Parabolae in the physical world

 

In nature, approximations of parabola and paraboloids (such as catenate curves) are found in many diverse situations. The best-known instance of the parabola in the history of physics is the trajectory of a particle or body in motion under the influence of a uniform gravitational field without air resistance (for instance, a baseball flying through the air, neglecting air friction).

The parabolic trajectory of projectiles was discovered experimentally by Galileo in the early 17th century, who performed experiments with balls rolling on inclined planes. He also later proved this mathematically in his book Dialogue Concerning Two New Sciences.For objects extended in space, such as a diver jumping from a diving board, the object itself follows a complex motion as it rotates, but the center of mass of the object nevertheless forms a parabola. As in all cases in the physical world, the trajectory is always an approximation of a parabola. The presence of air resistance, for example, always distorts the shape, although at low speeds, the shape is a good approximation of a parabola. At higher speeds, such as in ballistics, the shape is highly distorted and does not resemble a parabola.

Another situation in which parabola may arise in nature is in two-body orbits, for example, of a small planetoid or other object under the influence of the gravitation of the sun. Such parabolic orbits are a special case that are rarely found in nature. Orbits that form a hyperbola or an ellipse are much more common. In fact, the parabolic orbit is the borderline case between those two types of orbit. An object following a parabolic orbit moves at the exact escape velocity of the object it is orbiting, while elliptical orbits are slower and hyperbolic orbits are faster.

Approximations of parabola are also found in the shape of the main cables on a simple suspension bridge. The curve of the chains of a suspension bridge is always an intermediate curve between a parabola and a catenate, but in practice the curve is generally nearer to a parabola, and in calculations the second degree parabola is used.[9][10] Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise hyperbolic cable is deformed toward a parabola. Unlike an inelastic chain, a freely hanging spring of zero unstressed length takes the shape of a parabola.

Paraboloids arise in several physical situations as well. The best-known instance is the parabolic reflector, which is a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point, or conversely, collimates light from a point source at the focus into a parallel beam. The principle of the parabolic reflector may have been discovered in the 3rd century BC by the geometer Archimedes, who, according to a legend of debatable veracity, constructed parabolic mirrors to defend Syracuse against the Roman fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied to telescopes in the 17th century. Today, paraboloid reflectors can be commonly observed throughout much of the world in microwave and satellite dish receiving and transmitting antennas.

It can be seen in architecture as well. For instance the Oval Office of the The White House is essentially two Parabolas facing one another and as such an interesting effect happens. Two people, each standing at one of the focal points 21 feet apart, can whisper secretly to one another despite entertaining the company of others.

Paraboloids are also observed in the surface of a liquid confined to a container and rotated around the central axis. In this case, the centrifugal force causes the liquid to climb the walls of the container, forming a parabolic surface. This is the principle behind the liquid mirror telescope.

Aircraft used to create a weightless state for purposes of experimentation, such as NASA's “Vomit Comet,” follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall, which produces the same effect as zero gravity for most purposes.

Vertical curves in roads are usually parabolic by design.

Circle and its part

Actually the circle is the one kind of line means the type of line and also the set of points which are equidistant to any fix point of plane. This constant distance called the Radius of circle and that fix point are called the Center of circle.

The radius of a circle is the length of the line from the center to any point circle on its edge. The radius of circle is always constant. The plural form of the radius is - radii (pronounced "ray-dee-eye"). Sometimes the word 'radius' is used to refer to the line itself. In that sense you may see "draw a radius of the circle". In the more recent sense, it is the length of the line, and so is referred to as "the radius of the circle is 2 centimeters"

(1) DIAMETER

               when D = diameter of the circle

Radius is simply half the diameter.

(2) CIRCUMFERENCE

 

when C = circumference of the circle

         D = diameter of the circle

The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d.

(3) AREA ENCLOSED

             when A = the area of the circle

As proved by Archimedes, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius,which comes to π multiplied by the radius squared:

Equivalently, denoting diameter by d,

that is, approximately 79 percent of the circumscribing square (whose side is of length d).

The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality.

The history of Circle

The word "circle" derives from the Greek, kirkos "a circle," from the base ker- which means to turn or bend. The origins of the words "circus" and "circuit" are closely related.

 

The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern civilisation possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy, and calculus.

 

Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.

Length of an arc of a parabola

If a point X is located on a parabola which has focal length and if is the perpendicular distance from X to the axis of symmetry of the parabola, then the lengths of arcs of the parabola which terminate at X can be calculated from and as follows, assuming they are all expressed in the same units.

This quantity, , is the length of the arc between X and the vertex of the parabola.
The length of the arc between X and the symmetrically opposite point on the other side of the parabola is
The perpendicular distance, , can be given a positive or negative sign to indicate on which side of the axis of symmetry X is situated. Reversing the sign of reverses the signs of and without changing their absolute values. If these quantities are signed, the length of the arc between any two points on the parabola is always shown by the difference between their values of

This can be useful, for example, in calculating the size of the material needed to make a parabolic reflector or parabolic trough.

(Note: In the above calculation, the square-root, , must be positive. The quantity ln(a), sometimes written as log_e(a), is the natural logarithm of a, i.e. its logarithm to base "e".)

Mathematical definitions and properties

In Euclidean geometry

Definition

In Euclidean geometry, the ellipse is usually defined as the bounded case of a conic section, or as the set of points such that the sum of the distances to two fixed points (the foci) is constant. The ellipse can also be defined as the set of points such that the distance from any point in that set to a given point in the plane (a focus) is a constant positive fraction less than 1 (the eccentricity) of the perpendicular distance of the point in the set to a given line (called the directrix). Yet another equivalent definition of the ellipse is that it is the set of points that are equidistant from one point in the plane (a focus) and a particular circle, the directrix circle (whose center is the other focus).

The equivalence of these definitions can be proved using the Dandelin spheres

Equations.

The equation of an ellipse whose major and minor axes coincide with the Cartesian axes is
This means any noncircular ellipse is a squashed circle. If we draw an ellipse twice as long as it is wide, and draw the circle centered at the ellipse's center with diameter equal to the ellipse's longer axis, then on any line parallel to the shorter axis the length within the circle is twice the length within the ellipse. So the area enclosed by an ellipse is easy to calculate-- it's the lengths of elliptic arcs that are hard.

Focus

The distance from the center C to either focus is f = ae, which can be expressed in terms of the major and minor radii:

 

Eccentricity

 

The eccentricity of the ellipse (commonly denoted as either e or ) is

(where again a and b are one-half of the ellipse's major and minor axes respectively, and f is the focal distance) or, as expressed in terms using the flattening factor

Other formulas for the eccentricity of an ellipse are listed in the article on eccentricity of conic sections. Formulas for the eccentricity of an ellipse that is expressed in the more general quadratic form are described in the article dedicated to conic sections.

Directrix

Each focus F of the ellipse is associated with a line parallel to the minor axis called a directrix. Refer to the illustration on the right. The distance from any point P on the ellipse to the focus F is a constant fraction of that point's perpendicular distance to the directrix resulting in the equality, e=PF/PD. The ratio of these two distances is the eccentricity of the ellipse. This property (which can be proved using the Dandelin spheres) can be taken as another definition of the ellipse.
Besides the well–known ratio e=f/a, it is also true that e=a/d.

Circular directrix

The ellipse can also be defined as the set of points that are equidistant from one focus and a circle, the directrix circle, that is centered on the other focus. The radius of the directrix circle equals the ellipse's major axis, so the focus and the entire ellipse are inside the directrix circle.

Ellipse as hypotrochoid

The ellipse is a special case of the hypotrochoid when R = 2r.

Area

The area enclosed by an ellipse is πab, where a and b are one-half of the ellipse's major and minor axes respectively.

If the ellipse is given by the implicit equation , then the area is .

An ellipse (in red) as a special case of the hypotrochoid with R = 2r.



Drawing ellipses

Pins-and-string method

The characterization of an ellipse as the locus of points so that sum of the distances to the foci is constant leads to a method of drawing one using two drawing pins, a length of string, and a pencil. In this method, pins are pushed into the paper at two points which will become the ellipse's foci. A string tied at each end to the two pins and the tip of a pen is used to pull the loop taut so as to form a triangle. The tip of the pen will then trace an ellipse if it is moved while keeping the string taut. Using two pegs and a rope, this procedure is traditionally used by gardeners to outline an elliptical flower bed; thus it is called the gardener's ellipse.

Drawing an ellipse with two pins, a loop, and a pen. 

Trammel method

An ellipse can also be drawn using a ruler, a set square, and a pencil:

Draw two perpendicular lines M,N on the paper; these will be the major and minor axes of the ellipse. Mark three points A, B, C on the ruler. A->C being the length of the major axis and B->C the length of the minor axis. With one hand, move the ruler on the paper, turning and sliding it so as to keep point A always on line N, and B on line M. With the other hand, keep the pencil's tip on the paper, following point C of the ruler. The tip will trace out an ellipse.

The trammel of Archimedes or ellipsograph is a mechanical device that implements this principle. The ruler is replaced by a rod with a pencil holder (point C) at one end, and two adjustable side pins (points A and B) that slide into two perpendicular slots cut into a metal plate. The mechanism can be used with a router to cut ellipses from board material. The mechanism is also used in a toy called the "nothing grinder".

 

Trammel of Archimedes (ellipsograph) animation.

 

Parallelogram method

In the parallelogram method, an ellipse is constructed point by point using equally spaced points on two horizontal lines and equally spaced points on two vertical lines. Similar methods exist for the parabola and hyperbola.

Approximations to ellipses

An ellipse of low eccentricity can be represented reasonably accurately by a circle with its centre offset. To draw the orbit with a pair of compasses the centre of the circle should be offset from the focus by an amount equal to the eccentricity multiplied by the radius.

 

Ellipse construction applying the parallelogram method.

Elements of an ellipse

An ellipse is a smooth closed curve which is symmetric about its horizontal and vertical axes. The distance between antipodal points on the ellipse, or pairs of points whose midpoint is at the center of the ellipse, is maximum along the major axis or transverse diameter, and a minimum along the perpendicular minor axis or conjugate diameter.

The semi-major axis (denoted by a in the figure) and the semi-minor axis (denoted by b in the figure) are one half of the major and minor axes, respectively. These are sometimes called (especially in technical fields) the major and minor semi-axes, the major and minor semiaxes, or major radius and minor radius. The four points where these axes cross the ellipse are the vertices, points where its curvature is minimized or maximized.

The ellipse and some of its mathematical properties.

The foci of the ellipse are two special points F₁ and F₂ on the ellipse's major axis and are equidistant from the center point. The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis ( PF₁ + PF₂ = 2a ). Each of these two points is called a focus of the ellipse.

Refer to the lower Directrix section of this article for a second equivalent construction of an ellipse.

The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of the distance between the two foci, to the length of the major axis or e = 2f/2a = f/a. For an ellipse the eccentricity is between 0 and 1 (0<e<1). When the eccentricity is 0 the foci coincide with the center point and the figure is a circle. As the eccentricity tends toward 1, the ellipse gets a more elongated shape. It tends towards a line segment (see below) if the two foci remain a finite distance apart and a parabola if one focus is kept fixed as the other is allowed to move arbitrarily far away.

The distance ae from a focal point to the centre is called the linear eccentricity of the ellipse (f = ae).