The **grade** (also called **slope**, **incline**, **gradient**, **mainfall**, **pitch** or **rise**) of 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 the slope) and *rise* is the vertical distance.

- Nomenclature
- Equations
- Roads
- Environmental design
- Railways
- Compensation for curvature
- Continuous brakes
- See also
- References
- External links

Slopes of existing physical features such as canyons and hillsides, stream and river banks and beds are often described as grades, but typically grades are 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 the perpendicular cross slope.

There are several ways to express slope:

- as an
*angle*of inclination to the horizontal. (This is the angle α opposite the "rise" side of a triangle with a right angle between vertical rise and horizontal run.) - as a
*percentage*, the formula for which is which is equivalent to the tangent of the angle of inclination times 100. In Europe and the U.S. percentage "grade" is the most commonly used figure for describing slopes. - as a
*per mille*figure (‰), the formula for which is which could also be expressed as the tangent of the angle of inclination times 1000. This is commonly used in Europe to denote the incline of a railway. - as a
*ratio*of one part rise to so many parts run. For example, a slope that has a rise of 5 feet for every 1000 feet of run would have a slope ratio of 1 in 200. (The word "in" is normally used rather than the mathematical ratio notation of "1:200".) This is generally the method used to describe railway grades in Australia and the UK. It is used for roads in Hong Kong, and was used for roads in the UK until the 1970s. - as a
*ratio*of many parts run to one part rise, which is the inverse of the previous expression (depending on the country and the industry standards). For example, "slopes are expressed as ratios such as 4:1. This means that for every 4 units (feet or metres) of horizontal distance there is a 1 unit (foot or metre) vertical change either up or down."^{ [1] }

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; 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 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 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: . Or more simply, one can calculate 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.^{ [2] }

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 is expressed as a percentage, the angle can be determined as:

If the angle is expressed as a ratio *(1 in n)* then:

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 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, maximum grade for Federally funded highways is specified in a design table based on terrain and design speeds,^{ [3] } 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 the world are Baldwin Street in Dunedin, New Zealand, Ffordd Pen Llech in Harlech, Wales^{ [4] } and Canton Avenue in Pittsburgh, Pennsylvania.^{ [5] } 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^{ [6] } against the ruling that handed the title, briefly, to Ffordd Pen Llech. The Pittsburgh Department of Engineering and Construction recorded a grade of 37% (20°) for Canton Avenue.^{ [7] } The street has formed part of a bicycle race since 1983.^{ [8] }

The San Francisco Municipal Railway operates bus service among the city's hills. The steepest grade for bus operations is 23.1% by the * 67-Bernal Heights * on Alabama Street between Ripley and Esmeralda Streets.^{ [9] }

- 10% slope warning sign, Netherlands
- 7% descent warning sign, Finland
- 25% ascent warning sign, Wales
- A trolleybus climbing an 18% grade in Seattle
- ascent of German Bundesstraße 10
- A car parked on Baldwin Street, Dunedin, New Zealand
- Looking down Canton Avenue, Pittsburgh, Pennsylvania

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 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 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 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^{ [10] }) 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^{ [11] }) 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, the method of expression can affect the gradients selected.^{[ citation needed ]}

The steepest railway lines that do not use a rack system include:

- 13.5% (1 in 7.40) – Lisbon tram, Portugal
- 11.6% (1 in 8.62) – Pöstlingbergbahn, Linz, Austria
^{ [12] } - 11.0% (1 in 9.09) – Cass Scenic Railway, US (former logging line)
- 9.0% (1 in 11.11) – Ligne de Saint Gervais – Vallorcine, France
- 9.0% (1 in 11.11) – Muni Metro J Church, San Francisco, US
^{ [9] } - 8.8% (1 in 11.4) – Iași tram, Romania
^{ [13] } - 8.65% (1 in 11.95) – Portland Streetcar, Oregon, US
^{ [14] } - 8.33%(1 in 12) – Nilgiri Mountain Railway Tamil Nadu, India
- 8.0% (1 in 12.5) – Just outside the Tobstone Jct. Station in the Tombstone Junction Theme Park, Kentucky, US. The railroad line there had a ruling grade of 6% (1 in 16.7).
- 7.1% (1 in 14.08) – Erzberg Railway, Austria
- 7.0% (1 in 14.28) – Bernina Railway, Switzerland
- 6.5% (1 in 15.4) – Incline from the Causeway Street Tunnel up to the Lechmere Viaduct on the Green Line (MBTA), Boston, Massachusetts, US.
^{ [15] }This incline is the "steepest grade of tracks in the T system."^{ [16] } - 6.0% (1 in 16.7) – Arica, Chile to La Paz, Bolivia
- 6.0% (1 in 16.6) – Docklands Light Railway, London, UK
- 6.0% (1 in 16.6) – Ferrovia Centrale Umbra, Italy
^{ [17] } - 5.89% (1 in 16.97) – Madison, Indiana, US
^{ [18] } - 5.6% (1 in 18) – Flåm Line, Norway
- 5.3% (1 in 19) – Foxfield Railway, Staffordshire, UK
- 5.1% (1 in 19.6) – Saluda Grade, North Carolina, US
- 5.0% (1 in 20) – Khyber Pass Railway, Pakistan
- 4.5% (1 in 22.2) – The Canadian Pacific Railway's Big Hill, British Columbia, Canada (prior to the construction of the Spiral Tunnels)
- 4.0% (1 in 25) – Cologne-Frankfurt high-speed rail line, Germany
- 4.0% (1 in 25) – Bolan Pass Railway, Pakistan
- 4.0% (1 in 25) – (211.2 feet (64 m) per 1 mile (1,600 m) ) – Tarana – Oberon branch, New South Wales, Australia.
- 4.0% (1 in 25) – Matheran Light Railway, India
^{ [19] } - 4.0% (1 in 26) – Rewanui Incline, New Zealand. Fitted with Fell center rail but was not used for motive power, but only braking
- 3.6% (1 in 27) – Ecclesbourne Valley Railway, Heritage Line, Wirksworth, Derbyshire, UK
- 3.6% (1 in 28) – The Westmere Bank, New Zealand has a ruling gradient of 1 in 35, however peaks at 1 in 28
- 3.33% (1 in 30) – Umgeni Steam Railway, South Africa
^{ [20] } - 3.0% (1 in 33) – several sections of the Main Western line between Valley Heights and Katoomba in the Blue Mountains Australia.
^{ [21] } - 3.0% (1 in 33) – The entire Newmarket Line in central Auckland, New Zealand
- 3.0% (1 in 33) – Otira Tunnel, New Zealand, which is equipped with extraction fans to reduce chance of overheating and low visibility
- 3.0% (1 in 33) – The approaches to the George L. Anderson Memorial Bridge crossing the Neponset River, Boston, Massachusetts, US. The Ruling Gradient of the Braintree Branch of the Red Line (MBTA).
^{ [22] } - 2.7% (1 in 37) – Braganza Ghats, Bhor Ghat and Thull ghat sections in Indian Railways, India
- 2.7% (1 in 37) – Exeter Central to Exeter St Davids, UK (see Exeter Central railway station#Description)
- 2.7% (1 in 37) – Picton- Elevation, New Zealand
- 2.65% (1 in 37.7) – Lickey Incline, UK
- 2.6% (1 in 38) – A slope near Halden on Østfold Line, Norway – Ok for passenger multiple units, but an obstacle for freight trains which must keep their weight down on this international mainline because of the slope. Freight traffic has mainly shifted to road.
- 2.3% (1 in 43.5) – Schiefe Ebene, Germany
- 2.2% (1 in 45.5) – The Canadian Pacific Railway's Big Hill, British Columbia, Canada (after the construction of the Spiral Tunnels)
- 2.0% (1 in 48) – Beasdale Bank (West Coast Scotland mainline), UK
- 2.0% (1 in 50) – Numerous locations on New Zealand's railway network, New Zealand
- 1.51% (1 in 66) – (1 foot (0.3 m) per 1 chain (20 m)) New South Wales Government Railways, Australia, part of Main South line.
- 1.25% (1 in 80) – Wellington Bank, Somerset, UK
- 1.25% (1 in 80) – Rudgwick, UK (West Sussex) platform before regrading – too steep if a train is not provided with continuous brakes.
- 0.77% (1 in 130) – Rudgwick, UK platform after regrading – not too steep if a train is not provided with continuous brakes.

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 the gradient through the platform was eased from 1 in 80 to 1 in 130.

- Aspect (geography)
- Civil engineering
- Construction surveying
- Grading (engineering)
- Cut-and-cover
- Cut and fill
- Cut (earthmoving)
- Embankment (transportation)
- Grade separation
- Inclined plane
- List of steepest gradients on adhesion railways.
- Percentage
- Per mille
- Rake
- Roof pitch
- Ruling gradient
- Slope
- Slope stability
- Surface gradient
- Surveying
- Trench
- Tunnel
- Wheelchair ramp

In vector calculus, the **gradient** of a scalar-valued differentiable function *f* of several variables is the vector field whose value at a point is the vector whose components are the partial derivatives of at . That is, for , its gradient is defined at the point in *n-*dimensional space as the vector:

In mathematics, the **slope** or **gradient** of a line is a number that describes both the *direction* and the *steepness* of the line. Slope is often denoted by the letter *m*; there is no clear answer to the question why the letter *m* is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as "*y* = *mx* + *b*" and it can also be found in Todhunter (1888) who wrote it as "*y* = *mx* + *c*".

A **triangle** is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices *A*, *B*, and *C* is denoted .

An **inclined plane**, also known as a **ramp**, is a flat supporting surface tilted at an angle, with one end higher than the other, used as an aid for raising or lowering a load. The inclined plane is one of the six classical simple machines defined by Renaissance scientists. Inclined planes are widely used to move heavy loads over vertical obstacles; examples vary from a ramp used to load goods into a truck, to a person walking up a pedestrian ramp, to an automobile or railroad train climbing a grade.

The **angle of view** is the decisive variable for the visual perception of the size or projection of the size of an object.

In geometry, a **golden spiral** is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider by a factor of φ for every quarter turn it makes.

In mathematics, the **inverse trigonometric functions** are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

In trigonometry, **tangent half-angle formulas** relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the circle onto a line. Among these formulas are the following:

**Rifleman's rule** is a "rule of thumb" that allows a rifleman to accurately fire a rifle that has been calibrated for horizontal targets at uphill or downhill targets. The rule says that only the horizontal range should be considered when adjusting a sight or performing hold-over in order to account for bullet drop. Typically, the range of an elevated target is considered in terms of the *slant range*, incorporating both the *horizontal distance* and the *elevation distance*, as when a rangefinder is used to determine the distance to target. The slant range is not compatible with standard ballistics tables for estimating bullet drop.

**Geomorphometry**, or **geomorphometrics**, is the science and practice of measuring the characteristics of terrain, the shape of the surface of the Earth, and the effects of this surface form on human and natural geography. It gathers various mathematical, statistical and image processing techniques that can be used to quantify morphological, hydrological, ecological and other aspects of a land surface. Common synonyms for geomorphometry are **geomorphological analysis**, **terrain morphometry**, **terrain analysis**, and **land surface analysis**. **Geomorphometrics** is the discipline based on the computational measures of the geometry, topography and shape of the Earth's horizons, and their temporal change. This is a major component of geographic information systems (GIS) and other software tools for spatial analysis.

In mathematics, the **elasticity** or **point elasticity** of a positive differentiable function *f* of a positive variable at point *a* is defined as

The main **trigonometric identities** between trigonometric functions are proved, using mainly the geometry of the right triangle. For greater and negative angles, see Trigonometric functions.

In numerical optimization, the **nonlinear conjugate gradient method** generalizes the conjugate gradient method to nonlinear optimization. For a quadratic function

A **biarc** is a smooth curve formed from two circular arcs. In order to make the biarc smooth, the two arcs should have the same tangent at the connecting point where they meet.

In mathematics and statistics, a **circular mean** is a mean designed for angles and similar cyclic quantities, such as daytimes, and fractional parts of real numbers. This is necessary since most of the usual means may not be appropriate on angle-like quantities. For example, the arithmetic mean of 0° and 360° is 180°, which is misleading because 360° equals 0° modulo a full cycle. As another example, the "average time" between 11 PM and 1 AM is either midnight or noon, depending on whether the two times are part of a single night or part of a single calendar day. The circular mean is one of the simplest examples of circular statistics and of statistics of non-Euclidean spaces.

**Solution of triangles** is the main trigonometric problem of finding the characteristics of a triangle, when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.

**Hansen's problem** is a problem in planar surveying, named after the astronomer Peter Andreas Hansen (1795–1874), who worked on the geodetic survey of Denmark. There are two known points *A* and *B*, and two unknown points *P*_{1} and *P*_{2}. From *P*_{1} and *P*_{2} an observer measures the angles made by the lines of sight to each of the other three points. The problem is to find the positions of *P*_{1} and *P*_{2}. See figure; the angles measured are (*α*_{1}, *β*_{1}, *α*_{2}, *β*_{2}).

**Iso-damping** is a desirable system property referring to a state where the open-loop phase Bode plot is flat—i.e., the phase derivative with respect to the frequency is zero, at a given frequency called the "tangent frequency", . At the "tangent frequency" the Nyquist curve of the open-loop system tangentially touches the sensitivity circle and the phase Bode is locally flat which implies that the system will be more robust to gain variations. For systems that exhibit iso-damping property, the overshoots of the closed-loop step responses will remain almost constant for different values of the controller gain. This will ensure that the closed-loop system is robust to gain variations.

**Ffordd Pen Llech** is a public road in the town of Harlech which lies within Snowdonia National Park, North Wales. It was once considered the steepest street in the world, although that title reverted to the previous holder Baldwin Street on 8 April 2020.

- ↑ Strom, Steven; Nathan, Kurt; Woland, Jake (2013). "Slopes expressed as ratios and degrees".
*Site Engineering for Landscape Architects*(6th ed.). Wiley Publishing. p. 71. ISBN 978-1118090862. - ↑ "Traffic signs".
*www.gov.uk*. The Highway Code - Guidance. Retrieved 26 March 2016. - ↑
*A Policy on Geometric Design of Highways and Streets*(PDF) (4th ed.). Washington, DC: American Association of State Highway and Transportation Officials. 2001. pp. 507 (design speed), 510 (exhibit 8–1: Maximum grades for rural and urban freeways). ISBN 1-56051-156-7 . Retrieved 11 April 2014. - ↑ "Welsh town claims record title for world's steepest street".
*Guinness World Records*. 16 July 2019. - ↑ "Kiwi climb: Hoofing up the world's steepest street".
*CNN.com*. - ↑ "Baldwin street in New Zealand reinstated as the world's steepest street".
*Guinness World Records*. 8 April 2020. - ↑ "Canton Avenue, Beechview, PA".
*Post Gazette*. - ↑ "The steepest road on Earth takes no prisoners".
*Wired*. Autopia. December 2010. - 1 2 "General Information". San Francisco Metropolitan Transportation Agency. Archived from the original on 3 December 2016. Retrieved 20 September 2016.
- ↑ "Top five funicular railways".
*Sydney Morning Herald*. - ↑ "A wonderful railway".
*The Register*. Adelaide, Australia. 2 March 1920. p. 5. Retrieved 13 February 2013– via National Library of Australia. - ↑ "The New Pöstlingberg Railway" (PDF). Linz Linien GmbH. 2009. Archived from the original (PDF) on 22 July 2011. Retrieved 6 January 2011.
- ↑ "Pantele din Iaşi pun probleme ofertanţilor" (in Romanian). 5 March 2019.
- ↑ "Return of the (modern) streetcar – Portland leads the way" (Press release). Tramways & Urban Transit. Light Rail Transit Association. October 2001. Archived from the original on 27 September 2013. Retrieved 15 December 2018.
- ↑ "Boston's Light Rail Transit Prepares for the Next Hundred Years" (PDF).
*onlinepubs.trb.org*. Retrieved 23 February 2021. - ↑ "Lechmere, Science Park stations reopen".
*www.boston.com*. Archived from the original on 6 March 2007. Retrieved 23 February 2021. - ↑ "Il Piano Tecnologico di RFI" (PDF). Collegio Ingegneri Ferroviari Italiani. 15 October 2018. Retrieved 23 May 2019.
- ↑ "Madisonview".
*www.oldmadison.com*. Retrieved 7 April 2017. - ↑ "The Matheran Light Railway (extension to the Mountain Railways of India)". UNESCO World Heritage Centre.
- ↑ Martin, Bruno (September 2005). "Durban-Pietermaritzburg main line map and profile" (PDF).
*Transport in South and Southern Africa*. Retrieved 7 April 2017. - ↑ Valley Heights railway station
- ↑ "Improving the Southeast Expressway: A Conceptual Plan" (PDF).
*www.bostonmpo.org*. Retrieved 24 February 2021.

- "British railway gradients and their signs".
*Railsigns*.

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