Ellipse

The curve drawing algorithms of Xiaolin Wu (SIGGRAPH 91) are an example. DVD release date for the sixth season has yet to be announced, but is expected to arrive in 2006. A more challenging task is to perform these drawing operations with antialiasing, to create a smooth-looking curve. Original release or broadcast dates are listed in parentheses. 1984). All titles and information listed below are based on Region 1 DVD releases. An efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken (IEEE CG&A, Sept. Original broadcast or release dates and episode titles (where applicable) are listed in parentheses.

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. Most of these titles are out of print. Often such libraries are limited and can only draw an ellipse with either the major axis or the minor axis horizontal. The Flintstones remains the first cartoon to have original programming aired in prime time. 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. However, it was a repeat of cartoons that had aired on Sunday afternoons in 1956. Einstein's contributions to modern physics may not have been discovered if it were not for ellipses. This half-hour show ran for a few months on Friday nights on CBS in 1958.

Albert Einstein also used the ellipse to prove his theory of relativity by using an elliptical shaped mass. While The Flintstones is generally considered the first cartoon to air in prime time, it was preceded by The Gerald McBoing-Boing Show, a cartoon based on an Academy Award-winning animated short. 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. "Weird Al" Yankovic released a single and video entitled "Bedrock Anthem" from his album Alapalooza featuring many clips of The Flintstones animated TV show on the video. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus. The England cricketer Andrew Flintoff is nicknamed "Freddie" after Fred Flintstone, on account of both his similar surname and his "larger-than-life" character. 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. The Flintstones' car was removed to make room for Betty.

Later, Isaac Newton explained this as a corollary of his law of universal gravitation. However, since 1996, Betty has been in the bottle also. 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. The answer, at one time, was Betty Rubble. 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]. There has been a "Did You Know?" quiz circulating on the Internet for a number of years that asks which of the four main characters is not in Flintstone Vitamins. Since no other curve has such a property, it can be used as an alternative definition of an ellipse. An enduring license has been a line of children's multivitamins called "Flintstones Complete" (more popularly known as Flintstones Vitamins); the first seasons of the series were, in part, sponsored by One-a-Day Vitamins.

Then all rays are reflected to a single point — the second focus. Early ads used the closing tagline, "Yabba-dabba-delicious!", but the phrase was removed during the 1990s. Assume an elliptic mirror with a light source at one of the foci. Eventually, Fred catches on and gives chase after Barney as the ad ends. If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse. Commercials for these cereals featured a gimmick in the form of Barney (who seems much smarter in the commercials than in the show itself) tricking Fred out of his cereal, usually by way of disguising himself as something completely different. Similarly, any oblique projection onto a plane results in a conic section. The series spawned three breakfast cereals: the popular Fruity Pebbles and Cocoa Pebbles, and the discontinued Dino Pebbles.

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). The later series, The Simpsons, carries on a number of Flintstones traditions, including the incorporation of music into its storylines. 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 1996, Rhino Records released The Flintstones: Modern Stone Age Melodies, an album containing a number of musical selections taken from the series.[1] Missing from the collection are the two Ann-Margret songs, which have been released on the Bear Family Records label of Germany. The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions. One of these songs, "Meet the Flintstones", was later adopted as the series' theme song beginning with the third season. 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. During 1961, the cast members recorded an album of songs, in character, aimed at children.

which can also be written as:. The aforementioned "Littlest Lamb" also became a popular lullaby. A good approximation is Ramanujan's:. Two other songs became standards on their own and are not always identified as originating with the Flintstones -- the seemingly endless singalong "Happy Anniversary" which is often performed at anniversary parties, and the spiritual "Open Up Your Heart and Let the Sunshine In", the latter song being adopted as the series closing theme during the final season. The exact infinite series is:. One of the most fondly remembered songs of the series was "The Bedrock Twitch", performed by staff voice actor Daws Butler and later performed in one of the first live-action Flintstones film by The B-52s. The circumference of an ellipse is 4aE(e), where the function E is the complete elliptic integral of the second kind. In the final season, space-rockers, The Wayouts appeared.

The area enclosed by an ellipse is , where π is Archimedes' constant. James Darren, appearing as "Jimmy Darrock" performed "The Surfin' Craze", while The Beau Brummels performed "Laugh Laugh" which was a real-life hit for the group. 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°. Ann-Margret, appearing in a fourth season episode, performed her single "I Ain't Goin' to Be Your Fool No More" and the lullaby "The Littlest Lamb". In polar coordinates, an ellipse with one focus at the origin and the other on the negative x-axis is given by the equation. Hoagy Carmichael was the one of the first, introducing the original song "Yabba Dabba Doo!" in the second season premiere episode, "The Hit Songwriters" (in the same episode, Fred - on one of his tone deaf days - mangles Carmichael's "Stardust"). It is related to and (the ellipse's semi-axes) by the formula or, if using the eccentricity, . Many musical moments were provided by celebrity voice artists who lent both their vocal talents and their likenesses to characters.

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. Many of the original songs in the series were composed by Hoyt Curtin. The distance between the foci is 2ae. Mel Blanc also performed a few vocals, including a version of "Old Folks at Home", while Jean Van Der Pyl and Bea Benadaret sang two versions of "The Car Hop Song," one performing as a pair of young girls auditioning for a job with Fred and Barney's new hamburger stand, and a second version in character as Wilma and Betty. The ellipse shown in the image below has an eccentricity of approximately 0.8733. Alan Reed himself sang several tunes in his loud tone in the coming years, including "Christmas is My Favorite Time of Year" and "Dino the Dinosaur" in the series' often-replayed Christmas episode. The greater the eccentricity is, the larger the ratio of a to b is, and therefore the more elongated the ellipse is. Fred was from then on depicted as being tone deaf, but a couple times sang a few tunes that had limited vocal range.

The eccentricity is a positive number less than 1, or 0 in the case of a circle. In the second season, Fred was stripped of his singing ability (in the season premiere for season 2 "Hit Song Writers" Fred could not carry a tune when he attempted to sing "Stardust"). or where c (the linear eccentricity of the ellipse) equals the distance from the center to either focus. Also in the first season while Alan mostly created Fred's speaking voice in an unnaturally loud tone, on occasion he used his natural tone to voice Fred (episode 5 "Split Personality" for example). The eccentricity is related to a and b by the statement. Again, Fred's voice sounds so much like Torme's that it was speculated the singer provided the voice, although it is actually Reed singing these two songs. 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). A later episode, in which Fred takes on the persona of teen idol "Hye Fye" sees him performing "Listen to the Rockin' Bird" - a rewrite of the standard "Listen to the Mockingbird".

has normal (cosφ,sinφ). One of the first song performances in the series was the old spiritual "When the Saints Go Marching In" performed by Fred in the first season episode "Hot Lips Hannigan" in a vocal style strongly reminiscent of jazz crooner Mel Torme. A Gauss-mapped form:. In the first season Fred knew how to sing. where (h,k) is the center. The Flintstones was one of the more musical animated TV series, with many episodes featuring original, slightly rewritten, or actual popular recordings of the day, performed either by Fred, Barney, or a special guest star. 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. One episode's couch gag even featured the Simpsons running in and finding The Flintstones (as they were originally animated) sitting on the Simpsons' couch.

which use the trigonometric functions sine and cosine. Burns gives him a box of chocolates. The same ellipse is also represented by the parametric equations:. Homer subsequently says "Yabba Dabba Doo!" when Mr. 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). Burns greeting Homer, Marge and Maggie as Fred, Wilma and Pebbles. The derivation of this formula is quite instructive and not overly difficult. Another episode ("Lady Bouvier's Lover") has Mr.

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. In addition, the character Barney Gumble from The Simpsons is based on Barney Rubble. The constant a equals the length of the semimajor axis; the constant b equals the length of the semiminor axis. the Monorail") the starting sequence parodies the opening and theme song of The Flintstones, and in another episode ("Treehouse of Horror XII") Homer and Marge Simpson dress as Fred and Wilma Flintstone in the opening. The size of an ellipse is determined by two constants, conventionally denoted a and b. A number of episodes of The Simpsons made explicit or implicit references to The Flintstones---for example, in one episode of The Simpsons ("Marge vs. . And it was The Simpsons in 1997 that ultimately broke The Flintstones' record as the longest-running prime time animated series.

. Only the advent of The Simpsons decades later brought cartoons back to American prime time network television with the kind of success The Flintstones enjoyed. 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. The original show also was adapted into two feature non-animated films, in 1994 and 2000. 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. The show was revived in the 1970s with Pebbles and Bamm Bamm having grown into teenagers, and several different series and made-for-TV movies—including a series depicting Fred and Barney as police officers, another depicting the characters as children, and yet others featuring Fred and Barney encountering Marvel Comics superhero The Thing and comic strip character The Shmoo have appeared over the years. 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. The movie was released on DVD in North America in March 2005.

Likewise, the semiminor axis is one half the minor axis. The Man Called Flintstone was a musical spy caper that parodied James Bond and other secret agents. 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. Following the show's cancellation in 1966, a theatrical film based upon the series was released. The line which passes through the center (halfway between the foci), at right angles to the major axis, is called the minor axis. The remaining two seasons are expected to be released in 2006. The major axis is along the longest segment that passes through the ellipse. The first season of the original series, with the original opening credits, as well as "Rise and Shine" restored but not the cigarette ads, was released on DVD in late 2003; season 2 was released in December 2004; season 3 in March 2005; and season 4 in November 2005.

The line segment which passes through the foci and terminates on the ellipse is called the major axis. Nonetheless, a number of later Flintstones episodes in syndication used an alternate version of the closing credits in which Pebbles and Bamm-Bamm are shown singing "Open Up Your Heart and Let the Sunshine In". 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. According to information provided on the DVD release of the second season, this decision was made because at the time syndicated programs were often aired out of their original broadcast order, and it was felt having the show jump between the different opening credits sequences would confuse audiences. The string will form a triangle. New syndicated versions of the episodes in the 1990s restored the original first season credits and theme, albeit with cigarette and other advertising matter omitted. The pencil is placed on the paper inside the string, so the string is taut. As a result, the closing credits for all first season episodes in syndication were incorrect for many years.

The pins are placed at the foci and the pins and pencil are enclosed inside the string. The theme used for the first and second seasons, an instrumental called "Rise and Shine", was removed from all first and second season episodes in syndication from the 1960s through the early 1990s and replaced with the "Meet the Flintstones" opening, while a closing credits sequence taken from a later episode was substituted at the end. An ellipse can be drawn with two pins, a loop of string, and a pencil. The famous theme song "Meet the Flintstones" was not actually introduced until the third season (1962–1963), although early versions of the melody can be heard as background music in many episodes. 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. The series was initially aimed at adult audiences as the first season was sponsored by the cigarette company Winston and the characters appeared in several commercials for Winstons. Algebraically, an ellipse is a curve in the Cartesian plane defined by an equation of the form. Another story arc, occurring in the final season, centered on Fred and Barney's dealings with The Great Gazoo.

For a short elementary proof of this, see Dandelin spheres. A postscript to the arc occurred in the third episode of the fourth season, in which the Rubbles, depressed over being unable to have children of their own (making The Flintstones the first animated series in history to address the issue of infertility, though subtly), adopt Bamm-Bamm. 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. Beginning with the episode "The Surprise", aired midway through the third season, in which Wilma reveals her pregnancy to Fred, the arc continued through the trials and tribulations leading up to Pebbles' birth, and then continued with several episodes showing Fred and Wilma adjusting to the world of parenthood. The two fixed points are called foci (plural of focus). The most notable example was a series of episodes surrounding the birth of Pebbles. 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. Although most Flintstones episodes are standalone storylines, the series was significant in being the first American animated series to feature story arcs.

Elliptical redirects here, for the exercise machine, see Elliptical trainer.. Aside from the animation and fantasy setting, the show's scripts and format are typical of a 1950s American situation comedy, with the usual family issues resolved with a laugh at the end of each episode. After spending a brief period in development as The Gladstones, Hanna-Barbera settled upon The Flintstones. When the series itself was commissioned, the title was changed, possibly to avoid confusion with the Flagstons, characters in the popular comic strip, Hi and Lois. Originally, the series was to have been titled The Flagstones, and a brief demonstration film was created to sell the idea of a "modern stone age family" to sponsors and the network.

He also (as does Fred in this series) cries out for his wife, by asking her to stop the mechanism with the line, "Jane! Stop this crazy thing!". This running gag of having the lead character of the series ending up being helpless during the end credits in every episode due to the hijinks of a family pet would later be repeated by Hanna-Barbera in the series The Jetsons in which George Jetson ends up being caught on a treadmill that ends up spinning out of control. In the show's closing credits, Fred tries to "put the cat out for the night" but winds up getting locked out and yelling for his wife to come open the door: "Wilma! Come on, Wilma, open this door! Willllll-ma!" By the time the theme song "Meet the Flintstones" was used, Fred cut the yelling to: "Willllll-ma!" Although the cat, Baby Puss, was seen in the closing credits of every episode, it was rarely actually seen in any of the storylines. The similarities with The Honeymooners included the fact that Reed based Fred's voice upon Jackie Gleason's interpretation of Ralph Kramden, while Blanc, after a season of using a nasal, high-pitched voice for Barney, eventually adopted a style of voice similar to that used by Art Carney in his portrayal of Ed Norton.

The voice of Barney was provided by legendary voice actor Mel Blanc, though five episodes in the second season used actor Daws Butler while Blanc was recovering from a near-fatal car accident. It has been noted that Fred Flintstone physically resembled voice actor Alan Reed. In later seasons, the Flintstones cast expanded to include The Gruesomes, their strange next-door neighbors (inspired by the then-popular monster sitcoms The Addams Family and The Munsters), and The Great Gazoo, an alien exiled to Earth who helps Fred and Barney, usually against their will. Slate.

Fred Flintstone worked at a stone quarry and worked for several different bosses, the best known of which was the bald Mr. The Flintstones had a pet dinosaur named Dino (pronounced DEE-no, and which barked like a dog), and the Rubbles had a kangaroo-like animal named Hoppy. Later additions to the cast included the Flintstones' infant daughter Pebbles Flintstone and the Rubbles' abnormally strong adopted son Bamm Bamm Rubble. The series directly drew from The Honeymooners for its main quartet of characters: the blustering Fred Flintstone and his ever-patient wife Wilma Flintstone (née Slaghoople, though Pebble was also given on occasion) modeled after the Kramdens, and their friendly neighbors Barney Rubble and wife Betty Rubble (née Betty Jean McBricker) modeled after the Nortons.

Being set in the Stone Age allowed for endless gags and puns that involved rocks in one way or another, including the names of the various characters being "rock" puns; some such names included celebrities such as "Cary Granite", "Stony Curtis", and "Ann-Margrock.". Another commonly seen gadget in the series was a baby woolly mammoth being used as a vacuum cleaner. In a running gag, the animals powering such technology would look directly into the camera at the audience, shrug, and remark, "It's a living," or some similar phrase. For example, when the characters took photographs with an instant camera, the inside of the camera box would be shown to contain a bird carving the picture on a stone tablet with its bill.

One source of the show's humor was the ways animals were used for technology. Although the characters were set in the Stone Age, that never stopped the show's creators from producing a Christmas episode during the original series' run as well as several more Christmas specials in the decades that followed. The characters drive automobiles made out of stone or wood and animal skins and powered by gasoline, although foot power is required to start the vehicles. The show is set in a town called Bedrock in the Stone Age era, but with a society identical to that of the United States in the mid-to-late 20th century; in the Flintstones' fantasy version of the prehistoric past, dinosaurs, saber-toothed tigers, woolly mammoths and other long extinct animals coexist with cavemen, who use technology equivalent to that of the 20th century, largely through the use of various animals.

. The Flintstones, an American animated television series created by Hanna-Barbera Cartoons, is one of the most successful animated television series of all time, originally running in American prime time for six seasons, from 1960 to 1966, on the ABC network. The Man Called Flintstone (1966): Released in Canada April 2005 (US release canceled/postponed due to Columbia Pictures claiming ownership of the film). The Flintstones: The Complete Fifth Season (1964): Scheduled for release March 7, 2006.

The Flintstones: The Complete Fourth Season (1963): Released November 2005. Note: All episodes are in their original length except for "The Big Move," which is the syndicated version. The Flintstones: The Complete Third Season (1962): Released March 2005