Dodge Dakota

The Dakota is built at Warren Truck Assembly in Warren, Michigan. The curve drawing algorithms of Xiaolin Wu (SIGGRAPH 91) are an example. Two 4.7 L V8 engines are available as well. A more challenging task is to perform these drawing operations with antialiasing, to create a smooth-looking curve. There are one V6 and two V8 engines available: The standard engine is a 3.7 L PowerTech V6 (specs below). 1984). This model is 3.7 in longer and 2.7 in wider, and features a new front and rear suspension, and rack-and-pinion steering. An efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken (IEEE CG&A, Sept.

The redesigned 2005 Dakota shares its platform with the new Dodge Durango SUV. 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. Engines:. Often such libraries are limited and can only draw an ellipse with either the major axis or the minor axis horizontal. 2004 was the end of the old OHV V6 and the big R/T V8. 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. Most buyers ordered the V6 or V8 engines, which were considerably more powerful and, in the case of the V6, which was made standard for 2003, nearly as fuel-efficient with a manual transmission. Einstein's contributions to modern physics may not have been discovered if it were not for ellipses.

2002 was the final year for the four-cylinder engine in the Dakota, as Chrysler was ending production of the former AMC design. Albert Einstein also used the ellipse to prove his theory of relativity by using an elliptical shaped mass. The smaller V8 was replaced by a new high-tech V8 as well. 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. Four-door "Quad-Cab" models were added for 2000 with a slightly shorter bed, 63.1 in (160.2 cm), but riding on the Club Cab's 130.9 in (332.5 cm) wheelbase. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus. 1998 saw the introduction of the R/T model with the big 5.9 L 250 hp (186 kW) Magnum V8. 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.

It inherited the semi truck look of the larger Ram but remained largely the same underneath. Later, Isaac Newton explained this as a corollary of his law of universal gravitation. The second-generation Dakota was built from 1997 through 2004. 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. Engines:. 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]. It was the only major change for 1996, and would be carried over as the base engine in the new, larger 1997 model. Since no other curve has such a property, it can be used as an alternative definition of an ellipse.

In 1996, the first generation's final year, the K-based 2.5 L I4 engine was out of production and had been considered vastly underpowered compared to the competition, so Dodge borrowed the Jeep 2.5 L I4 (rated at 120 hp) and installed it as the base engine in the Dakota. Then all rays are reflected to a single point — the second focus. Both of the V-configuration engines were updated to Magnum specs the next year, providing a tremendous power boost. Assume an elliptic mirror with a light source at one of the foci. This engine produced 170 hp (127 kW). If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse. For 1991, the front of the Dakota received a more aerodynamic grille and hood, and Dodge added the 5.2 L V8 as an option, inspired by the earlier Shelby Dakota option. Similarly, any oblique projection onto a plane results in a conic section.

This model allowed the Dakota to boast capacity for six passengers, although the rear seat was best suited for children and shorter adults. 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). An extended "Club Cab" model was added for 1990, still with two doors. 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. Another important addition that year was Carroll Shelby's V8-powered Shelby Dakota, his first rear wheel drive vehicle in two decades. The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions. Just 2,482 were sold that first year. 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 first convertible pickup since the Ford Model T, it featured a fixed roll bar and complicated manual top. which can also be written as:. 1989 saw the unusual Dakota convertible. A good approximation is Ramanujan's:. Fuel injection was added to the 3.9 L V6 for 1988 but the output remained the same. The exact infinite series is:. Both 6.5 ft (2 m) and 8 ft (2.4 m) beds were offered. The circumference of an ellipse is 4aE(e), where the function E is the complete elliptic integral of the second kind.

Four wheel drive was available only with the V6. The area enclosed by an ellipse is , where π is Archimedes' constant. Straight-4 and V6 engines were offered along with either a 5-speed manual or 4-speed automatic transmission. 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°. The first generation of the Dakota was produced from 1987 through 1996. In polar coordinates, an ellipse with one focus at the origin and the other on the negative x-axis is given by the equation. . It is related to and (the ellipse's semi-axes) by the formula or, if using the eccentricity, .

One notable feature was the Dakota's rack and pinion steering, a first in work trucks. 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 Dakota has also long been the only midsize pickup with an optional V8 engine. The distance between the foci is 2ae. It is a conventional design with body-on-frame construction and leaf spring/live axle rear end. The ellipse shown in the image below has an eccentricity of approximately 0.8733. The Dakota has always been sized above the compact (Ford Ranger, Chevrolet S-10) and below the full-sized (Ford F-150, Chevrolet Silverado) pickups and Dodge's own Ram. The greater the eccentricity is, the larger the ratio of a to b is, and therefore the more elongated the ellipse is.

The Dakota was nominated for the North American Truck of the Year award for 2000. The eccentricity is a positive number less than 1, or 0 in the case of a circle. It was introduced in 1987 alongside the redesigned Dodge Ram 50. or where c (the linear eccentricity of the ellipse) equals the distance from the center to either focus. The Dakota is a midsize pickup truck from DaimlerChrysler's Dodge brand. The eccentricity is related to a and b by the statement. 2005 - 4.7 L HO PowerTech V8, 260 hp (194 kW) at 5200 rpm and 310 ft·lbf (420 N·m) at 5200 rpm. 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).

2005 - 4.7 L PowerTech V8, 230 hp (172 kW) at 4400 rpm and 290 ft·lbf (393 N·m) at 3600 rpm. has normal (cosφ,sinφ). 2005 - 3.7 L PowerTech V6, 210 hp (157 kW) at 5200 rpm and 235 ft·lbf (319 N·m) at 4000 rpm. A Gauss-mapped form:. 2004 - 3.7 L PowerTech V6, 210 hp (157 kW). where (h,k) is the center. 2000-2004 - 4.7 L PowerTech V8, 230 hp (175 kW). 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.

1998-2003 - 5.9 L Magnum V8, 250 hp (186 kW). which use the trigonometric functions sine and cosine. 1997-1999 - 5.2 L Magnum V8, 230 hp (172 kW). The same ellipse is also represented by the parametric equations:. 1997-2003 - 3.9 L Magnum V6, 175 hp (131 kW). 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). 1997-2002 - 2.5 L AMC I4, 120 hp (90 kW). The derivation of this formula is quite instructive and not overly difficult.

1996 - 2.5 L AMC I4, 120 hp (90 kW). 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. 1994-1996 - 5.2 L Magnum V8, 220 hp (164 kW). The constant a equals the length of the semimajor axis; the constant b equals the length of the semiminor axis. 1994-1996 - 3.9 L Magnum V6, 175 hp (131 kW). The size of an ellipse is determined by two constants, conventionally denoted a and b. 1991-1993 - 5.2 L Magnum V8, 230 hp (172 kW). .

1992-1993 - 3.9 L Magnum V6, 180 hp (134 kW).
. 1991 - 5.2 L LA V8, 170 hp (127 kW). 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. 1989-1995 - 2.5 L K I4, 99 hp (74 kW). 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. 1987-1991 - 3.9 L LA V6, 125 hp (93 kW). 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.

1987-1988 - 2.2 L K I4, SOHC, 96 hp (72 kW). Likewise, the semiminor axis is one half the minor axis. 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. The line which passes through the center (halfway between the foci), at right angles to the major axis, is called the minor axis. The major axis is along the longest segment that passes through the ellipse.

The line segment which passes through the foci and terminates on the ellipse is called the major axis. 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. The string will form a triangle. 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..