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56-713: The Codroy Valley is a glacial valley formed in the Anguille Mountains , a sub-range of the Long Range Mountains which run along Newfoundland's west coast fronting the Gulf of St. Lawrence . The valley runs inland at a perpendicular angle from the coast along a bearing of 45° (northeast), carrying the Codroy River and its tributaries to the gulf. The mouth of the Codroy Valley at the coast

112-415: A x 2 + b x + c {\displaystyle y=ax^{2}+bx+c} (with a ≠ 0 {\displaystyle a\neq 0} ) is a parabola with its axis parallel to the y -axis. Conversely, every such parabola 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

168-405: A c = 0 , {\displaystyle b^{2}-4ac=0,} or, equivalently, such that a x 2 + b x y + c y 2 {\displaystyle ax^{2}+bxy+cy^{2}} is the square of a linear polynomial . The previous section shows that any parabola with the origin as vertex and the y axis as axis of symmetry can be considered as the graph of

224-412: A V-shaped valley to be carved into a U-shaped valley. These valleys can be several thousand feet deep and tens of miles long. Glaciers will spread out evenly in open areas, but tend to carve deep into the ground when confined to a valley. Ice thickness is a major contributing factor to valley depth and carving rates. As a glacier moves downhill through a valley, usually with a stream running through it,

280-403: A parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles . It is frequently used in physics , engineering , and many other areas. The earliest known work on conic sections was by Menaechmus in the 4th century BC. He discovered a way to solve the problem of doubling the cube using parabolas. (The solution, however, does not meet

336-407: A U-shaped, glaciated valley is often stepwise where flat basins are interrupted by thresholds. Rivers often dig a V-shaped valley or gorge through the threshold. Surrounding smaller tributary valleys will often join the main valleys during glaciation periods, leaving behind features known as hanging valleys high in the trough walls after the ice melts. After deglaciation, snow and ice melt from

392-399: A connection with this curve, as Apollonius had proved. The focus–directrix property of the parabola and other conic sections was mentioned in the works of Pappus . Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a parabolic reflector could produce an image was already well known before the invention of

448-405: A function f ( x ) = a x 2  with  a ≠ 0. {\displaystyle f(x)=ax^{2}{\text{ with }}a\neq 0.} For a > 0 {\displaystyle a>0} the parabolas are opening to the top, and for a < 0 {\displaystyle a<0} are opening to the bottom (see picture). From

504-489: A parabola can then be transformed by the uniform scaling ( x , y ) → ( a x , a y ) {\displaystyle (x,y)\to (ax,ay)} into the unit parabola with equation y = x 2 {\displaystyle y=x^{2}} . 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

560-420: A parabola is the inverse of a cardioid . Remark 2: The second polar form is a special case of a pencil of conics with focus F = ( 0 , 0 ) {\displaystyle F=(0,0)} (see picture): r = p 1 − e cos ⁡ φ {\displaystyle r={\frac {p}{1-e\cos \varphi }}} ( e {\displaystyle e}

616-645: A set of points ( locus of points ) in the Euclidean plane: The midpoint V {\displaystyle V} of the perpendicular from the focus F {\displaystyle F} onto the directrix l {\displaystyle l} is called the vertex , and the line F V {\displaystyle FV} is the axis of symmetry of the parabola. If one introduces Cartesian coordinates , such that F = ( 0 , f ) ,   f > 0 , {\displaystyle F=(0,f),\ f>0,} and

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672-681: Is V = ( 0 , 0 ) {\displaystyle V=(0,0)} , and its focus is F = ( p 2 , 0 ) {\displaystyle F=\left({\tfrac {p}{2}},0\right)} . If one shifts the origin into the focus, that is, F = ( 0 , 0 ) {\displaystyle F=(0,0)} , one obtains the equation r = p 1 − cos ⁡ φ , φ ≠ 2 π k . {\displaystyle r={\frac {p}{1-\cos \varphi }},\quad \varphi \neq 2\pi k.} Remark 1: Inverting this polar form shows that

728-446: Is U-shaped ( opening to the top ). The horizontal chord through the focus (see picture in opening section) is called the latus rectum ; one half of it is the semi-latus rectum . The latus rectum is parallel to the directrix. The semi-latus rectum is designated by the letter p {\displaystyle p} . From the picture one obtains p = 2 f . {\displaystyle p=2f.} The latus rectum

784-729: Is called a fjord , from the Norwegian word for these features that are common in Norway. Outside of Norway, a classic U-shaped valley that is also a fjord is the Western Brook Pond Fjord in Gros Morne National Park in Newfoundland , Canada. Formation of a U-shaped valley happens over geologic time , meaning not during a human's lifespan. It can take anywhere between 10,000 and 100,000 years for

840-420: Is defined similarly for the other two conics – the ellipse and the hyperbola. The latus rectum is the line drawn through a focus of a conic section parallel to the directrix and terminated both ways by the curve. For any case, p {\displaystyle p} is the radius of the osculating circle at the vertex. For a parabola, the semi-latus rectum, p {\displaystyle p} ,

896-519: Is extremely windy and is the location of Wreckhouse , so-named by employees of the historic Newfoundland Railway for the wind's ability to blow railway cars off the tracks. The area was settled families of French , Irish , Mi'kmaq , English , and Scots . The Scots were Highlanders who arrived between the 1840s and 1860s, most of them secondary migrants who had been living on Cape Breton Island in Inverness County , Nova Scotia . Of

952-1063: Is not mentioned above. It is defined and discussed below, in § Position of the focus . Let us call the length of DM and of EM x , and the length of PM   y . The lengths of BM and CM are: Using the intersecting chords theorem on the chords BC and DE , we get B M ¯ ⋅ C M ¯ = D M ¯ ⋅ E M ¯ . {\displaystyle {\overline {\mathrm {BM} }}\cdot {\overline {\mathrm {CM} }}={\overline {\mathrm {DM} }}\cdot {\overline {\mathrm {EM} }}.} Substituting: 4 r y cos ⁡ θ = x 2 . {\displaystyle 4ry\cos \theta =x^{2}.} Rearranging: y = x 2 4 r cos ⁡ θ . {\displaystyle y={\frac {x^{2}}{4r\cos \theta }}.} For any given cone and parabola, r and θ are constants, but x and y are variables that depend on

1008-456: Is quite soft and it was unbelievable to many that it could be responsible for the severe carving of bedrock characteristic of glacial erosion. German geologist Penck and American geologist Davis were vocal supporters of this unprecedented glacial erosion. Progress was made in the 1970s and 1980s on the possible mechanisms of glacial erosion and U-shaped valleys via models proposed by various scientists. Numerical models have been created to explain

1064-442: 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 waves . This reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from

1120-436: Is that two conic sections (necessarily of the same type) are similar if and only if they have the same eccentricity. Therefore, only circles (all having eccentricity 0) share this property with parabolas (all having eccentricity 1), while general ellipses and hyperbolas do not. There are other simple affine transformations that map the parabola y = a x 2 {\displaystyle y=ax^{2}} onto

1176-408: Is the distance of the focus from the directrix. Using the parameter p {\displaystyle p} , the equation of the parabola can be rewritten as x 2 = 2 p y . {\displaystyle x^{2}=2py.} More generally, if the vertex is V = ( v 1 , v 2 ) {\displaystyle V=(v_{1},v_{2})} ,

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1232-400: Is the eccentricity). The diagram represents a cone with its axis AV . The point A is its apex . An inclined cross-section of the cone, shown in pink, is inclined from the axis by the same angle θ , as the side of the cone. According to the definition of a parabola as a conic section, the boundary of this pink cross-section EPD is a parabola. A cross-section perpendicular to the axis of

1288-992: The Carpathian Mountains , the Pyrenees , the Rila and Pirin mountains in Bulgaria , and the Scottish Highlands . A classic glacial trough is in Glacier National Park in Montana , USA in which the St. Mary River runs. Another well-known U-shaped valley is the Nant Ffrancon valley in Snowdonia , Wales . When a U-shaped valley extends into saltwater, becoming an inlet of the sea, it

1344-749: The Lake District , Yosemite Valley , and the Rottal and Engstlige  [ de ] valleys in Switzerland . Glacial troughs also exist as submarine valleys on continental shelves, such as the Laurentian Channel . These geomorphic features significantly influence sediment distribution and biological communities through their modification of current patterns. Geologists did not always believe that glaciers were responsible for U-shaped valleys and other glacial erosional features. Ice

1400-568: The Principal Cordillera of the Andes, glacial valley floors may be covered by thick lava flows . A glacial trough or glaciated mountain valley often ends in an abrupt head known as the 'trough end' or 'trough head'. This may have almost sheer rock walls and spectacular waterfalls . They are believed to have been formed where a number of small glaciers merge to produce a much larger glacier. Examples include: Warnscale Bottom in

1456-871: The eccentricity . If p > 0 , the parabola with equation y 2 = 2 p x {\displaystyle y^{2}=2px} (opening to the right) has the polar representation r = 2 p cos ⁡ φ sin 2 ⁡ φ , φ ∈ [ − π 2 , π 2 ] ∖ { 0 } {\displaystyle r=2p{\frac {\cos \varphi }{\sin ^{2}\varphi }},\quad \varphi \in \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \{0\}} where r 2 = x 2 + y 2 ,   x = r cos ⁡ φ {\displaystyle r^{2}=x^{2}+y^{2},\ x=r\cos \varphi } . Its vertex

1512-572: The reflecting telescope . Designs were proposed in the early to mid-17th century by many mathematicians , including René Descartes , Marin Mersenne , and James Gregory . When Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror . Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers. A parabola can be defined geometrically as

1568-407: The apex of the cone, D and E move along the parabola, always maintaining the relationship between x and y 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. It is proved in a preceding section that if a parabola has its vertex at the origin, and if it opens in

1624-449: The arbitrary height at which the horizontal cross-section BECD is made. This last equation shows the relationship between these variables. They can be interpreted as Cartesian coordinates of the points D and E, in a system in the pink plane with P as its origin. Since x is squared in the equation, the fact that D and E are on opposite sides of the y axis is unimportant. If the horizontal cross-section moves up or down, toward or away from

1680-399: The cone passes through the vertex P of the parabola. This cross-section is circular, but appears elliptical when viewed obliquely, as is shown in the diagram. Its centre is V, and PK is a diameter. We will call its radius  r . Another perpendicular to the axis, circular cross-section of the cone is farther from the apex A than the one just described. It has a chord DE , which joins

1736-513: The directrix has the equation y = − 1 4 {\displaystyle y=-{\tfrac {1}{4}}} . The general function of degree 2 is f ( x ) = a x 2 + b x + c      with      a , b , c ∈ R ,   a ≠ 0. {\displaystyle f(x)=ax^{2}+bx+c~~{\text{ with }}~~a,b,c\in \mathbb {R} ,\ a\neq 0.} Completing

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1792-752: The directrix has the equation y = − f {\displaystyle y=-f} , one obtains for a point P = ( x , y ) {\displaystyle P=(x,y)} from | P F | 2 = | P l | 2 {\displaystyle |PF|^{2}=|Pl|^{2}} the equation x 2 + ( y − f ) 2 = ( y + f ) 2 {\displaystyle x^{2}+(y-f)^{2}=(y+f)^{2}} . Solving for y {\displaystyle y} yields y = 1 4 f x 2 . {\displaystyle y={\frac {1}{4f}}x^{2}.} This parabola

1848-396: The directrix. The parabola is the locus of points in that plane that are equidistant from 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 a plane parallel to another plane that is tangential to the conical surface. The graph of a quadratic function y =

1904-665: The equation uses the Hesse normal form of a line to calculate the distance | P l | {\displaystyle |Pl|} ). For a parametric equation of a parabola in general position see § As the affine image of the unit parabola . The implicit equation of a parabola is defined by an irreducible polynomial of degree two: a x 2 + b x y + c y 2 + d x + e y + f = 0 , {\displaystyle ax^{2}+bxy+cy^{2}+dx+ey+f=0,} such that b 2 − 4

1960-752: The focus F = ( v 1 , v 2 + f ) {\displaystyle F=(v_{1},v_{2}+f)} , and the directrix y = v 2 − f {\displaystyle y=v_{2}-f} , one obtains the equation y = 1 4 f ( x − v 1 ) 2 + v 2 = 1 4 f x 2 − v 1 2 f x + v 1 2 4 f + v 2 . {\displaystyle y={\frac {1}{4f}}(x-v_{1})^{2}+v_{2}={\frac {1}{4f}}x^{2}-{\frac {v_{1}}{2f}}x+{\frac {v_{1}^{2}}{4f}}+v_{2}.} Remarks : If

2016-638: The focus is F = ( f 1 , f 2 ) {\displaystyle F=(f_{1},f_{2})} , and the directrix a x + b y + c = 0 {\displaystyle ax+by+c=0} , then one obtains the equation ( a x + b y + c ) 2 a 2 + b 2 = ( x − f 1 ) 2 + ( y − f 2 ) 2 {\displaystyle {\frac {(ax+by+c)^{2}}{a^{2}+b^{2}}}=(x-f_{1})^{2}+(y-f_{2})^{2}} (the left side of

2072-415: 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 that travels parallel to the axis of symmetry of a parabola and strikes its concave side

2128-568: The ice recedes or thaws, the valley remains, often littered with small boulders that were transported within the ice, called glacial till or glacial erratic . Examples of U-shaped valleys are found in mountainous regions throughout the world including the Andes , Alps , Caucasus Mountains , Himalaya , Rocky Mountains , New Zealand and the Scandinavian Mountains . They are found also in other major European mountains including

2184-403: The middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the " vertex " and is the point where 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 that is parallel to the directrix and passes through

2240-493: The mountain tops can create streams and rivers in U-shaped valleys. These are referred to as misfit streams. The streams that form in hanging valleys create waterfalls that flow into the main valley branch. Glacial valleys may also have natural, often dam-like, structures within them, called moraines . They are created due to excess sediment and glacial till moved and deposited by the glacier. In volcanic mountain ranges, such as

2296-398: The origin (0, 0) and the same semi-latus rectum p {\displaystyle p} can be represented by the equation y 2 = 2 p x + ( e 2 − 1 ) x 2 , e ≥ 0 , {\displaystyle y^{2}=2px+(e^{2}-1)x^{2},\quad e\geq 0,} with e {\displaystyle e}

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2352-430: The origin as vertex. A suitable rotation around the origin can then transform the parabola to one that has the y axis as axis of symmetry. Hence the parabola P {\displaystyle {\mathcal {P}}} can be transformed by a rigid motion to a parabola with an equation y = a x 2 ,   a ≠ 0 {\displaystyle y=ax^{2},\ a\neq 0} . Such

2408-587: The other by a similarity , that is, an arbitrary composition of rigid motions ( translations and rotations ) and uniform scalings . A parabola P {\displaystyle {\mathcal {P}}} with vertex V = ( v 1 , v 2 ) {\displaystyle V=(v_{1},v_{2})} can be transformed by the translation ( x , y ) → ( x − v 1 , y − v 2 ) {\displaystyle (x,y)\to (x-v_{1},y-v_{2})} to one with

2464-403: The parabola, which is the distance from the vertex to the focus. The focus and the point F are therefore equally distant from the vertex, along the same line, which implies that they are the same point. Therefore, the point F, defined above, is the focus of the parabola . This discussion started from the definition of a parabola as a conic section, but it has now led to a description as

2520-405: 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 θ , and angle PVF is complementary to angle VPF, therefore angle PVF is θ . Since the length of PV is r , the distance of F from the vertex of the parabola is r sin θ . It is shown above that this distance equals the focal length of

2576-426: The phenomenon of carving U-shaped valleys. Parabola In mathematics , a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits 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

2632-399: The points where the parabola intersects the circle. Another chord BC is the perpendicular bisector of DE and is consequently a diameter of the circle. These two chords and the parabola's axis of symmetry PM all intersect at the point M. All the labelled points, except D and E, are coplanar . They are in the plane of symmetry of the whole figure. This includes the point F, which

2688-407: The positive y direction, then its equation is y = ⁠ x / 4 f ⁠ , where f is its focal length. Comparing this with the last equation above shows that the focal length of the parabola in the cone is r sin θ . 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

2744-407: The process of glaciation . They are characteristic of mountain glaciation in particular. They have a characteristic U shape in cross-section, with steep, straight sides and a flat or rounded bottom (by contrast, valleys carved by rivers tend to be V-shaped in cross-section). Glaciated valleys are formed when a glacier travels across and down a slope, carving the valley by the action of scouring. When

2800-500: 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 by the method of exhaustion in the 3rd 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 to "application of areas" concept, that has

2856-462: The roughly 171 households at Codroy Valley in the 1880s, 67 (38%) belonged to people of Scottish descent. The Scottish Gaelic language was once commonly spoken here, with some families continuing to speak Gaelic at home until the 1960s. This Newfoundland and Labrador location article is a stub . You can help Misplaced Pages by expanding it . Glacial valley U-shaped valleys , also called trough valleys or glacial troughs , are formed by

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2912-470: The section above one obtains: For a = 1 {\displaystyle a=1} the parabola is the unit parabola with equation y = x 2 {\displaystyle y=x^{2}} . Its focus is ( 0 , 1 4 ) {\displaystyle \left(0,{\tfrac {1}{4}}\right)} , the semi-latus rectum p = 1 2 {\displaystyle p={\tfrac {1}{2}}} , and

2968-541: The shape of the valley is transformed. As the ice melts and retreats, the valley is left with very steep sides and a wide, flat floor. This parabolic shape is caused by glacial erosion removing the contact surfaces with greatest resistance to flow, and the resulting section minimises friction. There are two main variations of this U-shape. The first is called the Rocky Mountain model and it is attributed to alpine glacial valleys, showing an overall deepening effect on

3024-425: The square yields f ( x ) = a ( x + b 2 a ) 2 + 4 a c − b 2 4 a , {\displaystyle f(x)=a\left(x+{\frac {b}{2a}}\right)^{2}+{\frac {4ac-b^{2}}{4a}},} which is the equation of a parabola with Two objects in the Euclidean plane are similar if one can be transformed to

3080-435: The unit parabola, such as ( x , y ) → ( x , y a ) {\displaystyle (x,y)\to \left(x,{\tfrac {y}{a}}\right)} . But this mapping is not a similarity, and only shows that all parabolas are affinely equivalent (see § As the affine image of the unit parabola ). The pencil of conic sections with the x axis as axis of symmetry, one vertex at

3136-1105: The valley. The second variation is referred to as the Patagonia-Antarctica model, attributed to continental ice sheets and displaying an overall widening effect on its surroundings. The floors of these glacial valleys are where the most evidence can be found regarding glaciation cycles. For the most part, the valley floor is wide and flat, but there are various glacial features that signify periods of ice transgression and regression. The valley can have various steps, known as valley steps , and over-deepenings anywhere from ten to hundreds of meters deep. These then fill in with sediments to create plains or water to create lakes, sometimes referred to as "string-of-pearl" or ribbon lakes. Such water filled U-valley basins are also known as "fjord-lakes" or "valley-lakes" (Norwegian: fjordsjø or dalsjø ). Gjende and Bandak lakes in Norway are examples of fjord -lakes. Some of these fjord-lakes are very deep for instance Mjøsa (453 meters) and Hornindalsvatnet (514 m). The longitudinal profile of

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