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42-462: The chairlift has been replaced twice, in 1962 and 1992. Like its predecessors, the current chairlift loads near the lodge at 5,950 feet (1829 m) and unloads at 7,000 foot (2134 m), up an average gradient of 20%. Except for the very lowest part of the route, the lift is not protected by trees or land features and faces the full force of snow storms. Heavy winds frequently produce huge snowdrifts and copious and dense snow challenge lift crews to keep

84-409: A 71 percent grade instead of a 100 percent. But in practice the usual way to calculate slope is to measure the distance along the slope and the vertical rise, and calculate the horizontal run from that, in order to calculate the grade (100% × rise/run) or standard slope (rise/run). When the angle of inclination is small, using the slope length rather than the horizontal displacement (i.e., using

126-415: A physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal . It is a special case of the slope , where zero indicates horizontality . A larger number indicates higher or steeper degree of "tilt". Often slope is calculated as a ratio of "rise" to "run", or as a fraction ("rise over run") in which run is the horizontal distance (not the distance along

168-418: A projectile fired in a horizontal direction (i.e., with a zero vertical component) may leave the surface of the spherical Earth and indeed escape altogether. In the context of a 1-dimensional orthogonal Cartesian coordinate system on a Euclidean plane, to say that a line is horizontal or vertical, an initial designation has to be made. One can start off by designating the vertical direction, usually labelled

210-488: A projectile moving under gravity are independent of each other. Vertical displacement of a projectile is not affected by the horizontal component of the launch velocity, and, conversely, the horizontal displacement is unaffected by the vertical component. The notion dates at least as far back as Galileo. When the curvature of the Earth is taken into account, the independence of the two motion does not hold. For example, even

252-465: A radial, it may even be curved and be varying with time. On a smaller scale, a mountain to one side may deflect the plumb bob away from the true zenith . On a larger scale the gravitational field of the Earth, which is at least approximately radial near the Earth, is not radial when it is affected by the Moon at higher altitudes. Neglecting the curvature of the earth, horizontal and vertical motions of

294-459: A spinning earth, the plumb line deviates from the radial direction as a function of latitude. Only on the equator and at the North and South Poles does the plumb line align with the local radius. The situation is actually even more complicated because Earth is not a homogeneous smooth sphere. It is a non homogeneous, non spherical, knobby planet in motion, and the vertical not only need not lie along

336-687: A vehicle can ascend while maintaining a particular speed is sometimes termed that vehicle's "gradeability" (or, less often, "grade ability"). The lateral slopes of a highway geometry are sometimes called fills or cuts where these techniques have been used to create them. In the United States, the maximum grade for federally funded highways is specified in a design table based on terrain and design speeds, with up to 6% generally allowed in mountainous areas and hilly urban areas with exceptions for up to 7% grades on mountainous roads with speed limits below 60 mph (95 km/h). The steepest roads in

378-739: Is 23.1% by the 67-Bernal Heights on Alabama Street between Ripley and Esmeralda Streets. Likewise, the Pittsburgh Department of Engineering and Construction recorded a grade of 37% (20°) for Canton Avenue. The street has formed part of a bicycle race since 1983. Grade, pitch, and slope are important components in landscape design , garden design , landscape architecture , and architecture ; for engineering and aesthetic design factors. Drainage, slope stability, circulation of people and vehicles, complying with building codes, and design integration are all aspects of slope considerations in environmental design . Ruling gradients limit

420-654: Is expressed as a percentage, the angle can be determined as: If the angle is expressed as a ratio (1 in n) then: For degrees, percentage (%) and per-mille (‰) notations, larger numbers are steeper slopes. For ratios, larger numbers n of 1 in n are shallower, easier slopes. The examples show round numbers in one or more of the notations and some documented and reasonably well known instances. In vehicular engineering , various land -based designs ( automobiles , sport utility vehicles , trucks , trains , etc.) are rated for their ability to ascend terrain . Trains typically rate much lower than automobiles. The highest grade

462-424: The perpendicular cross slope . There are several ways to express slope: Any of these may be used. Grade is usually expressed as a percentage, but this is easily converted to the angle α by taking the inverse tangent of the standard mathematical slope, which is rise / run or the grade / 100. If one looks at red numbers on the chart specifying grade, one can see the quirkiness of using the grade to specify slope;

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504-510: The 1940s and early 1950s. Mounting disrepair, vandalism, neglect and unpaid taxes closed it February 17, 1955, at which time the Magic Mile was nonfunctional. The lodge reopened late that year under the management of Richard Kohnstamm, whose family still manages the resort. The Mile was made functional again, and in the following summer, ski racing camps began. In 1950, a cable car system began operation between 3,800 feet (1,200 m) and

546-466: The Great Western main line, nicknamed Brunel's Billiard Table, because the early locomotives (and their brakes) were feeble. Steep gradients were concentrated in short sections of lines where it was convenient to employ assistant engines or cable haulage , such as the 1.2 kilometres (0.75 miles) section from Euston to Camden Town . Extremely steep gradients require the use of cables (such as

588-639: The Guinness Book. Drawing on the U.S. National Elevation Dataset, 7x7 (magazine) identified ten blocks of public streets in San Francisco open to vehicular traffic in the city with grades over 30 percent. The steepest at 41 percent is the block of Bradford Street above Tompkins Avenue in the Bernal Heights neighborhood. The San Francisco Municipal Railway operates bus service among the city's hills . The steepest grade for bus operations

630-684: The Scenic Railway at Katoomba Scenic World , Australia, with a maximum grade of 122% (52°), claimed to be the world's steepest passenger-carrying funicular ) or some kind of rack railway (such as the Pilatus railway in Switzerland, with a maximum grade of 48% (26°), claimed to be the world's steepest rack railway ) to help the train ascend or descend. Gradients can be expressed as an angle, as feet per mile, feet per chain, 1 in n , x % or y per mille. Since designers like round figures,

672-571: The Timberline Lodge at 6,000 feet (1,800 m), a 3-mile (4.8 km) trip that took 10 minutes. Each 36-person "bus" pulled itself up the mountain along the cable. By 1962, the Magic Mile had long been a challenge to maintain, so the Timberline's operator replaced it with a double Riblet lift. The new lift had a midway station for loading and unloading, which allowed the lower mountain to remain open when harsh winter weather closed

714-411: The Y direction. The horizontal direction, usually labelled the X direction, is then automatically determined. Or, one can do it the other way around, i.e., nominate the x -axis, in which case the y -axis is then automatically determined. There is no special reason to choose the horizontal over the vertical as the initial designation: the two directions are on par in this respect. The following hold in

756-462: The buoyancy of an air bubble and its tendency to go vertically upwards may be used to test for horizontality. A water level device may also be used to establish horizontality. Modern rotary laser levels that can level themselves automatically are robust sophisticated instruments and work on the same fundamental principle. When the curvature of the Earth is taken into account, the concepts of vertical and horizontal take on yet another meaning. On

798-451: The classroom. For the teacher, writing perhaps on a white board, the y -axis really is vertical in the sense of the plumbline verticality but for the student the axis may well lie on a horizontal table. Although the word horizontal is commonly used in daily life and language (see below), it is subject to many misconceptions. In general or in practice, something that is horizontal can be drawn from left to right (or right to left), such as

840-563: The fact that the typical linear scales and dimensions of relevance in daily life are 3 orders of magnitude (or more) smaller than the size of the Earth. Hence, the world appears to be flat locally, and horizontal planes in nearby locations appear to be parallel. Such statements are nevertheless approximations; whether they are acceptable in any particular context or application depends on the applicable requirements, in particular in terms of accuracy. In graphical contexts, such as drawing and drafting and Co-ordinate geometry on rectangular paper, it

882-404: The gradient through the platform was eased from 1 in 80 to 1 in 130. Horizontal plane In astronomy , geography , and related sciences and contexts, a direction or plane passing by a given point is said to be vertical if it contains the local gravity direction at that point. Conversely, a direction or plane is said to be horizontal (or leveled ) if it is perpendicular to

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924-461: The horizontal plane is a vertical plane at P. Through any point P, there is one and only one horizontal plane but a multiplicity of vertical planes. This is a new feature that emerges in three dimensions. The symmetry that exists in the two-dimensional case no longer holds. In the 2-dimension case, as mentioned already, the usual designation of the vertical coincides with the y-axis in co-ordinate geometry. This convention can cause confusion in

966-463: The horizontal run by using the Pythagorean theorem, after which it is trivial to calculate the (standard math) slope or the grade (percentage). In Europe, road gradients are signed as a percentage. Grades are related using the following equations with symbols from the figure at top. The slope expressed as a percentage can similarly be determined from the tangent of the angle: If the tangent

1008-427: The lift open. The lift is generally closed when winds exceed 50–60 miles per hour (80–97 km/h) or dense fog reduces visibility below about 25 feet (7.6 m) — in all, about 40% of winter days. Construction of the original Magic Mile began in mid-1938, a few months after Timberline Lodge opened for business with a portable rope tow. (The tow remained in operation for at least several years.) The chairlift

1050-546: The load that a locomotive can haul, including the weight of the locomotive itself. On a 1% gradient (1 in 100) a locomotive can pull half (or less) of the load that it can pull on level track. (A heavily loaded train rolling at 20 km/h on heavy rail may require ten times the pull on a 1% upgrade that it does on the level at that speed.) Early railways in the United Kingdom were laid out with very gentle gradients, such as 0.07575% (1 in 1320) and 0.1515% (1 in 660) on

1092-539: The lodge for easy access, and for skier convenience from the top of the Pucci chairlift, which was installed in 1956. The Palmer chairlift, which opened July 1, 1980, was situated for convenient skier and snowboard transport from the top of the Mile. The Palmer was upgraded to a high-speed quad in 1996. In 1992, the Magic Mile chairlift was upgraded from a fixed grip double to a detachable high speed quad . The midway station

1134-433: The lower lift house. The rope runs at low speed to prevent the buildup of snow and ice. The chairs are redeployed in an automatic operation which takes about 30 minutes. 45°19′52″N 121°42′49″W  /  45.33103°N 121.71358°W  / 45.33103; -121.71358 Grade (slope) The grade (US) or gradient (UK) (also called stepth , slope , incline , mainfall , pitch or rise ) of

1176-753: The method of expression can affect the gradients selected. The steepest railway lines that do not use a rack system include: Gradients on sharp curves are effectively a bit steeper than the same gradient on straight track, so to compensate for this and make the ruling grade uniform throughout, the gradient on those sharp curves should be reduced slightly. In the era before they were provided with continuous brakes , whether air brakes or vacuum brakes , steep gradients made it extremely difficult for trains to stop safely. In those days, for example, an inspector insisted that Rudgwick railway station in West Sussex be regraded. He would not allow it to open until

1218-400: The numbers go from 0 for flat, to 100% at 45 degrees, to infinity as it approaches vertical. Slope may still be expressed when the horizontal run is not known: the rise can be divided by the hypotenuse (the slope length). This is not the usual way to specify slope; this nonstandard expression follows the sine function rather than the tangent function, so it calls a 45 degree slope

1260-400: The same root as vertex , meaning 'highest point' or more literally the 'turning point' such as in a whirlpool. Girard Desargues defined the vertical to be perpendicular to the horizon in his 1636 book Perspective . In physics, engineering and construction, the direction designated as vertical is usually that along which a plumb-bob hangs. Alternatively, a spirit level that exploits

1302-736: The sine of the angle rather than the tangent) makes only an insignificant difference and can then be used as an approximation. Railway gradients are often expressed in terms of the rise in relation to the distance along the track as a practical measure. In cases where the difference between sin and tan is significant, the tangent is used. In either case, the following identity holds for all inclinations up to 90 degrees: tan ⁡ α = sin ⁡ α 1 − sin 2 ⁡ α {\displaystyle \tan {\alpha }={\frac {\sin {\alpha }}{\sqrt {1-\sin ^{2}{\alpha }}}}} . Or more simply, one can calculate

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1344-413: The slope) and rise is the vertical distance. Slopes of existing physical features such as canyons and hillsides, stream and river banks , and beds are often described as grades, but typically the word "grade" is used for human-made surfaces such as roads, landscape grading, roof pitches , railroads , aqueducts , and pedestrian or bicycle routes. The grade may refer to the longitudinal slope or

1386-503: The surface of a smoothly spherical, homogenous, non-rotating planet, the plumb bob picks out as vertical the radial direction. Strictly speaking, it is now no longer possible for vertical walls to be parallel: all verticals intersect. This fact has real practical applications in construction and civil engineering, e.g., the tops of the towers of a suspension bridge are further apart than at the bottom. Also, horizontal planes can intersect when they are tangent planes to separated points on

1428-624: The surface of the Earth. In particular, a plane tangent to a point on the equator intersects the plane tangent to the North Pole at a right angle . (See diagram). Furthermore, the equatorial plane is parallel to the tangent plane at the North Pole and as such has claim to be a horizontal plane. But it is. at the same time, a vertical plane for points on the equator. In this sense, a plane can, arguably, be both horizontal and vertical, horizontal at one place , and vertical at another . For

1470-465: The two-dimensional case: Not all of these elementary geometric facts are true in the 3-D context. In the three-dimensional case, the situation is more complicated as now one has horizontal and vertical planes in addition to horizontal and vertical lines. Consider a point P and designate a direction through P as vertical. A plane which contains P and is normal to the designated direction is the horizontal plane at P. Any plane going through P, normal to

1512-402: The upper mountain, and the upper mountain to remain open for mid-to-late summer skiing when little snow remained on the lower mountain. The full ride up this second Magic Mile ride took about the same time as the original, 10 to 12 minutes, but tighter chair spacing and two riders per chair increased the capacity about 800 to 1,000 per hour. The bottom of the Mile was placed at the west side of

1554-561: The vertical direction. In general, something that is vertical can be drawn from up to down (or down to up), such as the y-axis in the Cartesian coordinate system . The word horizontal is derived from the Latin horizon , which derives from the Greek ὁρῐ́ζων , meaning 'separating' or 'marking a boundary'. The word vertical is derived from the late Latin verticalis , which is from

1596-715: The world according to the Guinness Book of World Records are Baldwin Street in Dunedin, New Zealand, Ffordd Pen Llech in Harlech, Wales and Canton Avenue in Pittsburgh, Pennsylvania. The Guinness World Record once again lists Baldwin Street as the steepest street in the world, with a 34.8% grade (1 in 2.87) after a successful appeal against the ruling that handed the title, briefly, to Ffordd Pen Llech . A number of streets elsewhere have steeper grades than those listed in

1638-452: The x-axis in the Cartesian coordinate system . The concept of a horizontal plane is thus anything but simple, although, in practice, most of these effects and variations are rather small: they are measurable and can be predicted with great accuracy, but they may not greatly affect our daily life. This dichotomy between the apparent simplicity of a concept and an actual complexity of defining (and measuring) it in scientific terms arises from

1680-472: Was as popular a summer tourist attraction then as it is now. The lift line was slightly east of the present chair. The upper bullwheel was inside Silcox Hut , which is 212 m (700 ft) ESE and 40 ft (12 m) lower in elevation. The bottom was east of the lodge about 377 m (1236 ft, one quarter mile) ENE at essentially the same elevation as the present chair. Timberline Lodge shut down for World War II and struggled financially through

1722-433: Was removed, and a new top station built slightly higher than its predecessor, lengthening its run to 5,500 feet (1,700 m). This Poma -built chair can move 3,000 passengers per hour, but is operated at 1,600 passengers per hour, with a ride time of just under 6 minutes. The use of a detachable chairlift reduces maintenance needed to clear the haul rope. When bad weather is expected, the chairs are removed and stored in

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1764-535: Was the first built by the Riblet company, which drew heavily on its designs for aerial trams for mining companies. Completed in late 1939, it loaded its first passengers on November 17, 1939, and was dedicated by the Crown Prince and Princess of Norway (later King Olaf ). The original chairlift was a single — each chair held one rider. The ride took 11 minutes and carried 225 passengers per hour. It

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