query: how to describe the shape of a parabola

pos: Parabola Parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits any of several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and

neg: Quadratic equation of any quadratic function has the same general shape, which is called a parabola. The location and size of the parabola, and how it opens, depend on the values of , , and . As shown in Figure 1, if , the parabola has a minimum point and opens upward. If , the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The "-coordinate" of the vertex will be located at formula_19, and the "-coordinate" of the vertex may be found by substituting this "-value" into the

neg: Parabola 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 catenary, but in practice the curve is generally nearer to a parabola, and in calculations the second degree parabola is used. Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise catenary-shaped cable is deformed toward a parabola. Unlike an inelastic chain, a freely hanging spring of zero unstressed length takes the shape of a parabola. Suspension-bridge cables are, ideally, purely in tension, without

neg: 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 doesn't resemble a parabola. Another hypothetical situation in which parabolas might arise, according to the theories of physics described in the 17th and 18th centuries by Sir Isaac Newton, is in two-body orbits; for example the path of a small planetoid or other object under the

neg: Paraboloid a moving parabola directed by a second parabola. With an elliptic paraboloid is a "paraboloid of revolution": a surface obtained by revolving a parabola around its axis. It is the shape of the parabolic reflectors used in mirrors, antenna dishes, and the like; and is also the shape of the surface of a rotating liquid, a principle used in liquid mirror telescopes and in making solid telescope mirrors (see rotating furnace). This shape is also called a "circular paraboloid". There is a point called the "focus" (or "focal point") on the axis of a circular paraboloid such that, if the

neg: Nose cone design defining shape rather than a circle. Just as it does on an ogive, this construction produces a nose shape with a sharp tip. For the blunt shape typically associated with a parabolic nose, see power series below. (The parabolic shape is also often confused with the elliptical shape.) For formula_26 : formula_27 For the case of the full parabola () the shape is tangent to the body at its base, and the base is on the axis of the parabola. Values of less than result in a slimmer shape, whose appearance is similar to that of the secant ogive. The

neg: Parabola (in formula_210) of the parabola determined by 3 points formula_211 with different -coordinates is (if two -coordinates are equal there is no parabola with directrix parallel to the -axis, which passes through the points) The "focal length" can be determined by a suitable parameter transformation (which does not change the geometric shape of the parabola). The focal length is Hence "Remark: "The advantage of this definition is, one obtains a simple parametric representation of an arbitrary parabola, even in the space, if the vectors formula_215 are vectors of the Euclidean space. A quadratic Bézier curve is a curve formula_216 defined

neg: Nose cone design as model rocketry) due to the blunt nose and tangent base. This is not a shape normally found in professional rocketry, which almost always flies at much higher velocities where other designs are more suitable. If equals , this is a hemisphere. This nose shape is not the blunt shape that is envisioned when people commonly refer to a "parabolic" nose cone. The parabolic series nose shape is generated by rotating a segment of a parabola around a line parallel to its latus rectum. This construction is similar to that of the tangent ogive, except that a parabola is the

neg: Smoothness that the five conic sections were really five different versions of the same shape. An ellipse tends to a circle as the eccentricity approaches zero, or to a parabola as it approaches one; and a hyperbola tends to a parabola as the eccentricity drops toward one; it can also tend to intersecting lines. Thus, there was "continuity" between the conic sections. These ideas led to other concepts of continuity. For instance, if a circle and a straight line were two expressions of the same shape, perhaps a line could be thought of as a circle of infinite radius. For such

neg: Parabola always maintaining the relationship between and shown in the equation. The parabolic curve is therefore the locus of points where the equation is satisfied, which makes it a Cartesian graph of the quadratic function in the equation. This discussion started from the definition of a parabola as a conic section, but it has now led to a description as a graph of a quadratic function. This shows that these two descriptions are equivalent. They both define curves of exactly the same shape. It is proved in a preceding section that if a parabola has its vertex at the origin, and

neg: Parabolic reflector in radar sets. In acoustics, parabolic microphones are used to record faraway sounds such as bird calls, in sports reporting, and to eavesdrop on private conversations in espionage and law enforcement. Strictly, the three-dimensional shape of the reflector is called a "paraboloid". A parabola is the two-dimensional figure. (The distinction is like that between a sphere and a circle.) However, in informal language, the word "parabola" and its associated adjective "parabolic" are often used in place of "paraboloid" and "paraboloidal". If a parabola is positioned in Cartesian coordinates with its vertex at the origin and its axis of symmetry along

neg: Quickprop Quickprop Quickprop is an iterative method for determining the minimum of the loss function of an artificial neural network, following an algorithm inspired by the Newton's method. Sometimes, the algorithm is classified to the group of the second order learning methods. It follows a quadratic approximation of the previous gradient step and the current gradient, which is expected to be close to the minimum of the loss function, under the assumption that the loss function is locally approximately square, trying to describe it by means of an upwardly open parabola. The minimum is sought in the vertex of the parabola.

neg: Quadratic formula the quadratic formula is called a root of the quadratic equation. Geometrically, these roots represent the values at which "any" parabola, explicitly given as , crosses the -axis. As well as being a formula that will yield the zeros of any parabola, the quadratic formula will give the axis of symmetry of the parabola, and it can be used to immediately determine how many real zeros the quadratic equation has. The earliest methods for solving quadratic equations were geometric. Babylonian cuneiform tablets contain problems reducible to solving quadratic equations. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050

neg: Bézier curve the empty product is 1. A quadratic Bézier curve is also a segment of a parabola. As a parabola is a conic section, some sources refer to quadratic Béziers as "conic arcs". With reference to the figure on the right, the important features of the parabola can be derived as follows: The derivative for a curve of order "n" is The "t" in the function for a linear Bézier curve can be thought of as describing how far B("t") is from P to P. For example, when "t=0.25", B("t") is one quarter of the way from point P to P.

neg: Tethered particle motion to exponent of the ratio between the elastic energy and the thermal energy: formula_16 where formula_17 is the spring constant, formula_18 is Boltzmann constant and formula_19 is the temperature. By taking the logarithm of the distribution formula_20 and fitting it to a parabola shape, one can get the spring constant of the polymer: formula_21 where formula_22 is the coefficient of formula_23 from the parabola fit. Advantages include a simple setup, cost, the fact that observations are made in the polymer's natural environment (no external forces are used), it is suitable for various microscopy methods (e.g. TIRFM, dark field, differential interference

neg: Nose cone design shape is no longer tangent at the base, and the base is parallel to, but offset from, the axis of the parabola. The "power series" includes the shape commonly referred to as a 'parabolic' nose cone, but the shape correctly known as a parabolic nose cone is a member of the parabolic series (described above). The power series shape is characterized by its (usually) blunt tip, and by the fact that its base is not tangent to the body tube. There is always a discontinuity at the joint between nose cone and body that looks distinctly non-aerodynamic. The shape can

neg: Sacro Cuore di Gesù agonizzante a Vitinia arranged vertically (the four curves making up each shape are also parabolic). These windows increase in overall size and proportional length from top row to bottom, and the rows number two, three, four and five. The parabola is bounded by two pink brick walls which slope backwards from the two corners of the overall façade, creating voids either side of the parabola, and it also has a floating canopy either side of its apex covering these voids. The side walls of the church are strongly zig-zag, with six triangular projections running from ground to roof on each side in the

neg: Fundamental diagram of traffic flow the same flow rate. In the first speed-flow diagram, the free flow branch is a horizontal line, which shows that the roadway is at free flow speed until the optimum flow is reached. Once the optimum flow is reached, the diagram switches to the congested branch, which is a parabolic shape. The second speed flow diagram is a parabola. The parabola suggests that the only time there is free flow speed is when the density approaches zero; it also suggests that as the flow increases the speed decreases. This parabolic graph also contains an optimum flow. The optimum flow also

neg: Van Hiele model until they have built up an extensive understanding of the systems of relationships between geometric ideas. These systems cannot be learned by rote, but must be developed through familiarity by experiencing numerous examples and counterexamples, the various properties of geometric figures, the relationships between the properties, and how these properties are ordered. The five levels postulated by the van Hieles describe how students advance through this understanding. The five van Hiele levels are sometimes misunderstood to be descriptions of how students understand shape classification, but the levels actually describe the way that students reason about shapes and other geometric ideas.

neg: Pattern patterns in nature including the mathematics of symmetry, waves, meanders, and fractals. Fractals are mathematical patterns that are scale invariant. This means that the shape of the pattern does not depend on how closely you look at it. Self-similarity is found in fractals. Examples of natural fractals are coast lines and tree shapes, which repeat their shape regardless of what magnification you view at. While self-similar patterns can appear indefinitely complex, the rules needed to describe or produce their formation can be simple (e.g. Lindenmayer systems describing tree shapes). In pattern theory, devised by Ulf Grenander, mathematicians attempt to describe

neg: Brick Expressionism halls, office buildings, water towers, "etc.") and residential buildings. Brick was also used for representative buildings, such as town halls, post offices, churches and villas. An important example is Alfred Fischer's Hans-Sachs-Haus in Gelsenkirchen, planned as multi-functional a building but eventually used as the city hall. Its comparatively simply brick facade and rounded corners characterise it as a synthesis between expressionism and New Objectivity. Also in Gelsenkirchen, in the Ückendorf area, is the main work of Josef Franke, the Parabola Church of Heilig-Kreuz (Holy Cross). Its vault has the shape of a tall parabola. The top of the square tower

neg: Headlamp clear-lens multi-reflector headlamps; these were developed by Stanley in Japan. The optics to distribute the light in the desired pattern are designed into the reflector itself, rather than into the lens. Depending on the development tools and techniques in use, the reflector may be engineered from the start as a bespoke shape, or it may start as a parabola standing in for the size and shape of the completed package. In the latter case, the entire surface area is modified so as to produce individual segments of specifically calculated, complex contours. The shape of each segment is designed such that

neg: Structural differential or more, it is totally arbitrary how many we want to symbolize the higher levels), chained in order. The highest, most reliable abstractions at a date are made by science, he claimed (e.g., science has conveyed the nature and danger of bacteria to us), and that is why he attached the last label back to the parabola. It is science that has told us that the sub-microscopic domain exists, and in general semantics the parabola represents that domain. In general semantics, the "natural order of evaluation" proceeds from lower orders of abstraction to higher orders of abstraction, and back again

neg: Evangelista Torricelli how they traveled through the air. "Perhaps his most notable achievement in the field of projectiles was to establish for the first time the idea of an envelope: projectiles sent out at [...] the same speed in all directions trace out parabolas which are all tangent to a common paraboloid. This envelope became known as the "parabola di sicurezza" (safety parabola)." Torricelli gave the first scientific description of the cause of wind: Torricelli is also famous for the discovery of the "Torricelli's trumpet" (also - perhaps more often - known as Gabriel's Horn) whose surface area is infinite, but whose

neg: Tensile structure hence changing its shape. The level of pretension applied determines the shape of a membrane structure. The alternative approximated approach to the form-finding problem solution is based on the total energy balance of a grid-nodal system. Due to its physical meaning this approach is called the Stretched Grid Method (SGM). A uniformly loaded cable spanning between two supports forms a curve intermediate between a catenary curve and a parabola. The simplifying assumption can be made that it approximates a circular arc (of radius "R"). By equilibrium: The horizontal and vertical reactions : By geometry: The length of the cable: The

neg: Parabola "y"-axis, one obtains three statements for a parabola. The following properties of a parabola deal only with terms "connect, intersect, parallel", which are invariants of similarities. So, it is sufficient to prove any property for the "unit parabola" with equation formula_72. Any parabola can be described in a suitable coordinate system by an equation formula_73. "Proof:" straightforward calculation for the unit parabola formula_52. "Application:" The 4-points-property of a parabola can be used for the construction of point formula_116, while formula_117 and formula_105 are given. "Remark:" the 4-points-property of a parabola is an affine version of the 5-point-degeneration of Pascal's theorem.

neg: Catenary Catenary In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends. The catenary curve has a U-like shape, superficially similar in appearance to a parabolic arch, but it is not a parabola. The curve appears in the design of certain types of arches and as a cross section of the catenoid—the shape assumed by a soap film bounded by two parallel circular rings. The catenary is also called the alysoid, chainette, or, particularly in the materials sciences, funicular. Mathematically, the catenary

neg: Catenary Catenary In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends. The catenary curve has a U-like shape, superficially similar in appearance to a parabolic arch, but it is not a parabola. The curve appears in the design of certain types of arches and as a cross section of the catenoid—the shape assumed by a soap film bounded by two parallel circular rings. The catenary is also called the alysoid, chainette, or, particularly in the materials sciences, funicular. Mathematically, the catenary

neg: Parabolic arch base. Also, it can span the widest area. It is commonly used in bridge design, where long spans are needed. When an arch carries a uniformly distributed vertical load, the correct shape is a parabola. When an arch carries only its own weight, the best shape is a catenary. A hen's egg can be fairly well described as two different paraboloids connected by part of an ellipse A few examples of parabolic arches.: Parabolic arches were used by architects Oscar Niemeyer, and Antoni Gaudí, who also used catenary arches. Parabolic arch A parabolic arch is an arch shaped like a

neg: Dynamical simulation to model, so most physics engines ignore deformation. A body that is assumed to be non-deformable is called a rigid body. Rigid body dynamics deals with the motion of objects that cannot change shape, size, or mass but can change orientation and position. To account for rotational energy and momentum, we must describe how force is applied to the object using a moment, and account for the mass distribution of the object using an inertia tensor. We describe these complex interactions with an equation somewhat similar to the definition of force above: where formula_3 is the central inertia tensor, formula_4

neg: Parabola rigid motion to a parabola with an equation formula_70. Such a parabola can then be transformed by the uniform scaling formula_71 into the unit parabola with equation formula_72. Thus, any parabola can be mapped to the unit parabola by a similarity. A synthetic approach, using similar triangles, can also be used to establish this result. The general result is that two conic sections (necessarily of the same type) are similar if and only if they have the same eccentricity. Thus, only circles (all having eccentricity 0) share this property with parabolas (all having eccentricity 1), while general ellipses and hyperbolas

neg: Parabola of safety Parabola of safety In classical mechanics and ballistics, the parabola of safety or safety parabola is the envelope of the parabolic trajectories of projectiles shot from a certain point with a given speed at different angles to horizon in a fixed vertical plane. The fact that this envelope is a parabola had been first established by Evangelista Torricelli and was later reproven by Johann Bernoulli using the infinitesimal calculus methods of Leibniz. The paraboloid of revolution obtained by rotating the safety parabola around the vertical axis is the boundary of the safety zone, consisting of all points that cannot be

neg: Colin Archer little for trim. Archer spent much time calculating how an efficient hull should be designed.He started with Chapman's displacement parabola curve, but with Scott Russell's positioning of the midship section well aft of amidships and thus with sharp bow waterlines. In 1876 he changed Chapman's parabola for the displacement curve, with Scott Russell's wave curves; the sine curve forward and the trochoid curve aft. The change gave designers more freedom in shaping the hull than Scott Russell's theory. We know now that the theory is not correct at all, but it did away with the old "Cod's Head-Mackerel Tail"-type. However,

neg:
Semicubical parabola is, those curves had been "rectified"), the semicubical parabola was the first algebraic curve (excluding the line and circle) to be rectified. Semicubical parabola In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve defined by an "equation" of the form Solving for formula_2 leads to the "explicit form" which is the cause for the term "semicubical parabola".
 (A parabola in the common sense can be described by an equation formula_4.)
 Solving (A) for formula_5 yields the second "explicit form" Equation (A) shows, that is a "parametric representation" of the curve. The arc length of the curve

neg: Production drawing or production team, the main purpose of which is to define the size, shape, location and production of the building or component'. Orthographic projections are often supplied, giving views of machine parts and their assembly in an accessible form akin to artistic rendering in perspective, sometimes in exploded form which illustrates how the whole may be constructed from sub-assemblies and sub-assemblies into individual components. The production drawings may describe the preferred order in which to assemble components and if the engineering drawings call for a screw fastener to be tightened to a specific torque the production drawings would typically describe

neg: Searching the conformational space for docking and the ligand in order to find an optimal pose. Programs include DOCK, FRED, GLIDE, SURFLEX, eHiTS and many more. Most methods describe the molecules in terms of a finite number of descriptors that include structural complementarity and binding complementarity. Structural complementarity is mostly a geometric description of the molecules, including solvent-accessible surface area, overall shape and geometric constraints between atoms in the protein and ligand. Binding complementarity takes into account features like hydrogen bonding interactions, hydrophobic contacts and van der Waals interactions to describe how well a particular ligand will bind to the protein. Both kinds of descriptors are

neg: Kaleidics conditions in some particular market and global knowledge of theories about how markets work, including those held by the regulator. Kaleidics The term kaleidics ( "kalos": "good", "beautiful"; εἶδος "eidos": "form", "shape") denotes the ever-changing shape and status of an economy. Uncertainty is the primary kaleidic factor. British economist George Lennox Sharman Shackle, who has been characterised as a post-Keynesian but also as influenced by Austrian economics, made an attempt to challenge classical rational choice theory. He used the term "kaleidostatics" (or "kaleido-statics"), derived from "kaleidoscope", to describe the status of the economy in the particular posture which prevails at

neg: Parabola GNU/Linux-libre Parabola live image, but adds speech and braille packages to make it possible for blind and visually impaired users to install Parabola eyes-free. The Parabola community has created a number of cartoon characters for the project. The characters are a gnu and a cat named "Bola", who is conceived after Parabola's main characteristics: "elegant, minimalist and lightweight". Parabola GNU/Linux-libre Parabola GNU/Linux-libre is an operating system for the i686, x86-64 and ARMv7 architectures. It is based on many of the packages from Arch Linux and Arch Linux ARM, but distinguishes from the former by offering only free software. It includes the

neg: Standard linear solid model Kelvin–Voigt models to accurately describe the overall behavior of a system under a given set of loading conditions. The behavior of a material applied to an instantaneous stress is shown as having an instantaneous component of the response. Instantaneous release of a stress also results in a discontinuous decrease in strain, as is expected. The shape of the time-dependent strain curve is true to the type of equation that characterizes the behavior of the model over time, depending upon how the model is loaded. Although this model can be used to accurately predict the general shape of the strain curve,

neg: Parabola E travelling parallel to the axis of symmetry will be reflected by the line so it travels along the line , as shown in red in the diagram (assuming that the lines can somehow reflect light). Since is the tangent to the parabola at E, the same reflection will be done by an infinitesimal arc of the parabola at E. Therefore, light that enters the parabola and arrives at E travelling parallel to the axis of symmetry of the parabola is reflected by the parabola toward its focus. The point E has no special characteristics. This conclusion about reflected light

neg: Parabola Films which screened at the Hot Docs film festival in Toronto. In 2017, Parabola Films secured funding from the Theatrical Documentary Program, sponsored by Telefilm Canada and Rogers Group, for its upcoming film "Billy". Also that year, Parabola filmmaker Sarah Chadwick created a documentary "1999", about suicides at a high school. It was filmed in Moncton, New Brunswick, and was funded by Eurimages. In 2018 Parabola Films completed another documentary, "Sweet Dreams for Chiyo", about a young girl living with diabetes. Parabola Films Parabola Films is a Montreal-based Canadian cinema production company founded by Sarah Spring and Selin Murat, a documentary

neg: Parabola a plane which is parallel to another plane that is tangential to the conical surface. A third description is algebraic. The graph of a quadratic function (for example, ) is a parabola, and, conversely, a parabola whose axis of symmetry is parallel to the -axis is the graph of a quadratic function. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point on the parabola that intersects the axis of symmetry is called the "vertex", and is the point where

neg: Parabola symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel ("collimated") beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy. This reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used

neg: Parabola function of degree 2 is Completing the square yields which is the equation of a parabola with Two objects in the Euclidean plane are "similar" if one can be transformed to the other by a "similarity", that is, an arbitrary composition of rigid motions (translations and rotations) and uniform scalings. A parabola formula_66 with vertex formula_24 can be transformed by the translation formula_68 to one with the origin as vertex. A suitable rotation around the origin can then transform the parabola to one that has the -axis as axis of symmetry. Hence the parabola formula_66 can be transformed by a

neg: Parabola "Proof:" can be performed for the unit parabola formula_52. A short calculation shows: line formula_112 has slope formula_133 which is the slope of the tangent at point formula_127. "Application:" The 3-points-1-tangent-property of a parabola can be used for the construction of the tangent at point formula_127, while formula_136 are given. "Remark:" The 3-points-1-tangent-property of a parabola is an affine version of the 4-point-degeneration of Pascal's theorem. "Proof:" straight forward calculation for the unit parabola formula_72. "Application:" The 2-points-2-tangents-property can be used for the construction of the tangent of a parabola at point formula_143 while formula_151 and the tangent at formula_140

neg: Agricultural literacy work, which is unique in the field of agricultural education. People have a wide variety views on agriculture and agricultural practices (see agrarianism). These values shape how an agricultural literacy instructor designs curriculum and works with clientele and how the clientele receives the messages of the instructor. Agricultural literacy Agricultural literacy is a phrase being used by several universities (e.g., Texas Tech, the University of Arizona, Colorado State University, and California Polytechnic State University) to describe programs to promote the understanding and knowledge necessary to synthesize, analyze, and communicate basic information about agriculture with students, producers, consumers, and the public.

neg: Parabola tangent to the parabola at its vertex, it follows that the point of intersection between any tangent to a parabola and the perpendicular from the focus to that tangent lies on the line that is tangential to the parabola at its vertex. See animated diagram and pedal curve. If light travels along the line , it moves parallel to the axis of symmetry and strikes the convex side of the parabola at E. It is clear from the above diagram that this light will be reflected directly away from the focus, along an extension of the segment . The above

neg: The Method of Mechanical Theorems parabola must be 1/3 to give it the opposite torque. This type of method can be used to find the area of an arbitrary section of a parabola, and similar arguments can be used to find the integral of any power of "x", although higher powers become complicated without algebra. Archimedes only went as far as the integral of "x", which he used to find the center of mass of a hemisphere, and in other work, the center of mass of a parabola. Consider the parabola in the figure to the right. Pick two points on the parabola and call

neg: Feminist movement Emily Martin describes the concept of how metaphors are gendered and ingrained into everyday life. Metaphors are used in everyday language and have become a way that people describe the world. Martin explains that these metaphors structure how people think and in regards to science can shape what questions are being asked. If the right questions are not being asked then the answers are not going to be the right either. For example, the aggressive sperm and passive egg is a metaphor that felt 'natural' to people in history but as scientists have reexamined this phenomenon they have come up

neg: Parabola by three points formula_217, formula_218 and formula_219, its "control points": This curve is an arc of a parabola (see ). For numerical integration one replaces the graph of a function by arcs of parabolas and integrates the parabola arcs. A parabola is determined by three points. The formula for one arc is The method is called Simpson's rule. The following quadrics contain parabolas as plane sections: A parabola can be used as a trisectrix, that is it allows the exact trisection of an arbitrary angle with straightedge and compass. Note that this is not in contradiction to the impossibility of

neg: Cissoid of Diocles whose centers lie on a parabola and which pass through the vertex of the parabola. Also, if two congruent parabolas are set vertex-to-vertex and one is rolled along the other; the vertex of the rolling parabola will trace the cissoid. : The cissoid of Diocles can also be defined as the inverse curve of a parabola with the center of inversion at the vertex. To see this, take the parabola to be "x"="y". In polar coordinates this becomes and the inverse curve then has equation which is a special case of the equation defining the cissoid of Diocles on polar

neg:
Semicubical parabola Semicubical parabola In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve defined by an "equation" of the form Solving for formula_2 leads to the "explicit form" which is the cause for the term "semicubical parabola".
 (A parabola in the common sense can be described by an equation formula_4.)
 Solving (A) for formula_5 yields the second "explicit form" Equation (A) shows, that is a "parametric representation" of the curve. The arc length of the curve was calculated by the English mathematician William Neile and published in 1657 (see section History). . "Proof:" The similarity formula_10 (uniform scaling)

neg: Parabola (magazine) is essentially, irrevocably changed. The magazine's subtitle has changed over the years. In its first years, it was "Parabola: Myth and the Quest for Meaning", then "Parabola: The Magazine of Myth and Tradition", later "Parabola: Myth, Tradition, and the Search for Meaning", and now its current form. Each issue focuses on a particular subject, with each article related to the main subject. The subjects of the first five years' issues are listed here. Issues prior to Vol. III No. 4 do not indicate the subject on the cover; later issues do. Authors contributing articles to Parabola are listed as contributing

neg: Parabola of safety hit by a projectile shot from the given point with the given speed. Philip Robinson, "On the Geometrical Approach to Projectile Motion", The Mathematical Gazette, Vol. 82, No. 493, 1998, pp. 118–122 Parabola of safety In classical mechanics and ballistics, the parabola of safety or safety parabola is the envelope of the parabolic trajectories of projectiles shot from a certain point with a given speed at different angles to horizon in a fixed vertical plane. The fact that this envelope is a parabola had been first established by Evangelista Torricelli and was later reproven by Johann Bernoulli using the infinitesimal

neg: Parabola and , which is , which comes to . But is also the slope (first derivative) of the parabola at E. Therefore, the line is the tangent to the parabola at E. The distances and are equal because E is on the parabola, F is the focus and C is on the directrix. Therefore, since B is the midpoint of , triangles △FEB and △CEB are congruent (three sides), which implies that the angles marked are congruent. (The angle above E is vertically opposite angle ∠BEC.) This means that a ray of light which enters the parabola and arrives at

neg: Parabola in physics, engineering, and many other areas. The earliest known work on conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolas. (The solution, however, does not meet the requirements of compass-and-straightedge construction.) The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes via the method of exhaustion in the third century BC, in his "The Quadrature of the Parabola". The name "parabola" is due to Apollonius who discovered many properties of conic sections. It means "application", referring

neg: Parabola GNU/Linux-libre Parabola GNU/Linux-libre Parabola GNU/Linux-libre is an operating system for the i686, x86-64 and ARMv7 architectures. It is based on many of the packages from Arch Linux and Arch Linux ARM, but distinguishes from the former by offering only free software. It includes the GNU operating system components common to many Linux distributions and the Linux-libre kernel instead of the generic Linux kernel. Parabola is listed by the Free Software Foundation as a completely free operating system, true to their Free System Distribution Guidelines. Parabola uses a rolling release model like Arch, such that a regular system update is all that

neg: Helen Beardsley Dennis Quaid. Helen Eileen Brandmeir was born in 1930 in Seattle, Washington. Her father owned a lumber mill, which came under severe stress during the Great Depression. Helen would later describe how her observations of her father would shape her personality, and in particular how it would teach her to value independence and problem-solving. Brandmeir trained as a nurse in Seattle. In 1949 at age 19 she married Richard North, then a chief petty officer in the United States Navy. Their marriage lasted for eleven years and produced eight children. In 1960, the Norths were stationed at Naval Air Station

neg: Lera Boroditsky In addition to scholarly work, Boroditsky also gives popular science lectures to the general public, and her work has been covered in news and media outlets. Boroditsky talks about how all the languages differ from one another, whether in grammatical differences or contain different sounds, vocabulary, or patterns. Boroditsky studies how the languages we speak shape the way we think. It was first detected by John Hovland that some languages, unlike English, do not use body-centered terms to describe position, but instead use cardinal directions, such as North, South, East and West. In order to speak a language using cardinal

neg: Parabola parabola is equidistant from the focus and from the directrix is taken as axiomatic. Consider the parabola . Since all parabolas are similar, this simple case represents all others. The right-hand side of the diagram shows part of this parabola. The point E is an arbitrary point on the parabola, with coordinates . The focus is F, the vertex is A (the origin), and the line (the -axis) is the axis of symmetry. The line is parallel to the axis of symmetry, and intersects the -axis at D. The point C is located on the directrix (which is not shown,

neg: Lissajous curve logos. Examples include: Lissajous curve In mathematics, a Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations which describe complex harmonic motion. This family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail by Jules Antoine Lissajous in 1857. The appearance of the figure is highly sensitive to the ratio . For a ratio of 1, the figure is an ellipse, with special cases including circles (, radians) and lines (). Another simple Lissajous figure is the parabola (, ). Other ratios produce

neg: Lissajous curve Lissajous curve In mathematics, a Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations which describe complex harmonic motion. This family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail by Jules Antoine Lissajous in 1857. The appearance of the figure is highly sensitive to the ratio . For a ratio of 1, the figure is an ellipse, with special cases including circles (, radians) and lines (). Another simple Lissajous figure is the parabola (, ). Other ratios produce more complicated curves,

neg: Parabola are given. "Remark 1:" The 2-points-2-tangents-property of a parabola is an affine version of the 3-point-degeneration of Pascal's theorem. "Remark 2:" The 2-points-2-tangents-property should not be confused with the following property of a parabola, which deals with 2 points and 2 tangents, too, but is "not" related to Pascal's theorem. The statements above presume the knowledge of the axis-direction of the parabola, in order to construct the points formula_153. The following property determines the points formula_153 by two given points and their tangents only, and the result is: the line formula_112 is parallel to the axis of the parabola. "Proof:"

neg: Parabola if it opens in the positive direction, then its equation is , where is its focal length. Comparing this with the last equation above shows that the focal length of the parabola in the cone is . In the diagram above, the point V is the foot of the perpendicular from the vertex of the parabola to the axis of the cone. The point F is the foot of the perpendicular from the point V to the plane of the parabola. By symmetry, F is on the axis of symmetry of the parabola. Angle VPF is complementary to , and

neg: Parabola the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola which is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar. Parabolas have the property that, if they are made of material that reflects light, then light which travels parallel to the axis of

neg: Ex's & Oh's written by Elle King with the song's producer, Dave Bassett, and King credits him with helping her shape the sound of the album. Its instrumentation includes the banjo, guitar and drums and is influenced by country music. The song's lyrics describe the narrator's past relationships with needy men and her tendency to "[chew] men up and eat them." King describes how her ex-boyfriends are like ghosts; they always want to come back to haunt her. She also explains how her feelings toward her boyfriends quickly shift and her carelessness over the relationship. According to Musicnotes.com, the song is written in

neg: Cardioid is which lies on the perimeter of the circle with midpoint formula_18 and radius formula_1 (see picture). For the example shown in the graph the generator circles have radius formula_47. Hence the cardioid has the polar representation and its inverse curve which is a parabola (s. parabola in polar coordinates) with the equation formula_50 in cartesian coordinates. "Remark: "Not every inverse curve of a parabola is a cardioid. For example, if a parabola is inverted across a circle whose center lies at the "vertex" of the parabola, then the result is a cissoid of Diocles. In the previous section if

neg: Orthoptic (geometry) Orthoptic (geometry) In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle. Examples: Generalizations: Any parabola can be transformed by a rigid motion (angles are not changed) into a parabola with equation formula_2. The slope at a point of the parabola is formula_3. Replacing formula_4 gives the parametric representation of the parabola with the tangent slope as parameter: formula_5 The tangent has the equation formula_6 with the still unknown formula_7, which can be determined by inserting the coordinates of the parabola point. One gets formula_8

neg:
Involute a semicubic parabola. From formula_38 one gets formula_39 and formula_40. Extending the string by formula_41 causes an essential simplification of the calculation and one gets Eliminating parameter formula_44 yields the equation of a "parabola": formula_45
 Hence: (Parallel curves of a parabola are not parabolas anymore !) "Remark:" The "evolute" of the parabola formula_48 is the semicubic parabola formula_49 (see section involute and evolute.) For the catenary formula_50 one gets formula_51 and because of formula_52 the length of the tangent vector is formula_53 and the integral formula_54 Hence the parametric representation of the corresponding involute is which describes a tractrix.
 Result:

neg: Parabola ways by the curve. For any case, formula_18 is the radius of the osculating circle at the vertex. For a parabola, the semi-latus rectum, formula_18, is the distance of the focus from the directrix. Using the parameter formula_18, the equation of the parabola can be rewritten as More generally, if the vertex is formula_24, the focus formula_25 and the directrix formula_26, one obtains the equation "Remark:" If the focus is formula_34 and the directrix formula_35 one obtains the equation For a parametric equation of a parabola in general position see . The implicit equation of a parabola is defined by

neg: The Shape of Time art has "an unmistakable erosion" (46) that wears down any work of art. This 'erosion' is particularly caused from the artist's failing to describe the process that was needed to obtain the finished piece as well as "the steps in the artist's elaboration of his conceptions." Some of these descriptions may be trivial but what lies between the trivial and the factual is the work. Solitary and Gregarious Artist Kubler says that "Today the artist is neither a rebel nor an entertainer."(53) he explains how art is affected by different personalities. He explains how more social personalities work well in

neg: Ski geometry a parabola or a clothoid, are often used to describe the curvature of a sidecut. Also, multiple radii are stitched together in a piecewise fashion as well. Alpine skis have undergone three distinct changes in sidecut design. The earliest skis from the 1800s all the way to modern examples in the late 1990s featured a very small amount of sidecut that produced a ski that was almost rectangular in profile as seen from the top. Starting with the Elan SCX in 1995, and very rapidly thereafter, these designs dramatically increased sidecut, a design known as the parabolic ski, or later

neg: Parabola Films Parabola Films Parabola Films is a Montreal-based Canadian cinema production company founded by Sarah Spring and Selin Murat, a documentary filmmaker. Parabola Films focuses on the production of videos which demonstrate the role of cinema in social change. The company collaborates with other film-making organizations who emphasize storytelling. Parabola Films' offering "Ariel" was shown at the International Documentary Film Festival Amsterdam, and at the Rencontres Internationales du Documentaire de Montréal 2013. It was funded by a grant from the Rogers Documentary and Cable Network Fund. It also aired on the CBC Television network. "Jeppe on a Friday" was funded through

neg: Order of approximation with only three data points, a parabola is an exact fit. While higher-order approximations exist and are crucial to a better understanding and description of reality, they are not typically referred to by number. Continuing the above, a third-order approximation would be required to perfectly fit four data points, and so on. See polynomial interpolation. These terms are also used colloquially by scientists and engineers to describe phenomena that can be neglected as not significant (e.g. "Of course the rotation of the Earth affects our experiment, but it's such a high-order effect that we wouldn't be able to measure it"

neg: Kempe's universality theorem be found. There is still, however, a wide field open to the mathematical artist to discover the simplest linkworks that will describe particular curves. A series of animations demonstrating the linkwork that results from Kempe's universality theorem are available for the parabola, self-intersecting cubic, smooth elliptic cubic and the trifolium curves. Several approaches have been taken to simplify the drawing linkages that result from Kempe's universality theorem. Some of the complexity arises from the linkages Kempe used to perform addition and subtraction of two angles, the multiplication of an angle by a constant, and translation of the rotation of a

neg: Perpendicular of symmetry is perpendicular to each of the latus rectum, the directrix, and the tangent line at the point where the axis intersects the parabola. From a point on the tangent line to a parabola's vertex, the other tangent line to the parabola is perpendicular to the line from that point through the parabola's focus. The orthoptic property of a parabola is that If two tangents to the parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents which intersect on the directrix are perpendicular. This implies that, seen from any point on its directrix,

neg: Focus (geometry) an ellipse; if "e"=1 the conic is a parabola; and if "e">1 the conic is a hyperbola. If the distance to the focus is fixed and the directrix is a line at infinity, so the eccentricity is zero, then the conic is a circle. It is also possible to describe all the conic sections as loci of points that are equidistant from a single focus and a single, circular directrix. For the ellipse, both the focus and the center of the directrix circle have finite coordinates and the radius of the directrix circle is greater than the distance between the

neg: Parabola Allegory Parabola Allegory The Parabola Allegory is a Rosicrucian allegory, of unknown authorship, dating from the latter part of the seventeenth century. It is sometimes attributed to German alchemist Henricus Madathanus. Bearing many similarities to "The Chymical Wedding of Christian Rosenkreutz", it is steeped in alchemical imagery. It deals with the journey of initiation of an unknown narrator, who, after many trials, enters the Rose Garden and bears witness to the dissolution and reconstitution of a pair of royal lovers into a King and Queen. Like "The Chymical Wedding", the "Parabola Allegory" has the haunting quality of a dream. It was

neg: Parabola proofs of the reflective and tangent bisection properties use a line of calculus. For readers who are not comfortable with calculus, the following alternative is presented. In this diagram, F is the focus of the parabola, and T and U lie on its directrix. P is an arbitrary point on the parabola. is perpendicular to the directrix, and the line bisects angle ∠FPT. Q is another point on the parabola, with perpendicular to the directrix. We know that = and = . Clearly, > , so > . All points on the bisector are equidistant from F and T, but

neg: Parabola can be done (like the properties above) for the unit parabola formula_52. "Application:" This property can be used to determine the direction of the axis of a parabola, if two points and their tangents are given. An alternative way is to determine the midpoints of two parallel chords, see section on parallel chords. "Remark:" This property is an affine version of the theorem of two "perspective triangles" of a non-degenerate conic. Steiner established the following procedure for the construction of a non-degenerate conic (see Steiner conic): This procedure can be used for a simple construction of points on the parabola

neg: Parabolic SAR Thus, unless a security can continue to generate more profits over time, it should be liquidated. The indicator generally works only in trending markets, and creates "whipsaws" during ranging or, sideways phases. Therefore, Wilder recommends first establishing the direction or change in direction of the trend through the use of parabolic SAR, and then using a different indicator such as the Average Directional Index to determine the strength of the trend. A parabola below the price is generally bullish, while a parabola above is generally bearish. A parabola below the price may be used as support, whereas a parabola above

neg: Solar architecture effect. A solar parabola is a parabolic mirror that concentrates the sunlight to reach high temperatures. In Auroville's collective kitchen, a large solar parabola on the roof provides heat for cooking. The solar parabola can also be used for industrial building. The Odeillo solar furnace, one of the largest solar parabola in the world, concentrates the sunlight 10,000 times and reaches temperatures above 3,200 K. No material resists, even diamond melts. It opens the vision of a futuristic metallurgy, using a clean and renewable source of energy. One of the first large commercial buildings to exemplify solar architecture is 4

neg: Marcus theory are kept at a constant distance by means of a stiff spacer (Fig.4). "A posteriori" one may presume that in the systems where the reaction partners may diffuse freely the optimum distance for the electron jump may be sought, i.e. the distance for which ΔG = 0 and ΔG = - λ. For λ is dependent on R, λ increases for larger R and the opening of the parabola smaller. It is formally always possible to close the parabola in Fig. 2 to such an extent, that the f-parabola intersects the i-parabola in the apex. Then always ΔG = 0

neg: Parabola an irreducible polynomial of degree two such that formula_39 or, equivalently, such that formula_40 is the square of a linear polynomial. The previous section shows: any parabola with the origin as vertex and the y-axis as axis of symmetry can be considered as the graph of a function For formula_42 the parabolas are opening to the top and for formula_43 opening to the bottom (see picture). From the section above one obtains: For formula_51 the parabola is the unit parabola with equation formula_52. Its focus is formula_53, the semi-latus rectum formula_54 and the directrix has the equation formula_55. The general

neg: Orthoptic (geometry) can be adopted from the ellipse case by replacing with (as in the case of the orthoptics, see above). To visualize the isoptics, see implicit curve. Orthoptic (geometry) In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle. Examples: Generalizations: Any parabola can be transformed by a rigid motion (angles are not changed) into a parabola with equation formula_2. The slope at a point of the parabola is formula_3. Replacing formula_4 gives the parametric representation of the parabola with the tangent slope as parameter: formula_5

neg: Newtonian telescope grinding the mirror and may have been the first to use a pitch lap to polish the optical surface. He chose a spherical shape for his mirror instead of a parabola to simplify construction; even though it would introduce spherical aberration, it would still correct chromatic aberration. He added to his reflector what is the hallmark of the design of a Newtonian telescope, a secondary diagonally mounted mirror near the primary mirror's focus to reflect the image at a 90° angle to an eyepiece mounted on the side of the telescope. This unique addition allowed the image to be viewed

neg: Newton's reflector To create the primary mirror Newton used a custom composition of metal consisting of six parts copper to two parts tin, an early composition of speculum metal. He devised means for shaping and grinding the mirror and may have been the first to use a pitch lap to polish the optical surface. He chose a spherical shape for his mirror instead of a parabola to simplify construction: he had satisfied himself that the chromatic, and not the spherical aberration, formed the chief faults of refracting telescopes. He added to his reflector what is the hallmark of the design of a

neg: The Quadrature of the Parabola geometric series, and proves that this is the area of the parabolic segment. This represents the most sophisticated use of the method of exhaustion in ancient mathematics, and remained unsurpassed until the development of integral calculus in the 17th century, being succeeded by Cavalieri's quadrature formula. A parabolic segment is the region bounded by a parabola and line. To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle. The base of this triangle is the given chord of the parabola, and the third vertex is the point on the parabola such that the tangent to the

neg: Microwave antenna Microwave antenna A microwave antenna is a physical transmission device used to broadcast microwave transmissions between two or more locations . In addition to broadcasting, antennas are also used in radar, radio astronomy and electronic warfare. A parabolic antenna is an antenna that uses a parabolic reflector, a curved surface with the cross-sectional shape of a parabola, to direct the radio waves. These devices range anywhere from 6" to more than 12' diameter depending on application and use. A horn antenna or microwave horn is an antenna that consists of a flaring metal waveguide shaped like a horn to direct

neg: Parabola GNU/Linux-libre (viewing itself as only competing against nonfree systems), free culture, democracy, and to follow Arch's philosophy. Under the covenant are included the GNU Free System Distribution Guidelines. There are two ways to install Parabola, either from scratch using installable ISO images, or by migrating from an existing Arch-based system. The latter process is almost as simple as switching to the Parabola repositories list. TalkingParabola is a derivative install CD based on TalkingArch. It is a respin of the Parabola ISO modified to include speech and braille output for blind and visually impaired users. TalkingParabola retains all the features of the

neg: Parabola butyraula Parabola butyraula Parabola is a genus of moth in the family Gelechiidae. It contains the species Parabola butyraula, which is found in South Africa (Mpumalanga). The wingspan is about 15 mm. The forewings are fuscous with a broad light ochreous-yellow streak along the dorsum throughout, from beyond the middle dilated so as to reach half across the wing. There is a fine strongly-curved violet-whitish line from three-fourths of the costa to the tornus, finely edged with dark fuscous posteriorly, margined anteriorly by an ochreous-yellow line edged with a few fuscous scales, and posteriorly on the upper half by a similar

neg: Parabola butyraula line terminated beneath by a blackish dot. The apical prominence is silvery-whitish. The hindwings are pale greyish, on the tornus tinged with whitish-ochreous. Parabola butyraula Parabola is a genus of moth in the family Gelechiidae. It contains the species Parabola butyraula, which is found in South Africa (Mpumalanga). The wingspan is about 15 mm. The forewings are fuscous with a broad light ochreous-yellow streak along the dorsum throughout, from beyond the middle dilated so as to reach half across the wing. There is a fine strongly-curved violet-whitish line from three-fourths of the costa to the tornus, finely edged with dark

neg: Superparabola Superparabola The superparabola is a geometric curve defined in the Cartesian coordinate system as a set of points withwhere "p", "a", and "b" are positive integers. The equation defines an open curve in the rectangle . The superparabola can vary in shape from a rectangular function , to a semi-ellipse ( , to a parabola , to a pulse function . Without loss of generality we can consider the canonical form of the superparabola When the function describes a continuous differentiable curve on the plane. The curve can be described parametrically on the complex plane as Derivatives of the superparabola

neg: Parabolic antenna Parabolic antenna A parabolic antenna is an antenna that uses a parabolic reflector, a curved surface with the cross-sectional shape of a parabola, to direct the radio waves. The most common form is shaped like a dish and is popularly called a dish antenna or parabolic dish. The main advantage of a parabolic antenna is that it has high directivity. It functions similarly to a searchlight or flashlight reflector to direct the radio waves in a narrow beam, or receive radio waves from one particular direction only. Parabolic antennas have some of the highest gains, meaning that they can produce

neg: The Quadrature of the Parabola The Quadrature of the Parabola The Quadrature of the Parabola () is a treatise on geometry, written by Archimedes in the 3rd century BC. Written as a letter to his friend Dositheus, the work presents 24 propositions regarding parabolas, culminating in a proof that the area of a parabolic segment (the region enclosed by a parabola and a line) is 4/3 that of a certain inscribed triangle. The statement of the problem used the method of exhaustion. Archimedes may have dissected the area into infinitely many triangles whose areas form a geometric progression. He computes the sum of the resulting

neg: Cissoid of Diocles accepts that cissoids of Diocles do exist, then there must exist at least one example of such a cissoid. This cissoid could then be translated, rotated, and expanded or contracted in size (without changing its proportional shape) at will to fit into any position. Then one would readily admit that such a cissoid can be used to correctly solve the Delian problem. The pedal curve of a parabola with respect to its vertex is a cissoid of Diocles. The geometrical properties of pedal curves in general produce several alternate methods of constructing the cissoid. It is the envelopes of circles

neg: Parabola GNU/Linux-libre of hardware problems surfaced, due to lack of free firmware. He said "That effectively means that many devices are not supported in Parabola. Eg., after installing it on my Acer Aspire One 531h netbook I couldn't use my Broadcom bluetooth module and Intel's WiFi/WiMax Link 5150 adapter. (I still could use my HTC Magic with a custom Gingerbread ROM as a USB 3G/WiFi modem, though having something always connected to USB port is pretty annoying on laptop and specifically on netbook.)" Czarkoff also criticized the lack of documentation available for Parabola. He concluded "The overall impression of the Parabola GNU/Linux

neg: Discriminant two of the equation. It is equal to and determines the shape of the conic section. If this discriminant is negative, the curve either has no real points, or is an ellipse or a circle, or, if degenerated, is reduced to a single point. If the discriminant is zero, the curve is a parabola, or, if degenerated, a double line or two parallel lines. If the discriminant is positive, the curve is a hyperbola, or, if degenerated, a pair of intersecting lines. A real quadric surface in the Euclidean space of dimension three is a surface that may be defined

neg: Parabola (magazine) Parabola (magazine) Parabola: Where Spiritual Traditions Meet, whose founder and editor was Dorothea M. Dooling, began publishing in Manhattan in 1976 as a quarterly magazine on the subjects of mythology and the world's religious and cultural traditions. It is published by The Society for the Study of Myth and Tradition, a not-for-profit organization. The name of the magazine is explained by the editors as follows: The parabola represents the epitome of a quest. It is the metaphorical journey to a particular point, and then back home, along a similar path perhaps, but in a different direction, after which the traveler

neg: Semicubical parabola maps the semicubical parabola formula_8 onto the curve formula_12 with formula_13. The "proof" follows from the tangent vector formula_17. Only for formula_18 this vector has zero length. Differentiating the "semicubical unit parabola" formula_19 one gets at point formula_20 of the "upper" branch the equation of the tangent: This tangent intersects the "lower" branch at exactly one further point with coordinates (Proving this statement one should use the fact, that the tangent meets the curve at formula_23 twice.) Determining the arclength of a curve formula_24 one has to solve the integral formula_25. For the semicubical parabola formula_26 one gets "Example:" For