In geometry and science , a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane , or the analog in higher- dimensional spaces. Cutting an object into slices creates many parallel cross-sections. The boundary of a cross-section in three-dimensional space that is parallel to two of the axes , that is, parallel to the plane determined by these axes, is sometimes referred to as a contour line ; for example, if a plane cuts through mountains of a raised-relief map parallel to the ground, the result is a contour line in two-dimensional space showing points on the surface of the mountains of equal elevation .
31-573: The Butterley Gangroad was an early tramway in Derbyshire of approximately 3 ft 6 in ( 1,067 mm ) gauge, which linked Hilt's Quarry and other limestone quarries at Crich with the Cromford Canal at Bullbridge . The first railway project of Derbyshire civil engineer Benjamin Outram (1764–1805), the line was originally a horse-drawn and gravity-driven plateway ,
62-406: A four-dimensional object passed through our three-dimensional space, we would see a three-dimensional cross-section of the four-dimensional object. In particular, a 4-ball (hypersphere) passing through 3-space would appear as a 3-ball that increased to a maximum and then decreased in size during the transition. This dynamic object (from the point of view of 3-space) is a sequence of cross-sections of
93-414: A cylinder is used in this sense, the above paragraph would read as follows: A plane section of a right circular cylinder of finite length is a circle if the cutting plane is perpendicular to the cylinder's axis of symmetry, or an ellipse if it is neither parallel nor perpendicular to that axis. If the cutting plane is parallel to the axis the plane section consists of a pair of parallel line segments unless
124-598: A cylinder of height h and radius r has A ′ = π r 2 {\displaystyle A'=\pi r^{2}} when viewed along its central axis, and A ′ = 2 r h {\displaystyle A'=2rh} when viewed from an orthogonal direction. A sphere of radius r has A ′ = π r 2 {\displaystyle A'=\pi r^{2}} when viewed from any angle. More generically, A ′ {\displaystyle A'} can be calculated by evaluating
155-424: A form of tramway that Outram popularised. Unlike modern edgeways , where flanges on the wheel guide it along the track, plateways used L-shaped rails where a flange on the rail guided the wheels. The line was constructed in 1793, with the construction of Fritchley Tunnel , now believed to be the world's oldest railway tunnel, being required to go under a road junction at Fritchley . A steam locomotive using
186-570: A logistics centre and the Volkswagen factory. Cross section (geometry) In technical drawing a cross-section, being a projection of an object onto a plane that intersects it, is a common tool used to depict the internal arrangement of a 3-dimensional object in two dimensions. It is traditionally crosshatched with the style of crosshatching often indicating the types of materials being used. With computed axial tomography , computers can construct cross-sections from x-ray data. If
217-409: A plane intersects a solid (a 3-dimensional object), then the region common to the plane and the solid is called a cross-section of the solid. A plane containing a cross-section of the solid may be referred to as a cutting plane . The shape of the cross-section of a solid may depend upon the orientation of the cutting plane to the solid. For instance, while all the cross-sections of a ball are disks,
248-510: A plane section is taken parallel to the xy -plane, the result is an isoquant showing the various combinations of labor and capital usage that would result in the level of output given by the height of the plane section. Alternatively, if a plane section of the production function is taken at a fixed level of y —that is, parallel to the xz -plane—then the result is a two-dimensional graph showing how much output can be produced at each of various values of usage x of one input combined with
279-502: A stationary engine, or pulled by small, light locomotives. Tramways can exist in many forms; sometimes simply tracks temporarily placed on the ground to transport materials around a factory, mine or quarry. Many use narrow-gauge railway technology, but because tramway infrastructure is not intended to support the weight of vehicles used on railways of wider track gauge , the infrastructure can be built using less substantial materials, enabling considerable cost savings. The term "tramway"
310-758: A walking mechanism, known as the Steam Horse locomotive , was trialled on the line in 1813. In the 1840s, upgrading took place to accommodate steam locomotives, and part of the original line was moved. The railway remained in use until 1933. This England rail transport related article is a stub . You can help Misplaced Pages by expanding it . 53°04′23″N 1°27′59″W / 53.07315°N 1.46629°W / 53.07315; -1.46629 Tramway (industrial) Tramways are lightly laid industrial railways , often not intended to be permanent. Originally, rolling stock could be pushed by humans, pulled by animals (especially horses and mules), cable-hauled by
341-408: Is "visible" from the perspective of the viewer. For a convex body , each ray through the object from the viewer's perspective crosses just two surfaces. For such objects, the integral may be taken over the entire surface ( A {\displaystyle A} ) by taking the absolute value of the integrand (so that the "top" and "bottom" of the object do not subtract away, as would be required by
SECTION 10
#1732780253320372-605: Is not used in North America, but is commonly used in the United Kingdom and elsewhere where British railway terminology and practices influenced management practices, terminologies and railway cultures, such as Australia , New Zealand , and those parts of Asia, Africa and South America that consulted with British engineers when undergoing modernization. In New Zealand, they are commonly known as " bush tramways " and are often not intended to be permanent. In Australia
403-514: Is that of a plane section , which is the curve of intersection of a plane with a surface . Thus, a plane section is the boundary of a cross-section of a solid in a cutting plane. If a surface in a three-dimensional space is defined by a function of two variables, i.e., z = f ( x , y ) , the plane sections by cutting planes that are parallel to a coordinate plane (a plane determined by two coordinate axes) are called level curves or isolines . More specifically, cutting planes with equations of
434-571: The Divergence Theorem applied to the constant vector field r ^ {\displaystyle \mathbf {\hat {r}} } ) and dividing by two: In analogy with the cross-section of a solid, the cross-section of an n -dimensional body in an n -dimensional space is the non-empty intersection of the body with a hyperplane (an ( n − 1) -dimensional subspace). This concept has sometimes been used to help visualize aspects of higher dimensional spaces. For instance, if
465-444: The 4-ball. In geology , the structure of the interior of a planet is often illustrated using a diagram of a cross-section of the planet that passes through the planet's center, as in the cross-section of Earth at right. Cross-sections are often used in anatomy to illustrate the inner structure of an organ, as shown at the left. A cross-section of a tree trunk, as shown at left, reveals growth rings that can be used to find
496-410: The center of an ellipsoid forms an elliptic region, while the corresponding plane sections are ellipses on its surface. These degenerate to disks and circles, respectively, when the cutting planes are perpendicular to a symmetry axis. In more generality, the plane sections of a quadric are conic sections. A cross-section of a solid right circular cylinder extending between two bases is a disk if
527-442: The cross-section is parallel to the cylinder's base, or an elliptic region (see diagram at right) if it is neither parallel nor perpendicular to the base. If the cutting plane is perpendicular to the base it consists of a rectangle (not shown) unless it is just tangent to the cylinder, in which case it is a single line segment . The term cylinder can also mean the lateral surface of a solid cylinder (see cylinder (geometry) ). If
558-409: The cross-sections of a cube depend on how the cutting plane is related to the cube. If the cutting plane is perpendicular to a line joining the centers of two opposite faces of the cube, the cross-section will be a square, however, if the cutting plane is perpendicular to a diagonal of the cube joining opposite vertices, the cross-section can be either a point, a triangle or a hexagon. A related concept
589-425: The cutting plane is tangent to the cylinder, in which case, the plane section is a single line segment. A plane section can be used to visualize the partial derivative of a function with respect to one of its arguments, as shown. Suppose z = f ( x , y ) . In taking the partial derivative of f ( x , y ) with respect to x , one can take a plane section of the function f at a fixed value of y to plot
620-454: The fixed value of the other input y . Also in economics, a cardinal or ordinal utility function u ( w , v ) gives the degree of satisfaction of a consumer obtained by consuming quantities w and v of two goods. If a plane section of the utility function is taken at a given height (level of utility), the two-dimensional result is an indifference curve showing various alternative combinations of consumed amounts w and v of
651-412: The following surface integral: where r ^ {\displaystyle \mathbf {\hat {r}} } is the unit vector pointing along the viewing direction toward the viewer, d A {\displaystyle d\mathbf {A} } is a surface element with an outward-pointing normal, and the integral is taken only over the top-most surface, that part of the surface that
SECTION 20
#1732780253320682-423: The form z = k (planes parallel to the xy -plane) produce plane sections that are often called contour lines in application areas. A cross section of a polyhedron is a polygon . The conic sections – circles , ellipses , parabolas , and hyperbolas – are plane sections of a cone with the cutting planes at various different angles, as seen in the diagram at left. Any cross-section passing through
713-567: The ground they were less likely to be blocked by debris, but they obstructed other traffic, and the wagons could not be used beyond the limits of the rails – whereas plateways had the advantage that trucks with unflanged wheels could be wheeled freely on wharves and in factories. Edge rails were the forerunners of the modern railway track. These early lines were built to transport minerals from quarries and mines to canal wharves. From about 1830, more extensive trunk railways appeared, becoming faster, heavier and more sophisticated and, for safety reasons,
744-428: The level curve of z solely against x ; then the partial derivative with respect to x is the slope of the resulting two-dimensional graph. A plane section of a probability density function of two random variables in which the cutting plane is at a fixed value of one of the variables is a conditional density function of the other variable (conditional on the fixed value defining the plane section). If instead
775-440: The plane section is taken for a fixed value of the density, the result is an iso-density contour . For the normal distribution , these contours are ellipses. In economics , a production function f ( x , y ) specifies the output that can be produced by various quantities x and y of inputs, typically labor and physical capital. The production function of a firm or a society can be plotted in three-dimensional space. If
806-471: The public highway, sharing with other road users. Initially horse-drawn, they were developed to use electric power from an overhead line . A development of the tramway in the United Kingdom was the trolleybus , which dispensed with tracks but drew electricity from overhead wires . Between 2001 and 2020, two trams built to carry automotive parts (the " CarGoTram ") operated in Dresden , Germany between
837-520: The requirements placed on them by Parliament became more and more stringent. See rail tracks . These restrictions were excessive for the small mineral lines and it became possible in the United Kingdom for them to be categorised as light railways subject to certain provisos laid down by the Light Railways Act 1896 . Meanwhile, in the United Kingdom the term tramway became the term for passenger vehicles (a tram ) that ran on tracks in
868-503: The shaft of a wheelbarrow—in turn from Low German traam , meaning a beam. The tracks themselves were sometimes known as gangways , dating from before the 12th century, being usually simply planks laid upon the ground literally "going road". In south Wales and Somerset the term "dramway" is also used, with vehicles being called drams. An alternative term, " wagonway " (and wainway or waggonway), originally consisted of horses, equipment and tracks used for hauling wagons. Usually
899-516: The term was widely used in connection with logging, no longer extant. Today in the state of Queensland , however, there remain several thousand kilometres of sugar-cane tramways . Passengers do not generally travel aboard tramways, although employees sometimes use them, either officially or unofficially. The term was originally applied to wagons running on primitive tracks in mediaeval Great Britain and Europe . The name seems to date from about 1517 and to be derived from an English dialect word for
930-411: The two goods all of which give the specified level of utility. Cavalieri's principle states that solids with corresponding cross-sections of equal areas have equal volumes. The cross-sectional area ( A ′ {\displaystyle A'} ) of an object when viewed from a particular angle is the total area of the orthographic projection of the object from that angle. For example,
961-424: The wheels would be guided along grooves. In time, to combat wear, the timber would be reinforced with an iron strip covering. This developed to use L-shaped steel plates, the track then being known as a plateway . An alternative appeared in 1789, the so-called " edge-rail ", which allowed wagons to be guided by having the wheels flanged instead of running, flangeless, in grooves. Since these rails were raised above