Ellipse

The curve drawing algorithms of Xiaolin Wu (SIGGRAPH 91) are an example. Ragg is a sturdy wool fibre made into yarn and used in many rugged applications like gloves. A more challenging task is to perform these drawing operations with antialiasing, to create a smooth-looking curve. This process was invented in the Heavy Woollen District of West Yorkshire and created a micro-economy in this area for many years. 1984). Such yarns are typically used as weft yarns with a cotton warp. An efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken (IEEE CG&A, Sept. The recycled wool may be mixed with raw wool, wool noil, or another fibre such as cotton to increase the average fibre length.

Jack Bresenham at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. As this process makes the wool fibres shorter, the remanufactured fabric is inferior to the original. Often such libraries are limited and can only draw an ellipse with either the major axis or the minor axis horizontal. To make shoddy, existing wool fabric is cut or torn apart and respun. Drawing an ellipse is a common graphics primitive in standard display libraries, such as the QuickDraw and GDI interfaces on the Macintosh and Windows systems. Shoddy is recycled or remanufactured wool. Einstein's contributions to modern physics may not have been discovered if it were not for ellipses. Ancient Greeks lined their helmets with felt and Roman legionnaires used breastplates made of wool felt.

Albert Einstein also used the ellipse to prove his theory of relativity by using an elliptical shaped mass. Wool felt covers piano hammers and it is used to absorb odors and noise in heavy machinery and stereo speakers. The general solution for a harmonic oscillator in two or more dimensions is also an ellipse, but this time with the origin of the force located at the center of the ellipse. In addition to clothing, wool has been used for carpeting, felt, and upholstery. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus. (source). More generally, in the gravitational two-body problem, if the two bodies are bound to each other (i.e., the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. These small scale farmers may raise any type of sheep they wish, so the selection of fleeces is quite wide.

Later, Isaac Newton explained this as a corollary of his law of universal gravitation. There is also a thriving 'home flock' contingent of small scale farmers who raise small hobby flocks of specialty sheep for the handspinning market. In the 17th century, Johannes Kepler explained that the orbits along which the planets travel around the Sun are ellipses, which is Kepler's first law. In the United States, Texas, New Mexico and Colorado also have large commercial sheep flocks and their mainstay is the Rambouillet (or French Merino). Indian astronomer Aryabhata discovered that the orbits of the planets around the sun are ellipses in 499, which he described in his book, the Aryabhatiya [1]. Breeds such as Lincoln and Romney produce coarser fibres and wool of these sheep is usually used for making carpets. Since no other curve has such a property, it can be used as an alternative definition of an ellipse. Most Australian wool comes from the merino breed.

Then all rays are reflected to a single point — the second focus. Australia, China and New Zealand are leading commercial producers of wool. Assume an elliptic mirror with a light source at one of the foci. Global wool production is approximately 1.3 million tonnes per annum of which 60% goes into apparel. If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse. Australia's colonial economy was based on sheep raising and the Australian wool trade eventually overtook that of the Germans by 1845. Similarly, any oblique projection onto a plane results in a conic section. German wool—based on sheep of Spanish origin—did not overtake British wool until comparatively late.

The stretched ellipse will have different properties (perhaps changed eccentricity and semi-major axis length, for instance), but it will still be an ellipse (or a degenerate ellipse: a circle or a line). Spain allowed export of Merino lambs only with royal permission. An ellipse may be uniformly stretched along any axis, in or out of the plane of the ellipse, and it will still be an ellipse. In the Renaissance, Medicis of Florence built their wealth and banking system on wool trade with the aid of the Arte della Lana, the wool guild. The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions. In 1699 English crown forbade its American colonies to trade wool with anyone else but England itself. More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral. The smuggling of wool out of the country, known as owling, was at one time punishable by the cutting off of a hand.

which can also be written as:. Over the centuries, various British laws controlled the wool trade or required the use of wool even in burials. A good approximation is Ramanujan's:. English wool exports - which bordered on European monopoly - were a significant source of income to the crown. The exact infinite series is:. In medieval times, the wool trade was serious business. The circumference of an ellipse is 4aE(e), where the function E is the complete elliptic integral of the second kind. Prior to invention of shears - probably in the Iron Age - they probably plucked the wool out by hand or by bronze combs.

The area enclosed by an ellipse is , where π is Archimedes' constant. As the raw material has been readily available since the widespread domestication of sheep and similar animals, the use of wool for clothing and other fabrics dates back to some of the earliest civilizations. An ellipse can also be thought of as a projection of a circle: a circle on a plane at angle φ to the horizontal projected vertically onto a horizontal plane gives an ellipse of eccentricity sin φ, provided φ is not 90°. . In polar coordinates, an ellipse with one focus at the origin and the other on the negative x-axis is given by the equation. The finer diameters are generally more valuable. It is related to and (the ellipse's semi-axes) by the formula or, if using the eccentricity, . The fibre diameter of wool varies from 15 micrometres (superfine merino) to 30 or more micrometres for the coarser wools.

The semi-latus rectum of an ellipse, usually denoted (lowercase L), is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis. The quality of fleece is determined by a technique known as wool classing, whereby a qualified woolclasser tries to group wools of similar gradings together to maximise the return for the farmer or sheep owner. The distance between the foci is 2ae. The latter four are packaged and sold separately. The ellipse shown in the image below has an eccentricity of approximately 0.8733. After shearing, the wool is separated into five main categories: fleece (which makes up the vast bulk), pieces, bellies, crutchings and locks. The greater the eccentricity is, the larger the ratio of a to b is, and therefore the more elongated the ellipse is. The grease is generally removed for processing by scouring with detergent and alkali.

The eccentricity is a positive number less than 1, or 0 in the case of a circle. In this state it can be worked into yarn or knitted into water-resistant mittens, such as those of the Aran Island fishermen. or where c (the linear eccentricity of the ellipse) equals the distance from the center to either focus. Wool straight off a sheep contains a high level of grease (thus "greasy wool") which contains valuable lanolin. The eccentricity is related to a and b by the statement. Wool is generally a creamy white colour, although some breeds of sheep produce natural colors such as black, brown (also called moorit) and grey. The shape of an ellipse is usually expressed by a number called the eccentricity of the ellipse, conventionally denoted e (not to be confused with the mathematical constant e). The relative amounts of kemp to wool vary from breed to breed, and make some fleeces more desirable for spinning, felting or carding into batts for quilts or other insulating products.

has normal (cosφ,sinφ). On sheep, the hair part of the fleece is called kemp. A Gauss-mapped form:. Hair, by contrast, has little if any scale and no crimp and little ability to bind into yarn. where (h,k) is the center. A fine wool like merino may have up to a hundred crimps per inch, where the coarser wools like karakul may have as few as one to two crimps per inch. If an ellipse is not centered at the origin of an x-y coordinate system, but again has its major axis along the x-axis, it may be specified by the equation. The amount of crimp corresponds with the fineness of the wool fibres.

which use the trigonometric functions sine and cosine. Insulation also works both ways; bedouins and tuaregs use wool clothes to keep the heat out. The same ellipse is also represented by the parametric equations:. Because of the crimp, wool fabrics have a greater bulk than other textiles and retain air, which causes the product to retain heat. The following diagram shows an ellipse demonstrating the Pythagoras equation a² = b² + c² as a special case of the non-parametric equation above (x=0, y=b). They help the individual fibres attach to each other so that they stay together. The derivation of this formula is quite instructive and not overly difficult. Both the scaling and the crimp make it possible to spin and felt the fleece.

An ellipse centered at the origin of an x-y coordinate system with its major axis along the x-axis is defined by the equation. Most of the fibre from domestic sheep has two qualities that distinguish it from hair or fur: it has scales which overlap like shingles on a roof and it is crimped; in some fleeces the wool fibres have more than 20 bends per inch. The constant a equals the length of the semimajor axis; the constant b equals the length of the semiminor axis. Wool is the fibre produced as the outer coat of sheep. The size of an ellipse is determined by two constants, conventionally denoted a and b. This article deals with the wool produced from domestic sheep. . Wool is the fibre derived from the hair of animals of the Caprinae family, principally sheep and goats, but the hair of other mammals such as alpacas may also be called wool.

. United States: .77%. Then the axes of the ellipse will lie along the eigenvectors of A, and the squares of the lengths of the axes are the inverses of the eigenvalues. South Africa: 1%. An ellipse centred at the origin can be viewed as the image of the unit circle under a linear map associated with a symmetric matrix A = PDP^T, D being a diagonal matrix with the eigenvalues of A, both of which are real positive, along the main diagonal, and P being a real unitary matrix having as columns the eigenvectors of A. Sudan: 2%. If the two foci coincide, then the ellipse is a circle; in other words, a circle is a special case of an ellipse, one where the eccentricity is zero. India: 2%.

Likewise, the semiminor axis is one half the minor axis. United Kingdom: 2%. A semimajor axis is one half the major axis: the line segment from the center, through a focus, and to the edge of the ellipse. Iran: 2%. The line which passes through the center (halfway between the foci), at right angles to the major axis, is called the minor axis. Turkey: 2%. The major axis is along the longest segment that passes through the ellipse. Argentina: 3%.

The line segment which passes through the foci and terminates on the ellipse is called the major axis. New Zealand: 11%. If the pencil is moved around so that the string stays taut, the sum of the distances from the pencil to the pins will remain constant, satisfying the definition of an ellipse. China: 18%. The string will form a triangle. Australia: 25% of global wool clip (475 million kg greasy, 2004/2005). The pencil is placed on the paper inside the string, so the string is taut.

The pins are placed at the foci and the pins and pencil are enclosed inside the string. An ellipse can be drawn with two pins, a loop of string, and a pencil. such that B² < 4AC, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the ellipse, exists. Algebraically, an ellipse is a curve in the Cartesian plane defined by an equation of the form.

For a short elementary proof of this, see Dandelin spheres. An ellipse is a type of conic section: if a conical surface is cut with a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse. The two fixed points are called foci (plural of focus). In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant.

Elliptical redirects here, for the exercise machine, see Elliptical trainer..