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Topic: Level curve

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	Level sets*
		A level curve of a function f (x, y) is the curve of points (x, y) where f (x, y) is some constant value.
		A level curve is simply a cross section of the graph of z = f (x, y) taken at a constant value, say z = c.
		To emphasize that level curves and level surfaces are similar concepts, I have redone the CVT for level curves to put it in the format that we will use for level surfaces.
	www.math.umn.edu /~nykamp/m2374/readings/levelset/index.html (956 words)

	Space-Filling Curves
		A space-filling curve is typically defined as the limit of a sequence of curves.
		Curves drawn while it is on will include a background grid that shows how the unit square is subdivided into cells.
		The Sierpinski_C curve and the Sierpinski curve are, in a sense, complements of each other: the shape of the areas excluded by one curve is the shape included by the other.
	www.cs.utexas.edu /users/vbb/misc/sfc/Oindex.html (511 words)

	Implementation of an Improved Contour Plotting Algorithm
		A third difficulty demonstrated by Figure 1 is that level curves with a small radius of curvature have noticeable corners at the points where the line segments approximating them meet.
		Another approach to improving the level curve tracing algorithm in order to get more accurate results for degenerate cells is based on the assumption that these cells either contain a saddle point at the height of the level curve or have a ridge or a valley running through them.
		The heights of the level curves and the fineness of the grid are identical to those used in Figure 1.
	members.bellatlantic.net /~vze2vrva/thesis.html#2_1_2 (7905 words)

	Level Curves and Surfaces (Site not responding. Last check: 2007-09-30)
		A level curve, or surface, is a set on which f is constant.
		Any circle centered at the origin is a level curve, or level set, since it is everywhere perpendicular to the gradient.
		Moving to a higher dimension, the sphere is a level surface in 3 space.
	www.mathreference.com /ca-mv,level.html (332 words)

	Level set - Wikipedia, the free encyclopedia
		In mathematics, a level set of a real-valued function f of n variables is a set of the form
		A level set is also called a fiber.
		The red curves follow the direction of the gradient.
	en.wikipedia.org /wiki/Level_set (367 words)

	Curve sketching (Site not responding. Last check: 2007-09-30)
		The level curve of f for the value 1 is the set of all pairs (x, y) such that xy = 1, or, equivalently, y = 1/x.
		By drawing the level curves of a function for various values, we construct a topographic map of the function that gives us a good picture of the nature of the function.
		The level curves corresponding to heights of 10 and 30 are thick, whereas those corresponding to heights of 0 and 20 are not.
	www.chass.utoronto.ca /~osborne/MathTutorial/SKE.HTM (1127 words)

	Project Links \| Gradient
		The resulting curve is called a level curve (or contour curve) of the function at.
		Thus, when the level curves are close together, the value of the function (e.g., the altitude or elevation) is changing more rapidly than when the level curves are further apart.
		Any curve satisfying the equation where is constant, may be regarded as the level curve of a suitable function such that is in the range of.
	links.math.rpi.edu:16080 /devmodules/gradient/html/geometricproperties_theory.html (512 words)

	[No title]
		For example, a level 1 c-curve consists of two perpendicular lines and a level 2 c-curve is one-half of a square.
		When the procedure c-curve is executed with the end points p1 and p2 and with a level argument 1, two line segments result; one from x1y1 to xmym and one from xmym to x2y2.
		Sketch level 0, 1, 2 and 3 c-curves.
	www.bridgeport.edu /sed/fcourses/cs325/Assignments/PS2af02.doc (497 words)

	Interannual sea level variability in the Southern Ocean within the context of global climate change
		During the past decade, the sea level at Kerguelen was lowest in 1994 (-30 mm relative to the mean over the period 1993-2000), followed by an abrupt rise in 1997, reaching a peak of 50 mm above normal at the end of 1997, a period coinciding precisely with the strongest ENSO of the last century.
		From early 1998, the sea level gradually fell, coming close to the quasi-decennial mean level at the end of 2000.
		To situate the observed sea level variability at Kerguelen within the global context, T/P sea level trends of the world's oceans are shown in figure 2a.
	sealevel.jpl.nasa.gov /science/invest-park.html (1210 words)

	A Level Curve Applet
		The Level Curves window shows the graph of the function being cut by a plane at height z0.
		The level curve at this height is in orange.
		A worksheet for using the Level Curves applet is available.
	www.slu.edu /classes/maymk/banchoff/LevelCurve.html (168 words)

	SOFIA - Paper - Geology and Hydrogeology of the Florida Keys - Holocene Geology
		The Florida Keys are in the geographic region where curves of relative sea level during the late Holocene can be expected to show a history of continual submergence up to the present day (Zone III of Clark et al., 1978, and Peltier et al., 1978).
		The observed curve is that of Robbin (1984), and it is in general agreement with both the modeling by Clark et al.
		The curve plots slightly below the curve of Scholl and Stuiver (1968), in which sea level was at about 1.6 m at 3.5 ka and 0.5 m at 1.7 ka.
	sofia.usgs.gov /publications/papers/keys_geohydro/holocene.html (699 words)

	lab07.html
		If you imagine the level curves as wide stairs, then it is clear that the fastest way to ascend is to go perpendicular to the stairs, rather than to angle off.
		The analogue of level curves is level surfaces.
		Instead of attempting to visualize the graph of the function, which is a three dimensional object in four dimensions (just as the graph of a function of two variables is a two-dimensional object in three dimensions), we settle on analyzing the path in domain space which corresponds to the path of most rapid ascent.
	www.uwec.edu /smithaj/Spring00/216/Lab07/Lab071.html (734 words)

	Inflation and the Phillips Curve
		The Phillips Curve was an empirical phenomenon looking for a theory and, around that time, there were two competing theories of inflation, both of which were expressed by Keynes in various places: "demand-pull" inflation and "cost-push" inflation - terms, as Machlup (1960) has shown, that can have a far from obvious meaning.
		The non-linearity of the Phillips Curve is justified by appeals to frictional unemployment and institutional difficulties at the extremes.
		Thus, the migration of the so-called "short-run" Phillips Curve (as in the move in Figure 14) was explained in terms of ever-higher inflationary expectations.
	cepa.newschool.edu /het/essays/keynes/inflation.htm (5546 words)

	The Gradient Vector is Perpendicular to the Level Curve (Site not responding. Last check: 2007-09-30)
		The level curves are plots at the height f[x,y]=c.
		The gradient vector is perpendicular to the level curve!
		is level perpendicular the The to vector curve!
	www.ma.iup.edu /projects/CalcDEMma/veccalc/veccalc9.html (259 words)

	Level curves (Site not responding. Last check: 2007-09-30)
		R, by graphing its level curves, i.e., the curves where f (x, y) is a constant value.
		A level curve is simply a cross section of z = f (x, y) taken at a constant value of z.
		We can plot the level curves for a bunch of different constants c together in a level curve plot.
	www.math.umn.edu /~nykamp/m2374/readings/levelcurve (202 words)

	[No title]
		Figure 8.2 contours the intersection of sea level with the Pleistocene limestone surface at 5500 YBP (-6.2 m), 3500 YBP (-1.5 m), and the present (0 m).
		The high-frequency sea level oscillations are visible in the Brazilian situation because the overall regression is leaving behind a visible, exposed, prograding strandplain record of the late Holocene.
		An important difference between the present phase of accelerated sea level rise and that of 5500 to 3200 YBP is that the present follows a prolonged period of growth of coastal sediment bodies creating large coastal and nearshore sediment reservoirs.
	www.geology.iupui.edu /research/sedlab/publications/Sealevel1.htm (8422 words)

	A readjusted sea level curve of the Eemian in the Netherlands
		A readjusted sea level curve of the Eemian in the Netherlands
		In an article with the title "Sea level changes in the Netherlands during the Eemian" (1983) Zagwijn published a curve of the high tide sealevels.
		We are aware that reconstruction of true rates of rise and fall in the sea level during the Eemian is still not possible, as Zagwijn already mentioned in 1983.
	web.inter.nl.net /users/Meijer.T/tm/paginas/ae-pub-slevel.htm (836 words)

	Abandoned Stations : Nevins St lower level
		The BMT De Kalb Ave station was built as if this had still not been ruled out: its main track level is at the level of the IRT lower level, for the outbound track connection, and its upper mezzanine level has no structure blocking the path of a track off the IRT main level.
		The upper level curve is still visible in the wall of the Atlantic Ave station side platform, as extended northward in 1964.
		The ramp to the lower level is seen here from the front of a train running from Hoyt St to Nevins St. There would have been a switch at this point, and trains could have shifted to the right and down that darkened portal to the lower level.
	www.columbia.edu /~brennan/abandoned/nevins.html (1241 words)

	rgb curve adjust
		rgb curve adjust is a plug that lets you exert unprecedented control over color in you image.
		and since it may sometimes be difficult to locate a specific color, rgb curve adjust lets you select a reference color that is displayed as a reference line in each color channel.
		these are probably the same pros that will tell you that rgb curve adjust is essentially the only plug you need (not true) and that it is one of the fastest plugs around (true).
	www.imovieplugins.com /plugs/rgbcurveadjust.html (440 words)

	sea level changes text patterns
		The ensuing sea-level curve is generally characterized by a gradual rise in sea level, although the exact character and the continuity of that rise continue to be debated.
		The curve contains significant gaps in the depositional record from terrace to terrace which he interpreted to reflect sudden rises in sea level due to the rapid addition of meltwater into the world's oceans.
		An alternative possibility is that the reefs formed on older terraces and that each reef represents the accretion of coral as sea level passed by a pre-existing notch in the otherwise smooth island slope.
	geology.uprm.edu /Morelock/GEOLOCN_/patterns.htm (3307 words)

	What type of sea-level curve - Re-assessment of Archaeological Potential of Continental Shelves
		Eustatic curves are most commonly used as they purport to give global sea level change and so are used in the absence of local or regional data.
		Eustatic curves infact represent “global ocean volume” which albeit is a major control on sea level it is only one of a number (the others include glacio- and hydro-isostasy, active tectonism, sedimentation and erosion and “other” climatic effects).
		Consequently, there are a number of different types of “sea level curves” which can be used for reconstruction, each of which can give a significantly different palaeogeographic map.
	www.arch.soton.ac.uk /Research/Aggregates/shelve-th1-sealevel.htm (366 words)

	[No title]
		There are three choices (the radio buttons) of ways to input the level curves: \n 1.
		This plots a single level curve at this height.\n 2.
		This plots N level curves at heights evenly spaced between the min and max of the function on the chosen domain.
	www4.ncsu.edu /~lkn/MA242/MapleWorksheetsForClass/LevelCurves.maplet (735 words)

	Lab #2 Level Curves and Equipotential Lines MA 1024P (Site not responding. Last check: 2007-09-30)
		To use vector calculus from last week as well as Maple to understand and study level curves, especially equipotential lines due to an electric dipole.
		Notice this means the level curve for a given k is a subset of the domain.
		This function f is then a function of two variables and thus may have level curves associated with it.
	users.wpi.edu /~goulet/calc4P/lab2.htm (1476 words)

	Living with Coastal Change
		The oxygen isotope curve reveals these ice age cycles and serves as a proxy for sea level changes.
		Stillstands are times of stable sea level when the ocean is neither rising nor falling rapidly.
		The present platform and sea cliff along the California coast were cut by wave action during the sea level stillstand of the last 6,000 years.
	earthguide.ucsd.edu /coastalchange/st4_climatechange/sealevel.html (264 words)

	No Title
		Here, a curve is not a geometric set of points, rather it is a procedure for taking number values in the unit interval, and generating the corresponding point.
		shows the first few approximations to the Gosper curve, where we stop after a certain number of levels: a level-0 curve is simply a straight line; a level-1 curve consists of two level-0 curves; a level 2 curve consists of two level-1 curves, and so on.
		In the middle piece, we draw two scale copies of the initial curve, one rotated by 60 degrees, and the other rotated by -60 degrees, where all of the curves are drawn so that their endpoints meet.
	theory.lcs.mit.edu /~meyer/6001/FT97/psets/ps3/hops-new/hops-new.html (3122 words)

	Write-Up #1 (Site not responding. Last check: 2007-09-30)
		Having in mind that the curve for each n is the curve of level zero of the equation
		I decided to study why is that the curve of level zero of the above surface when n=1 looks as if the surface intersects itself.
		In a class situation, a student might at this point try to have a clear picture of what the surface looks like near its zero level curve when looking at the graph of the surface.
	jwilson.coe.uga.edu /emt668/EMAT6680.F99/Moura/writeup1/WriteUp1.htm (273 words)

	Level curve evolution of Chelushka-Belushka (Site not responding. Last check: 2007-09-30)
		Level curve evolution of Chelushka-Belushka under the geometric heat equation
		Movie of the 75 intensity level set evolution for 500 iterations of mean curvature deformation
		Movie of the 150 intensity level set evolution for 500 iterations of mean curvature deformation
	www.cim.mcgill.ca /~mdesco/levelCurves.html (40 words)

	Distance From a Point to a Straight Line
		It is noteworthy that all points on one of the level lines have the same distance to any other level line, because the lines are parallel and the distances are measured via the perpendiculars.
		In particular, if the equation is normalized to start with, a unit step in the direction normal to the line changes its value by exactly 1!.
		With every unit step, the values on the level lines grow by 1 in the direction of the normal, but they decrease by 1 in the opposite direction.
	www.cut-the-knot.org /Curriculum/Calculus/DistanceToLine.shtml (692 words)

	Introduction to Macroeconomics - Chapter 23
		If we recognize the average price level is not fixed then we must transform the Keynesian aggregate expenditures curve with prices fixed into an aggregate demand curve with prices variable.
		An increase in the average level of prices, which causes a decrease in consumption expenditures, causes the Keynesian aggregate expenditure curve to shift down.
		An increase in the average level of prices, which causes an increase in the real interest rate and a reduction in consumption and investment expenditures, causes the Keynesian aggregate expenditure curve to shift down.
	mason.gmu.edu /~tlidderd/104/ch23Lect.html (858 words)

	A Matlab toolbox implementing Level Set Methods
		As can be seen, the corners of the curve are slightly smoothed but except that the zero level set of phi is preserved during the diffusion.
		Note that, phi does not have to be a signed distance function, but if it is this increases the stability and quality of the evolution (especially if a vector field-based force and a force in normal direction are combined).
		In this example, all level sets of the image are smoothed under curvature-based forces.
	barissumengen.com /level_set_methods/index.html (1228 words)

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