Horizontal and vertical

This article is about the physical and mathematical concepts. For the concept in economics, see Horizontal and vertical integration. For the Asian writing systems, see Horizontal and vertical writing in East Asian scripts.

The usage of the inter-related terms horizontal and vertical varies with context, There are important symmetries and asymmetries between the two terms which change as one goes from two to three dimensions, from a flat earth scenario to the spherical earth one.

In two dimensions

In two dimensions 1. The vertical direction is designated. 2. The horizontal is normal to the vertical. Through any point P, there is exactly one vertical and exactly one horizontal. Alternatively, one can start by designating the horizontal direction.

In the context of a two-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 the Y direction.[1] The horizontal direction, usually labelled the X direction,[2] 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 the two-dimensional case:

a) Through any point P in the plane, there is one and only one vertical line within the plane and one and only one horizontal line within the plane. This symmetry breaks down as one moves to the three-dimensional case.

b) A vertical line is any line parallel to the vertical direction. A horizontal line is any line normal to a vertical line.

c) Horizontal lines do not cross each other.

d) Vertical lines do not cross each other.

Not all of these elementary geometric facts are true in the 3-D context.

In three dimensions

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 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.

Observations

The plumb line and spirit level

Spirit level bubble on a marble shelf tests for horizontality

In physics, in engineering, and in construction, the direction designated as vertical is usually that along which a plumb-bob hangs. Alternatively, a spirit level that exploits the buoyancy of an air bubble and its tendency to go vertically upwards may be used to test for horizontality. Modern rotary laser levels that can level themselves automatically and are robust sophisticated instruments work on the same fundamental principle.[3][4]

The flat earth approximation

In the flat earth scenario,[5] where the earth is notionally a large (infinite) flat surface with gravitational field at a right angle to the surface, the earth surface is horizontal and any plane parallel to the earth surface is also horizontal. Vertical planes, e.g., walls, may be parallel to each other or they may intersect at a vertical line. Horizontal surfaces do not intersect. Furthermore, a plane cannot both be a horizontal plane at one place and a vertical plane somewhere else.

The spherical Earth

Strictly, vertical walls are never parallel on the surface of a spherical planet

When the curvature of the earth is taken into account, the concepts of vertical and horizontal take on yet another meaning. On 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. [6]

On a spherical planet, horizontal planes intersect. In the example shown, the blue line represents the tangent plane at the North pole, the red the tangent plane at an equatorial point. The two intersect at a right angle.

Also, horizontal planes can intersect when they are tangent planes to separated points on 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.

Further complications

For a spinning earth, the plumb line deviates from the radial direction as a function of latitude.[7] Only 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 homomogenous smooth sphere. It is a non homogenous, non spherical, knobbly planet in motion, and the vertical not only need not lie along 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.[8]

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.[9][10]

Walls and floors

Brick wall showing horizontal mortar courses and vertical perpendiculars. By contrast, a level floor only contains horizontal lines.

On a (horizontal) floor, one can draw a horizontal line but not a vertical line in the sense of a plumb bob line. But on a (vertical) wall, one can draw both vertical and horizontal lines. In this sense, a vertical wall allows more options. This is reflected in the tools a bricklayer uses: a plumb line for verticality and a spirit level to check that the mortar courses are horizontal. On the other hand, in contrast to a wall a horizontal floor allows more options when one considers compass directions. One can draw on a floor lines going north, south, east, and west, in fact, along any compass direction. A wall allows fewer options. For instance, on a wall which runs along a longitude, an insect cannot crawl east.

In the class room

The y-axis on the wall is vertical but the one on the table is horizontal

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 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.

Independence of horizontal and vertical motions

Neglecting the curvature of the earth, horizontal and vertical motions of a projectile moving under gravity are independent of each other.[11] 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.[12]

When the curvature of the earth is taken into account, the independence of the two motion does not hold. For example, even if a projectile fired in a horizontal direction (i.e., with a zero vertical component) it will leave the surface of the spherical earth and may indeed escape altogether.[13]

See also

References and notes

  1. ↑ For an example of identification of the y-axis with the vertical, and the x-axis with the horizontal, see G.S.Rehill's Interactive Maths Series, "Building a Strong foundation in Mathematics", accessible on http://www.mathsteacher.com.au/year8/ch15_graphs/05_hor/ver.htm.
  2. ↑ For a definition of "Horizontal axis" see Math Dictionary at http://www.icoachmath.com/math_dictionary/Horizontal_Axis.html
  3. ↑ See Laser Levels
  4. ↑ For how a spirit level works, see http://www.physicsforums.com/showthread.php?t=562730
  5. ↑ See Theory and Problems of Theoretical Mechanics" by Murray R Spiegel, 1987, pg 62
  6. ↑ Encyclopedia.com. In very long bridges, it may be necessary to take the earth's curvature into account when designing the towers. For example, in the New York's Verrazano Narrows Bridge, the towers, which are 700 ft (215 m) tall and stand 4260 ft (298 m) apart, are about 1.75 in (4.5 cm) farther apart at the top than they are at the bottom.
  7. ↑ "Working in the Rotating Reference Frame of the Earth" (PDF).
  8. ↑ Such a deflection was measured by Nevil Maskelyne. See Maskelyne, N. (1775). "An Account of Observations Made on the Mountain Schiehallion for Finding Its Attraction". Phil. Trans. Royal Soc. 65 (0): 500–542. doi:10.1098/rstl.1775.0050. Charles Hutton used the observed value to determine the density of the earth.
  9. ↑ Cornish, Neil J. "The Lagrangian Points" (PDF). Montana State University – Department of Physics. Retrieved 29 July 2011.
  10. ↑ For an example of curved field lines, see The gravitational field of a cube by James M. Chappell, Mark J. Chappell, Azhar Iqbal, Derek Abbott for an example of curved gravitational field. arXiv:1206.3857 [physics.class-ph] (or arXiv:1206.3857v1 [physics.class-ph] for this version)
  11. ↑ Salters Hornerns Advanced Physics Project, As Student Book, Edexcel Pearson, London 2008,pg 48.
  12. ↑ See Galilleo;s discussion of how bodies rise and fall under gravity on a moving ship in his Dialogue Concerning the Two Chief World Systems(trans. S. Drake). University of California Press, Berkeley, 1967, pp. 186–187.
  13. ↑ See Harris Benson University Physics, New York 1991, page 268.

General references

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