Elevation and mathematics

Summary: Various aspects of elevation can be calculated using mathematical techniques.

Trigonometry has long been used to measure height. Elevation is often the height of a point relative to sea level, and its measurement is called “hypsometry.” Elevation affects air pressure, temperature, and gravity, all of which have noteworthy effects on people. Astronomers and mathematicians such as Blaise Pascal and Edmund Halley investigated relationships between barometric pressure and elevation.

Historical surveys of elevation include those who used barometers, like John Charles Frémont, who was at one time professor of mathematics of the Navy, and physician Christopher Packe. However, this method is sensitive to a number of variables. In the twenty-first century, detailed elevation data are available. Mount Everest is known as Earth’s highest elevation. Topographical maps represent elevation by using contour lines, each line following a path of constant elevation. Transits were developed in the nineteenth century, and they can be used to calculate changes in elevation. Contour integrals and generalized contours for functions of two variables are investigated in multivariable calculus classrooms. Mathematicians and computer scientists have helped create realistic computer models of land elevation, called “digital elevation models.” They have explored ideas like irregular-mesh grids or shifting nested grids in surface reconstruction. Other types of elevation studies also benefit from mathematical techniques, like using the ocean wave spectrum to investigate sea surface elevation peaks, or statistical techniques to investigate the impacts of elevation changes. Mathematician and astronomer Nilakantha Somayagi investigated the elevation of lunar cusps in the sixteenth century. The term “angle of elevation” in high school classrooms represents the angle between where an observer is standing and the line of sight to an object. The angle of elevation is found in many contexts, including in the Pyramids of Egypt, in the astrolabe, and in global positioning systems.

Topographic Maps

A topographic map is a two-dimensional map that conveys elevation information as well as other features of an area. Contour lines are the key to capturing elevation changes from a three-dimensional world on a two-dimensional map. A contour line is a path that follows a constant elevation. Early uses of contours date to the eighteenth and nineteenth centuries and include the work of engineer Jean-Louis Dupain-Triel and astronomer and mathematician John Couch Adams.

A contour line is drawn each time a predetermined elevation change is achieved. For example, a map may use 100-foot elevation increments, with one contour line following points having an elevation of 100 feet and the next marking an elevation of 200 feet. Consecutive contour lines always differ by 100 feet in elevation. As the mapped terrain climbs more steeply, the contour lines on the map will be closer together. The lines can mark elevations that increase and decrease, representing terrain that rises and falls intermittently. Contour lines can represent elevations that are zero, or negative numbers as when mapping an ocean floor.

A topographic map of an area with constant elevation at its boundary, such as an island bounded by the sea, will not have contour lines extending off the map’s edge. In such cases, all contour lines will appear as closed curves. A curve is closed if it loops back to where it started. Typically, contour lines appear as simple closed curves that do not cross themselves. The pattern of contour lines as nonintersecting rings lying one within another is common on topographic maps. Also common is to have two separate sets of nonintersecting rings contained within a single contour line, as when two hills are surrounded by a larger path of constant elevation.

The U.S. Geological Survey (USGS) has created a complete large-scale topographic map of the United States in more than 56,000 pieces. The National Elevation Dataset is noted as the “the primary elevation product of the USGS.” The data set is updated regularly, and historic data sets are also available for investigations.

There is an ever-growing growing need for digitized maps, which allow a computer user to read elevation at any spot on the map. Some digitized maps enable the user to view a landscape from different perspectives, creating a three-dimensional view of the area’s elevation changes, similar to what would be seen at the actual location. Data from existing topographic maps and aerial photography are used to create digitized maps. Improvements in technology will continue to affect the science of map making.

Effects of High Elevation

As elevation increases, air temperature drops because of a decrease in air pressure. At about 18,000 feet above sea level, for example, the air pressure is half that at sea level. In the troposphere, the lowest layer of Earth’s atmosphere, a general rule of thumb is that air temperature drops 6.5 degrees Celsius for every 1000 meters of elevation gain, or roughly one degree Fahrenheit for every 280 feet of elevation gain in standard conditions. This phenomenon, which can be modeled with an equation, can be seen directly when an observer standing at a low elevation on a warm day views a tall mountain covered with snow.

Another consequence of this cooling is that water vapor in the air condenses, sometimes causing increased rainfall on the windward side of a mountain range and a “rain shadow” downwind from the mountains. Many deserts lie just downwind from a mountain range. For example, sand dunes in Death Valley, California, lie in the rain shadow of Mount Whitney, the highest peak in the continental United States.

Because of these differences in temperature and precipitation, tall mountains can have multiple climatic zones, with different plant species thriving near the summit than at lower elevations. Some animal species, such as Roosevelt elk, migrate seasonally to take advantage of elevation effects, climbing to cooler locations in the summer and descending to warmer valleys in winter.

The lower atmospheric pressure at high elevations makes breathing more difficult. Mountain climbers at high elevations use special apparatus to breathe. Some competitive distance runners train at high elevations in order to challenge their cardiovascular systems. When they race at a lower elevation, the air feels relatively dense and oxygen-rich, giving them a competitive advantage.

With the less-dense atmosphere at high elevations, the sun’s rays can penetrate more easily, making sunburn possible even on cold days. Engines of naturally aspirated cars get less horsepower at higher elevations. Projectiles travel farther, a phenomenon known to golfers and baseball players. Standard equations for projectile motion sometimes assume a sea-level location; adjustments must be made to account for elevation.

The effect of gravity is reduced with travel to high elevations; mass remains the same but weight decreases slightly, primarily because of the increase in distance from Earth’s center of mass. A person’s weight would be less atop Mount Chimborazo than anywhere else on Earth.

Bibliography

Smith, Arthur. “Angles of Elevation of the Pyramids of Egypt.” Mathematics Teacher 75, no. 2 (1982) .

Thrower, Norman. Maps & Civilization: Cartography in Culture and Society. 3rd ed. Chicago: University of Chicago Press, 2008.

U.S. Geological Survey. “National Elevation Dataset.” http://ned.usgs.gov.