Color Visualization of Cyclic Magnitudes

Color Visualization of Cyclic Magnitudes

Alfredo Restrepoa and Viviana Estupinanb

aLaboratorio de SenalesbDepartamento de Ing. Electrica y Electronica de la Universidad de los Andes

Carrera 1 No. 18A-70; aoficina ML-427, Bogota 111711, Colombia

ABSTRACT

We exploit the perceptual, circular ordering of the hues in a technique for the visualization of cyclic variables.The hue is thus meaningfully used for the indication of variables such as the azimuth and the units of themeasurement of time. The cyclic (or circular) variables may be both of the continuous type or the discretetype; among the first there is azimuth and among the last you find the musical notes and the days of the week.A correspondence between the values of a cyclic variable and the chromatic hues, where the natural circularordering of the variable is respected, is called a color code for the variable. We base such a choice of hues onan assignment of of the unique hues red, yellow, green and blue, or one of the 8 even permutations of thisordered list, to 4 cardinal values of the cyclic variable, suitably ordered; color codes based on only 3 cardinalpoints are also possible. Color codes, being intuitive, are easy to remember. A possible low accuracy whenreading instruments that use this technique is compensated by fast, ludic and intuitive readings; also, the useof a referential frame makes readings precise. An achromatic version of the technique, that can be used bydichromatic people, is proposed.

Keywords: Cyclic variable, circular variable, hue circle, Newton color circle, time indication, direction map,orientation map, musical notes, moon cycle

1. INTRODUCTION

In a sense, the hours of the day are a cyclic or circular magnitude, after the 23rd hour comes the ”0th” hour ofthe next day. By abstracting the fact that that it is an hour of the next day, the hour 24 is also the hour 0; theabstraction of this fact makes the hours of the day a cyclic magnitude. The hue circle also gives an instance of acyclic variable and, with a suitable correspondence between the hues and the values of cyclic variables, such astime and azimuth, you get a method to indicate their values with the hue of color. We give a method to devicea correspondence that is natural and intuitive; with the method, you can design easy-to-read, color clock faces(see Fig. 1), for example.

We consider cyclic variables in some detail in Section 1.6 and, in Section 2, we present a theoretical charac-terization of them. One of the early, meaningful uses of hue to visualize a cyclic magnitude was to visualize theorientation columns of old-world monkeys,1 in cortical area V1. We consider the visualization of cyclic variablesin Section 3 and, in Section 4, we argue that, in the more general cases of data visualization using color, the huecomponent of a color should be used mainly to display a circular aspect of the magnitudes.

In the hue circle you find ”all” perceivable hues arranged on a circle, respecting the perceptual continuitybetween them: similar hues are near each other in the circle. Also, for example, the oranges are in betweenunique red and unique yellow; a circularly monotone correspondence between the hours and hues is one thatrespects betweenness: since the hour 0 is between the hours 23 and 1, so should their corresponding colours. Thiscontinuity and monotonicity of a correspondence, together with the concept of the cardinal values of a cyclicvariable, are the main aspects of the method proposed here for the visualization of cyclic variables using color.

[email protected], [email protected],(+57 1) 3394949 x 2827, http://labsenales.uniandes.edu.co

Figure 1. Two color clock faces; in each case, a reference frame surrounds the color indicators. Above, the central circularpatch indicates the seconds, the right semiannular patch indicates the minutes and the left semiannular patch indicatesthe hours; the indicated time is 12:45:15. Below, the three rectangular color patches within the rectangular referenceframe, indicate the hours (left), the minutes (middle) and the seconds (right). The indicated time is 09:10:00

270

0 = 360

90

180

Figure 2. The correspondence between angles and circle points, at the cardinal angles.

1.1 On Color Codes for Cyclic Variables

Chromatic hue is a cyclic magnitude but, unlike Newton’s thinking, the hue circle consists not only of thespectral hues but also of the extra-spectral hues (a collection of purples that includes unique red). The hue circlemeaningfully (perceptually) portrays the set of the hues, i.e. the set of the chromatic hues may be put into ameaningful one–to-one correspondence with the number, angle interval [0◦ , 360◦), which in turn can be bijectedto the points on a circle; thus, you get {hues} → ○. This correspondence can be naturally seen as a coloredcircle, see Fig. 5. The correspondence is improved by e.g. assigning the 4 unique hues to the angles 0◦, 90◦, 180◦

and 270◦, in one of several possible ways. Alternatively, using only three cardinal points, the hues of spectralred (which is a bit orange in comparison to unique red), unique green and unique blue, may be assigned to theangles 0◦, 120◦ and 240◦.

In general, by suitably choosing 4 values of any given circular magnitude as cardinal values, a meaningfulmap, {values of cyclic variable} → ○, from the cyclic variable to the circle of angles can be found and,by composing one such order homomorphism with the inverse of that from the hues to the circle, a color code{values of cyclic variable} →○→ {hues} results, that assigns the cardinal values of the variable to the uniquehues. Thus you code angles, times, azimuthal directions, etc., using colors.

1.2 The Hue Circle

It was Helmholtz who first pointed out that Newton was wrong in thinking that all hues (his circle of hues) werespectral: the purples are not2 and form what is known as the spectral gap in the hue circle. Newton was rightthough in thinking that the hues had a circular order.

The spectral colors, i.e. those of sunlight, at different positions after being refracted by a glass prism, are(spectral) red, the oranges, unique yellow, the citrines (lemon yellows), unique green, the cyans, unique blue, andthe violets, ordered as wavelength decreases, roughly from 730 nm to 370 nm. More precisely, a color is spectralif it corresponds to a light beam of only one wavelength. Since this is achievable only with a laser, in most casesyou can only have a beam of narrow spectrum; moreover, for a surface reflecting a broadband illuminant this isimpossible: the energy at a unique wavelength is zero.

A red is unique if you see no traces of orange or of violet in it; similarly, a yellow is unique if it is not orangenor certain, a green is unique if it is not cyan nor citrine and a blue is unique if it is neither violet nor cyan.Unique red is extra-spectral, that is, it cannot be obtained with a unique wavelength and it is required a mixtureof spectral red and violet to be obtained. The hues that are not unique are said to be binary. The four familiesof binary hues are the oranges, the cetrines, the cyans and the purples. Within each family of binary hues, there

is one that is central, for example, central orange is that orange that is nor more red than yellow nor more yellowthan red.

There is a continuity of the circle of hues, two points that are nearby in the circle correspond to similarhues and, moving very slowly a point on the circle, its hue will change imperceptibly; yet, each hue will appearprecisely once and, after seeing them all, you return to the initial hue. On the other hand, two hues that barelycan be discriminated, correspond to what is known in psychophysics as a just noticeable difference or JND.

1.3 The Circle of the Angles

As mentioned in Section 1.1, a most natural way to arithmetize the circle is with the numerical value of cor-responding angles. On a circle, given a point of reference and a direction of reference, each point on the circlecorresponds to a unique angle between 0◦ and∗ 360−◦; also, 0◦ = 360◦ and, in general, two numbers ω1 and ω2,representing angles, will refer to the same angle whenever ω1 − ω2 be an integer multiple of 360; e.g. 180◦ and-180◦, or, 10◦ and 370◦. The angle subtended on a clock face by a minute of time is 360◦/60 = 6◦ and the anglesubtended by an hour is 360◦/12 = 30◦ or 360◦/24 = 15◦.

You get arithmetic operations with angles for example if you consider the the unit circle of the complex plane,where the points on the circle are given by complex numbers of the form eωi where two real numbers ω1 and ω2

will be considered equivalent whenever eω1i = eω2i. Here, you say that the angles are measured in radians ratherthan in degrees.

It is useful to consider more than one reference point on a circle; so, the angles that are multiples of 90◦ arecalled cardinal angles. This notion can be generalized so that you may have not 4 cardinal points but any integernumber N of cardinal points, most commonly 2 or 3; this is done by considering the angles that are integermultiples of 360◦

N . Thus, you may have two cardinal points 0◦ and 180◦; three cardinal points 0◦, 120◦ and 240◦;four cardinal points 0◦, 90◦, 180◦ and 270◦, etc. See Fig. 2.

1.4 Directions and Orientations

If you consider a ray going from the center of the circle to a point on the circle, you get a direction that can becoded with the corresponding angle. If you consider both one such ray and its opposite, you get an orientation3

(not to be confused with the orientation of a topological space) that is coded with a number in the interval [0,180). Orientation is in fact different from direction, yet both are cyclic magnitudes; this has to do with the factthat there is only one closed 1-manifold, and that it is orientable.

1.5 Musical Notes

In the same way that, after hour 23 comes again hour 0, of the next day, after 7 (or 12, with semitones) notesC, D, E, F, G, A, B, comes again note C of the next octave. In this sense, the music notes are also a cyclicmagnitude. The indication with colors of the notes being played can help children learning to recognize thenotes; it also can be of use to deaf people in getting some feeling for western music.

1.6 More on Cyclic Variables

Loosely speaking, a variable is of a cyclic (or circular) nature if it repeats itself, as those in in Sections 1.2 - 1.5.Such is also the case of the azimuthal direction in front of you as you spin on your feet; if you do not reverse theturning direction but carry on, you will repeat yourself. In one turn, the series of directions is swept completelyand without repeating, nor jumping over any value, arriving at the end back to the starting point.

There is not a clear ”origin” for a circular space and three or four cardinal points do better the job of baptizingthe space, giving a consistent ”direction of travel” so that, given two values of a cyclic value, the choice regardingwhich of the two values is previous and which is posterior is determined. Thus, given two points on a circle, thereare two ways of going from one to the other: one respecting this canonical direction and the other, backwards.For example, for the hue circle, you may say that red is previous to yellow which in turn is previous to greenwhich in turn is previous to blue which in turn is previous to red. In some cases, you also speak of a clockwise

∗By 360− we mean a number less than 360 but otherwise arbitrarily close to 360.

Figure 3. A hue circle that is also a chromatic diagram (since opposing hues are complementary) is shown. The uniquehues are at angles spaced by multiples of 90◦, as cardinal points

and of a counterclockwise directions. A canonical direction of travel on a circle orients the circle; once you havegiven an orientation to a cyclic variable, given three values of the variable, there is one that is in between theother two.

Given two cyclic variables C and Z, where Z has as many possible values as C does, you may assign to eachvalue of C a value of Z without making repetitions (i.e. in a one-to-one fashion) that respects the betweenness ofC (i.e. whenever c2 is between c1 and c3, for the corresponding assignments, you have that z2 is between z1 andz3). This will be called a cyclic order homomorphism.The betweenness may be respected in two ways: respectingor reversing the orientation of C in Z; thus, the relation of betweenness is more basic than that of (topological)orientation. The values z of Z that are assigned to the values of C are called labels of the corresponding valuesc of C. b If Z is the circle of the hues, you have color labels and the assignment is a color code. In the caseof homomorphisms that are self-assignments, you can implement rotations and also correspondences that shrinkand expand different segments of the circle.

There are continuous cyclic variables such as the hue, the angle and the azimuth (the set of possible valuesis a continuum) and there are discrete cyclic variables such as the days of the week and the musical notes inan octave. In the discrete case, the set of possible values is countable and, most of the times, finite. Unlike theazimuth, the time and other cyclic physical variables that can me measured with an instrument, the chromatichue is a perceptual variable or, more precisely, a psychophysical variable.

2. THE MATHEMATICS OF RELATIONS

In mathematics, a (binary) relation ρ between the elements of a set A is any subset of the Cartesian productA×A, that is, ρ ⊂ A×A and, instead of (a, b) ∈ ρ, you also write a ρ b. The relation is reflexive if ∀a ∈ A, aρa.The relation is symmetric if ∀a, b ∈ A, whenever aρb then bρa. The relation is antisymmetric if ∀a, b ∈ A, aρband bρa then a = b. The relation is transitive if ∀a, b, c ∈ A, aρb and bρc then aρc.

The relation is connected if ∀a, b ∈ A, either aρb or bρa. The relation is a linear order if it is transitive,antisymmetric, reflexive and connected. The relation is an equivalence relation if it is transitive, symmetric,reflexive and connected.

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Figure 4. If b is after a and before c, and d is after a and before b, then b is after d, and before c and a is after c andbefore d.

An example of a linear order is the relation ≤ in the interval [0, 1] of the real numbers. Most physicalmagnitudes are assumed to take values in the set of the real numbers, with its standard linear order; however,as we have already said, some magnitudes are better modeled with a circle. Many times, a circularly ordered setis obtained from a linearly ordered set having a max and a min, by identifying the max with the min; however,such a cyclic order can be mathematized in a more straightforward way with the help of a ternary relation. Aternary relation between the elements of a set A is any subset of the Cartesian product† A×A×A.

2.1 Cyclic Orders

Following Megiddo,4 we say that a complete cyclic ordering for the elements of a set A containing at least threeelements, is a subset σ of the Cartesian product A3 = A×A×A such that‡:

i. (Antisymmetry) If (a, b, c) ∈ σ then a 6= b, a 6= c and b 6= c.ii. (Completeness) For each three points a, b, c ∈ A, either (a, b, c) ∈ σ or (a, c, b) ∈ σ, but not both.iii. (Rotational symmetry) For each three points a, b, c ∈ A, (a, b, c) ∈ σ if and only if (b, c, a) ∈ σ if and only if(c, a, b) ∈ σ. Note that (b, c, a) and (c, a, b) are the (nontrivial) even permutations of (a, b, c).

In addition, if A has more than three elements, theniv. (Transitivity) For each four points a, b, c, d ∈ σ, if (b, c, a) ∈ σ and (a, d, b) ∈ R then (d, b, c) ∈ σ and(c, a, d) ∈ σ. (See Figure 4.)

Note that a cyclic ordering, in addition to providing the notion of betweenness, provides a direction and, ifσ is a cyclic ordering for the elements of a set A then the ternary relation σ′ := {(x, y, z) ∈ A3 : ∃(a, b, c) ∈σ, (x, y, z) = (a, c, b)} is called the mirror σ’ of σ.

2.2 Circular order homomorphisms

Let A and B be sets with complete cyclic orders σ and τ , respectively; then, a function f : A→ B is said to bea cyclic order isomorphism if ∀a, b, c ∈ A, (a, b, c) ∈ σ ⇐⇒ (f(a), f(b), f(c)) ∈ τ . If f : A → B is such that∀a, b, c ∈ A, (a, b, c) ∈ σ′ ⇐⇒ (f(a), f(b), f(c)) ∈ τ , then f is said to be a cyclic order antimorphism.

†Strictly speaking, (A×A)×A 6= A× (A×A) but it is clear in what sense the product is associative.‡You can read (a, b, c) ∈ σ as ”b is after a and before c”

f : A→ B is said to be cyclic monotonic if it is either a cyclic order isomorphism or a cyclic order antimor-phism.

3. EXAMPLES

Consider first the display of the value of a cyclic variable such as time or azimuth, at a given time and place. Inthe following section, we consider the display of cyclic variables as they vary over 1- or 2-dimensional sets, e.g.wind direction maps.

Suppose three or (in most cases) four cardinal values of a cyclic variable to be visualized have been alreadychosen; as pointed out before, they will be indicated with spectral red, unique green and unique blue, or (in thecase of 4 cardinal values) the four unique hues red, yellow, green and blue. The cardinal values are meaningfulpoints that partition the remaining set of possible values of the variable into meaningful intervals, to be indicatedwith the four families of binary colors. For example, the angles 0◦, 90◦, 180◦ and 270◦, or the angles 0◦, 120◦

and 240◦ are natural choices of cardinal points in the case of angles.

An ordering (first, second, third and fourth) of the cardinal values gives a direction to the circle of the valuesof the cyclic variable. Any two cardinal values are either consecutive (the first and the second, the first and thefourth, the second and the third, or the third and the fourth) or opposing (the first and the third, or the secondand the fourth). It is assumed that the ordering of the four cardinal points is such that, if the circle of values isbroken apart into two halves, with cuts at the first and third cardinal points, one of the halves will contain thesecond cardinal point and the other half will contain the fourth cardinal value. Given an appropriate ordering ofthe four cardinal points, any even permutation of it will also give an appropriate ordering of the cardinal points;however, no odd permutation will be appropriate.

To indicate by means of a color the value taken by a cyclic variable, choose a color code as follows. Afterdetermining 4 cardinal values of the cyclic variable, suitably ordered, assign them to the four unique hues, alsosuitably ordered. Then you assign the intermediate values (those in between cardinal points) of the cyclic variableto corresponding families of binary hues, orange, cetrine, cyan and violet. For a choice of only three cardinalpoints, the 3 families of colors to be assigned to intermediate values of the cyclic variable are oranges, yellowand cetrines; cyans; and violets (and purples).

In the display device it may be convenient to have a reference frame, as shown in Fig. 5. The frame willmake as explicit as desired the color code. The frame will be convenient mainly for getting familiar with thecolor code. If the frame is contiguous to the color indication patch, there will be a zone of continuity of colorthat will also help in reading the variable.

An alternative choice of hues to be assigned to the cardinal values of a cyclic variable could be central orange,central cyan, central cetrine and central purple; in this order, or in any even permutation of it. Central orange isthat orange that is not perceived as more yellow than red nor viceversa, and likewise for the other central hues.

The use of color codes for the display of cyclic information allows thus for ludic instruments of fast andintuitive reading. Depending on the saturation and the luminance of the hues, the displays can be made eithersober or flashy.

Consider for example, time pieces, as in Figures 9 and 10; musical note (and/or key) indicators, as in Fig. 8;lunar clocks, as in Fig. 6; hue compasses, as in Fig. 7; instruments for the measure of angles or the indicationof orientation or direction (e.g. for a cockpit) as in Fig. 11, etc.

In an achromatic color code, which suffers from ambiguity but that is suitable for color blind people, twoopposing cardinal values of a cyclic variable are assigned to the colors black and white. As you traverse theachromatic circle, you go from black to white, and back to black. Since each gray is repeated twice, the code isambiguous; nevertheless, the value being represented by a given gray could be inferred from the context. Thisambiguity is not that rare, it is similar to the case when, seeing a 3 on a 12 hr. clock face, we know most of thetime whether it is 3 a.m. or 3 p.m. See Fig. 12. Again, the use of a reference frame is recommended.

Figure 5. Above, a reference frame that is a hue circle with the unique hues at angles spaced by multiples of 90◦. Below,in the reference frame, unique blue, unique green and spectral red are separated by intervals of 120◦, more suitable fora choice of three cardinal values. In each case, the hue of the central circle indicates the value that a cyclic variable istaking.

Figure 6. A chromatic lunar clock. According to the reference frame, it is showing ”full moon”.

Figure 7. A compass indicating the azimuthal direction ”East”.

Figure 8. Left: the 12 semitones of the chromatic music scale are shown in the reference frame, while the central indicatorpatch is indicating the note C. Right: the corresponding tones are indicated on a colored keyboard.

Figure 9. At left a 12-hour color clock face and at right a 24-hour color clock face. In both cases, the central circularpatch indicates the seconds, the right semiannular patch indicates the minutes and the left semiannular patch indicatesthe hours. At left, the indicated time is 12:45:15; at right, 00:45:15.

Figure 10. Several designs of color clock faces are possible; here, three rectangular color patches within the referenceframe, indicate the hours (right), the minutes (middle) and the seconds (right). The indicated time is 09:10:00.

Figure 11. An indicator for an instrument that measures angles. According to the reference frame, the central colorcircular patch is indicating 270◦ = -90◦.

Figure 12. An achromatic lunar clock. The central, indicator circle shows ”nearly full moon”; whether it is crescent orwaning, should be deduced from the context.

Figure 13. Annulus

4. COLORED LINES AND COLORED SURFACES

At least since the times of Fermat and Descartes, we visualize the values a linear magnitude takes on, as afunction of time or space, with the help of a plot on a Cartesian plane; that is, we plot lthe variable as a verticaldistance, the height from a reference, 0-level, that changes with the horizontal position in the plane.

On the other hand, if you need to visualize a cyclic variable (e.g. the phase of a Fourier transform) as afunction of a linear variable (e.g. the frequency) you may plot a line on a horizontal cylinder but also, moreconveniently, using a color code, you may draw a colored line where the hue indicates the value of the phase andthe horizontal distance indicates the frequency. Likewise, if you need to plot a cyclic variable as a function ofanother cyclic variable (e.g. the phase of the Fourier transfoem of a discrete signal) you may either draw a blackline on a torus or, given a color code and more conveniently, draw a colored circle on a flat screen or paper.

In the previous cases, the domain of the function being plotted is one dimensional. If you need to visualize acyclic variable (e.g. wind direction) as a function of a 2-dimensional variable, (e.g. the map of a country), givena color code, you could use a colored surface, instead of a surface with many little arrows indicating the winddirection. If, in addition to the wind direction you need to plot the wind velocity, you could use the luminanceof the colors, dark for slow, light for fast, for example. The wind velocity is in this sense, a vector magnitudewith one cyclic component and one linear component.

Let I be an interval of the real line. A function f : I → S1 has a one-dimensional graph§ that lives in theannulus (a topological cylinder), see Fig. 13, given by I × S1. If you assign a color to each point of S1, for eachpoint t ∈ I, f(t) gives a color and you may think of the graph of f as of a colored interval. For each point tof I there is one and only one point (t, f(t)) of the graph and, if you think of the annulus as of a collection ofconcentric circles, one for each t ∈ I, for each such circle there is precisely one point of the graph of f ; see Figure13. For f continuous, the graph is a line that goes from the, say, inner circular boundary of the annulus to theouter boundary circle; it may go in a straight fashion but it can ”oscillate” changing direction with respect toan orientation of S1; also, it may be homotopically¶ trivial or not.

Likewise, a function g : I × I → S1 has a graph that is a surface in the solid torus, see Fig. 14, given byI × I × S1. For each point of I × I there is precisely one point, or color if you prefer, of S1 in the graph. Thecolored surface can be flattened out and the graph of the function can thus be visualized as a colored I × I.

§The graph of a function φ : A→ B is the subset of A×B given by {(a, b) : ∃a ∈ A, b = φ(a)}¶Depending on how many times the line circles around in a net fashion.

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Figure 14. Solid torus

5. FINAL REMARKS

Research in the art of scientific visualization using color is worthwhile. Color has been a topic prone to confusionat all times and good visualization techniques will also teach aspects of color science to the public at large. Morecolor is used every day to visualize information, making the nonscientific watcher a common target. Color is afundamental tool in visualization, yet it poses nontrivial problems to the art of conveying data through chartsand plots. When used to visualize linear‖ magnitudes, the paradigm of warm and cool colors is commonly used,with warm colors for high values of the magnitude and cool colors for low values of the magnitude. Incidentally,it may be argued that not only there are warm colors: oranges and yellow, and cool colors: blue, cyans, green,but also sweet colors (purples) and sour colors (lemon yellows or citrines); in this sense, warm and cool colorsare actually disconnected sets of colors.

We argue that hue is a cyclic magnitude and that it should be used mainly to visualize cyclic magnitudes. Itcan be argued that there are as well spherical3 magnitudes which perhaps can be meaningfully visualized withcolors in the boundary of Runge’s ball, i.e. with maximally colorful colors.17 It is our stand that if all huesin the hue circle are to be used in a visualization plot, the proper use of hue is to display the value of circularor cyclic variables; otherwise, the use of hue can be misleading when used to display linear variables, mainlyat the transition between cool and warm hues. The cool-warm paradigm is closely related to the rainbow colormap where the spectral colors, according to wavelength, do represent a linear interval (that considers mostly huebut not much luminance or saturation); however, perceptually, the spectral colors are more like a ”circle with agap”. Also, the rainbow color map commonly gives rise to artificial contours in continuous data visualization.Another problem when using the color spectrum to visualize linear magnitudes is that violet, at the cool end ofthe spectrum, perceptually, is also a transition to red, which is in the warm end of the spectrum. Several people,see e.g.5 and6 have given thought to these problems. The problem has been stated as in ”Interval data sets havemeasurable distances (degrees Celsius, height) and ratio data sets also have a zero point (degrees Kelvin, heightabove sea level). Although attempts to display interval and ratio data often use color maps, user studies haveshown that contrast effects and other perceptual distortions make the user incapable of coming up with accurateabsolute value judgments.”7

Unlike the hue, the luminance and saturation of a color8 are linear variables. A basic approach is then tochange the saturation or the luminance, while keeping the hue constant, when visualizing linear variables, and

‖A linear magnitude, in contrast to a cyclic magnitude, is for example the temperature: by continuously changing it ina given direction you cannot go back to the initial point, only by reversing the direction can you reach an already passedpoint.

to change only the hue when visualizing cyclic variables. Such an approach should give good results and solvethe noted problems, it however may also be too restrictive as the cool-warm paradigm may still have a role toplay. All the visualizing potential of color, a tridimensional magnitude, should be used. When making 2D9 colorplots,10 self induced contrast11 plays a role as well. Variations in saturation help to display spatial chromaticvariations. We can read high spatial-frequency contrast, but not low spatial-frequency contrast with luminance,and viveversa for blue-yellow chrominance,12 this should be also taken into account.

The use of color in the industry of measure instruments is likely to grow. Kahil13 has proposed the use ofcolor to indicate time and azimuth, as well as Hooper14 who has proposed a hue compass; likewise, Miroslav15

has patented a watch that uses color, as well as Ernst,16 although not always following the guidelines presentedhere.

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