh\ \mu$

satisfies the equation

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

which is the equation of a hyperbola relative its canonical coordinate system.

When μ varies over the interval $-\infty < \mu < \infty$ one gets with this formula all points $(x\ ,\ y)$ on the right branch of the hyperbola.
The left branch for which $x < 0$ is in the same way obtained as