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70-472: Several earlier color order systems had placed colors into a three-dimensional color solid of one form or another, but Munsell was the first to separate hue, value, and chroma into perceptually uniform and independent dimensions, and he was the first to illustrate the colors systematically in three-dimensional space. Munsell's system, particularly the later renotations, is based on rigorous measurements of human subjects' visual responses to color, putting it on

140-418: A ) {\displaystyle (x_{a},y_{a})} and ( x b , y b ) {\displaystyle (x_{b},y_{b})} Linear interpolation is quick and easy, but it is not very precise. Another disadvantage is that the interpolant is not differentiable at the point x k . The following error estimate shows that linear interpolation is not very precise. Denote

210-414: A , y a ) and ( x b , y b ), and the interpolant is given by: This previous equation states that the slope of the new line between ( x a , y a ) {\displaystyle (x_{a},y_{a})} and ( x , y ) {\displaystyle (x,y)} is the same as the slope of the line between ( x a , y

280-521: A color space or model and can be thought as an analog of, for example, the one-dimensional color wheel , which depicts the variable of hue (similarity with red, yellow, green, blue, magenta , etc.); or the 2D chromaticity diagram (also known as color triangle ), which depicts the variables of hue and spectral purity . The added spatial dimension allows a color solid to depict the three dimensions of color: lightness (gradations of light and dark, tints or shades ), hue, and colorfulness , allowing

350-439: A color is spectral red (which is located at one end of the spectrum), it will be seen as black. If the size of the portion of total emission or reflectance is increased, now covering from the red end of the spectrum to the yellow wavelengths, it will be seen as red. If the portion is expanded even more, covering the green wavelengths, it will be seen as orange or yellow. If it is expanded even more, it will cover more wavelengths than

420-458: A firm experimental scientific basis. Because of this basis in human visual perception, Munsell's system has outlasted its contemporary color models, and though it has been superseded for some uses by models such as CIELAB ( L*a*b* ) and CIECAM02 , it is still in wide use today. The system consists of three independent properties of color which can be represented cylindrically in three dimensions as an irregular color solid : Munsell determined

490-556: A function s : [ a , b ] → R {\displaystyle s:[a,b]\to \mathbb {R} } such that f ( x i ) = s ( x i ) {\displaystyle f(x_{i})=s(x_{i})} for i = 1 , 2 , … , n {\displaystyle i=1,2,\dots ,n} (that is, that s {\displaystyle s} interpolates f {\displaystyle f} at these points). In general, an interpolant need not be

560-430: A functional that represents the entire family of interpolants satisfying those constraints, including those that are discontinuous or partially defined. These functionals identify the subspace of functions where the solution to a constrained optimization problem resides. Consequently, TFC transforms constrained optimization problems into equivalent unconstrained formulations. This transformation has proven highly effective in

630-416: A given function by another function from some predetermined class, and how good this approximation is. This clearly yields a bound on how well the interpolant can approximate the unknown function. If we consider x {\displaystyle x} as a variable in a topological space , and the function f ( x ) {\displaystyle f(x)} mapping to a Banach space , then

700-962: A good approximation, but there are well known and often reasonable conditions where it will. For example, if f ∈ C 4 ( [ a , b ] ) {\displaystyle f\in C^{4}([a,b])} (four times continuously differentiable) then cubic spline interpolation has an error bound given by ‖ f − s ‖ ∞ ≤ C ‖ f ( 4 ) ‖ ∞ h 4 {\displaystyle \|f-s\|_{\infty }\leq C\|f^{(4)}\|_{\infty }h^{4}} where h max i = 1 , 2 , … , n − 1 | x i + 1 − x i | {\displaystyle h\max _{i=1,2,\dots ,n-1}|x_{i+1}-x_{i}|} and C {\displaystyle C}

770-495: A high chroma, while semichromes like yellow, orange, and cyan have a slightly lower chroma. In color spheres and the HSL color space , the maximum chroma colors are located around the equator at the periphery of the color sphere. This makes color solids with a spherical shape inherently non- perceptually uniform , since they imply that all full colors have a lightness of 50%, when, as humans perceive them, there are full colors with

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840-469: A hue circle, are complementary colors , and mix additively to the neutral gray of the same value. The diagram below shows 40 evenly spaced Munsell hues, with complements vertically aligned. Value , or lightness , varies vertically along the color solid, from black (value 0) at the bottom, to white (value 10) at the top. Neutral grays lie along the vertical axis between black and white. Several color solids before Munsell's plotted luminosity from black on

910-685: A lightness from around 30% to around 90%. A perceptually uniform color solid has an irregular shape. In the beginning of the 20th century, industrial demands for a controllable way to describe colors and the new possibility to measure light spectra initiated intense research on mathematical descriptions of colors. The idea of optimal colors was introduced by the Baltic German chemist Wilhelm Ostwald . Erwin Schrödinger showed in his 1919 article Theorie der Pigmente von größter Leuchtkraft (Theory of Pigments with Highest Luminosity) that

980-460: A medium grey is specified by "N 5/". In computer processing, the Munsell colors are converted to a set of "HVC" numbers. The V and C are the same as the normal chroma and value. The H (hue) number is converted by mapping the hue rings into numbers between 0 and 100, where both 0 and 100 correspond to 10RP. As the Munsell books, including the 1943 renotation, only contains colors for some points in

1050-456: A single schematic, using it as an aid in the composition and analysis of visual art. Interpolation In the mathematical field of numerical analysis , interpolation is a type of estimation , a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science , one often has a number of data points, obtained by sampling or experimentation , which represent

1120-422: A slanted double cone by August Kirschmann in 1895. These systems became progressively more sophisticated, with Kirschmann’s even recognizing the difference in value between bright colors of different hues. But all of them remained either purely theoretical or encountered practical problems in accommodating all colors. Furthermore, none was based on any rigorous scientific measurement of human vision; before Munsell,

1190-691: A smaller error than linear interpolation, while the interpolant is smoother and easier to evaluate than the high-degree polynomials used in polynomial interpolation. However, the global nature of the basis functions leads to ill-conditioning. This is completely mitigated by using splines of compact support, such as are implemented in Boost.Math and discussed in Kress. Depending on the underlying discretisation of fields, different interpolants may be required. In contrast to other interpolation methods, which estimate functions on target points, mimetic interpolation evaluates

1260-405: Is a common way to approximate functions. Given a function f : [ a , b ] → R {\displaystyle f:[a,b]\to \mathbb {R} } with a set of points x 1 , x 2 , … , x n ∈ [ a , b ] {\displaystyle x_{1},x_{2},\dots ,x_{n}\in [a,b]} one can form

1330-422: Is a constant. Gaussian process is a powerful non-linear interpolation tool. Many popular interpolation tools are actually equivalent to particular Gaussian processes. Gaussian processes can be used not only for fitting an interpolant that passes exactly through the given data points but also for regression; that is, for fitting a curve through noisy data. In the geostatistics community Gaussian process regression

1400-424: Is a type of color solid that contains all the colors that humans are able to see . The optimal color solid is bounded by the set of all optimal colors. The emission or reflectance spectrum of a color is the amount of light of each wavelength that it emits or reflects, in proportion to a given maximum, which has the value of 1 (100%). If the emission or reflectance spectrum of a color is 0 (0%) or 1 (100%) across

1470-440: Is also known as Kriging . Other forms of interpolation can be constructed by picking a different class of interpolants. For instance, rational interpolation is interpolation by rational functions using Padé approximant , and trigonometric interpolation is interpolation by trigonometric polynomials using Fourier series . Another possibility is to use wavelets . The Whittaker–Shannon interpolation formula can be used if

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1540-558: Is no intrinsic upper limit to chroma. Different areas of the color space have different maximal chroma coordinates. For instance light yellow colors have considerably more potential chroma than light purples, due to the nature of the eye and the physics of color stimuli. This led to a wide range of possible chroma levels—up to the high 30s for some hue–value combinations (though it is difficult or impossible to make physical objects in colors of such high chromas, and they cannot be reproduced on current computer displays). Vivid solid colors are in

1610-421: Is proportional to higher powers of the distance between the data points. These methods also produce smoother interpolants. Polynomial interpolation is a generalization of linear interpolation. Note that the linear interpolant is a linear function . We now replace this interpolant with a polynomial of higher degree . Consider again the problem given above. The following sixth degree polynomial goes through all

1680-566: Is studied, that no regular contour will serve. Each horizontal circle Munsell divided into five principal hues : R ed, Y ellow, G reen, B lue, and P urple, along with 5 intermediate hues (e.g., YR ) halfway between adjacent principal hues. Each of these 10 steps, with the named hue given number 5, is then broken into 10 sub-steps, so that 100 hues are given integer values. In practice, color charts conventionally specify 40 hues, in increments of 2.5, progressing as for example 10R to 2.5YR. Two colors of equal value and chroma, on opposite sides of

1750-423: Is the field values that are conserved (not the integral of the field). Apart from linear interpolation, area weighted interpolation can be considered one of the first mimetic interpolation methods to have been developed. The Theory of Functional Connections (TFC) is a mathematical framework specifically developed for functional interpolation . Given any interpolant that satisfies a set of constraints, TFC derives

1820-541: Is unlikely to be used, as linear interpolation (see below) is almost as easy, but in higher-dimensional multivariate interpolation , this could be a favourable choice for its speed and simplicity. One of the simplest methods is linear interpolation (sometimes known as lerp). Consider the above example of estimating f (2.5). Since 2.5 is midway between 2 and 3, it is reasonable to take f (2.5) midway between f (2) = 0.9093 and f (3) = 0.1411, which yields 0.5252. Generally, linear interpolation takes two data points, say ( x

1890-470: The USGS for matching soil color , in prosthodontics during the selection of tooth color for dental restorations , and breweries for matching beer color . The original Munsell color chart remains useful for comparing computer models of human color vision. General information Data and conversion Other tools Color solid A color solid is the three-dimensional representation of

1960-567: The displacement interpolation problem used in transportation theory . Multivariate interpolation is the interpolation of functions of more than one variable. Methods include nearest-neighbor interpolation , bilinear interpolation and bicubic interpolation in two dimensions, and trilinear interpolation in three dimensions. They can be applied to gridded or scattered data. Mimetic interpolation generalizes to n {\displaystyle n} dimensional spaces where n > 3 {\displaystyle n>3} . In

2030-494: The electric field , for instance, since the line integral gives the electric potential difference at the endpoints of the integration path. Mimetic interpolation ensures that the error of estimating the line integral of an electric field is the same as the error obtained by interpolating the potential at the end points of the integration path, regardless of the length of the integration path. Linear , bilinear and trilinear interpolation are also considered mimetic, even if it

2100-417: The "cold" sharp edge. Each hue has a maximum chroma color, also known as maximum chroma point, semichrome, or full color; there are no colors of that hue with a higher chroma. They are the most chromatic, vibrant colors that we are able to see. Although we are, for now, unable to produce them, these are the colors that would be located in an ideal color wheel. They were called semichromes or full colors by

2170-460: The German chemist and philosopher Wilhelm Ostwald in the early 20th century. If B is the complementary wavelength of wavelength A, then the straight line that connects A and B passes through the achromatic axis in a linear color space, such as LMS or CIE 1931 XYZ. If the emission or reflection spectrum of a color is 1 (100%) for all the wavelengths between A and B, and 0 for all the wavelengths of

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2240-795: The MacAdam limit (the optimal colors, the boundary of the Optimal color solid) are computed, because all the other (non-optimal) colors exist inside the boundary. Color volume is the set of all available color at all available hue , saturation , lightness , and brightness . It's the result of a 2D color space or 2D color gamut (that represent chromaticity ) combined with the dynamic range . The term has been used to describe HDR 's higher color volume than SDR (i.e. peak brightness of at least 1,000 cd/m higher than SDR's 100 cd/m limit and wider color gamut than Rec. 709 / sRGB ). The color solid can also be used to clearly visualize

2310-402: The Munsell space, it is non-trivial to specify an arbitrary color in Munsell space. Interpolation must be used to assign meanings to non-book colors such as "2.8Y 6.95/2.3", followed by an inversion of the fitted Munsell-to-xyY transform. The ASTM has defined a method in 2008, but Centore 2012 is known to work better. The idea of using a three-dimensional color solid to represent all colors

2380-483: The bottom to white on the top, with a gray gradient between them, but these systems neglected to keep perceptual lightness constant across horizontal slices. Instead, they plotted fully saturated yellow (light), and fully saturated blue and purple (dark) along the equator. Chroma , measured radially from the center of each slice, represents the “purity” of a color (related to saturation ), with lower chroma being less pure (more washed out, as in pastels ). Note that there

2450-545: The boundary of the optimal color solid in the CIE 1931 color space for lightness levels from Y = 10 to 95 in steps of 10 units. This enabled him to draw the optimal color solid at an acceptable degree of precision. Because of his achievement, the boundary of the optimal color solid is called the MacAdam limit (1935). On modern computers, it is possible to calculate an optimal color solid with great precision in seconds. Usually, only

2520-545: The color solid, then, is gray all along its length, varying from black at the bottom to white at the top, it is a grayscale . All pure (saturated) hues are located on the surface of the solid, varying from light to dark down the color solid. All colors that are desaturated in any degree (that is, that they can be though of containing both black and white in varying amounts) comprise the solid's interior, likewise varying in brightness from top to bottom. The optimal color solid, Rösch – MacAdam color solid, or simply visible gamut ,

2590-463: The color wheel, contrasting (or complementary) hues are located opposite each other in most color solids. Moving toward the central axis, colors become less and less saturated, until all colors meet at the central axis as a neutral gray. Moving vertically in the color solid, colors become lighter (toward the top) and darker (toward the bottom). At the upper pole, all hues meet in white; at the bottom pole, all hues meet in black. The vertical axis of

2660-476: The domain of digital signal processing, the term interpolation refers to the process of converting a sampled digital signal (such as a sampled audio signal) to that of a higher sampling rate ( Upsampling ) using various digital filtering techniques (for example, convolution with a frequency-limited impulse signal). In this application there is a specific requirement that the harmonic content of the original signal be preserved without creating aliased harmonic content of

2730-457: The ends to zero in the middle. The first type produces colors that are similar to the spectral colors and follow roughly the horseshoe-shaped portion of the CIE xy chromaticity diagram (the spectral locus ), but are generally more chromatic , although less spectrally pure. The second type produces colors that are similar to (but generally more chromatic and less spectrally pure than) the colors on

2800-438: The entire visible spectrum, and it has no more than two transitions between 0 and 1, or 1 and 0, then it is an optimal color. With the current state of technology, we are unable to produce any material or pigment with these properties. Thus two types of "optimal color" spectra are possible: Either the transition goes from zero at both ends of the spectrum to one in the middle, as shown in the image at right, or it goes from one at

2870-424: The function at intermediate points, such as x = 2.5. {\displaystyle x=2.5.} We describe some methods of interpolation, differing in such properties as: accuracy, cost, number of data points needed, and smoothness of the resulting interpolant function. The simplest interpolation method is to locate the nearest data value, and assign the same value. In simple problems, this method

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2940-406: The function which we want to interpolate by g , and suppose that x lies between x a and x b and that g is twice continuously differentiable. Then the linear interpolation error is In words, the error is proportional to the square of the distance between the data points. The error in some other methods, including polynomial interpolation and spline interpolation (described below),

3010-421: The horseshoe-shaped spectral locus of the chromaticiy diagram). In linear color spaces that contain all colors visible by humans, such as LMS or CIE 1931 XYZ , the set of half-lines that start at the origin (black, (0, 0, 0)) and pass through all the points that represent the colors of the visible spectrum, and the portion of a plane that passes through the violet half-line and the red half-line (both ends of

3080-437: The integral of fields on target lines, areas or volumes, depending on the type of field (scalar, vector, pseudo-vector or pseudo-scalar). A key feature of mimetic interpolation is that vector calculus identities are satisfied, including Stokes' theorem and the divergence theorem . As a result, mimetic interpolation conserves line, area and volume integrals. Conservation of line integrals might be desirable when interpolating

3150-412: The interpolant has to go exactly through the data points is relaxed. It is only required to approach the data points as closely as possible (within some other constraints). This requires parameterizing the potential interpolants and having some way of measuring the error. In the simplest case this leads to least squares approximation. Approximation theory studies how to find the best approximation to

3220-487: The intervals, and chooses the polynomial pieces such that they fit smoothly together. The resulting function is called a spline. For instance, the natural cubic spline is piecewise cubic and twice continuously differentiable. Furthermore, its second derivative is zero at the end points. The natural cubic spline interpolating the points in the table above is given by In this case we get f (2.5) = 0.5972. Like polynomial interpolation, spline interpolation incurs

3290-454: The most-saturated colors that can be created with a given total reflectivity are generated by surfaces having either zero or full reflectance at any given wavelength, and the reflectivity spectrum must have at most two transitions between zero and full. Schrödinger's work was further developed by David MacAdam and Siegfried Rösch  [ Wikidata ] . MacAdam was the first person to calculate precise coordinates of selected points on

3360-627: The notations (sample definitions) for the modern Munsell Book of Color . Though several replacements for the Munsell system have been invented, building on Munsell's foundational ideas—including the Optical Society of America's Uniform Color Scales , and the International Commission on Illumination ’s CIELAB ( L*a*b* ) and CIECAM02 color models—the Munsell system is still widely used, by, among others, ANSI to define skin color and hair color for forensic pathology ,

3430-442: The number of data points is infinite or if the function to be interpolated has compact support. Sometimes, we know not only the value of the function that we want to interpolate, at some points, but also its derivative. This leads to Hermite interpolation problems. When each data point is itself a function, it can be useful to see the interpolation problem as a partial advection problem between each data point. This idea leads to

3500-427: The optimal color solid is centrally symmetric. In most color spaces, the surface of the optimal color solid is smooth, except for two points (black and white); and two sharp edges: the " warm " edge, which goes from black, to red, to orange, to yellow, to white; and the "cold" edge, which goes from black, to blue, to cyan , to white. This is due to the following: If the portion of the emission or reflection spectrum of

3570-414: The original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gain in simplicity may outweigh the loss from interpolation error and give better performance in calculation process. This table gives some values of an unknown function f ( x ) {\displaystyle f(x)} . Interpolation provides a means of estimating

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3640-472: The original signal above the original Nyquist limit of the signal (that is, above fs/2 of the original signal sample rate). An early and fairly elementary discussion on this subject can be found in Rabiner and Crochiere's book Multirate Digital Signal Processing . The term extrapolation is used to find data points outside the range of known data points. In curve fitting problems, the constraint that

3710-403: The other half of the color space, then that color is a maximum chroma color, semichrome, or full color (this is the explanation to why they were called semi chromes). Thus, maximum chroma colors are a type of optimal color. As explained, full colors are far from being monochromatic. If the spectral purity of a maximum chroma color is increased, its chroma decreases, because it will approach

3780-450: The problems of linear interpolation. However, polynomial interpolation also has some disadvantages. Calculating the interpolating polynomial is computationally expensive (see computational complexity ) compared to linear interpolation. Furthermore, polynomial interpolation may exhibit oscillatory artifacts, especially at the end points (see Runge's phenomenon ). Polynomial interpolation can estimate local maxima and minima that are outside

3850-424: The range of approximately 8. A color is fully specified by listing the three numbers for hue, value, and chroma in that order. For instance, a purple of medium lightness and fairly saturated would be 5P 5/10 with 5P meaning the color in the middle of the purple hue band, 5/ meaning medium value (lightness), and a chroma of 10 (see swatch). An achromatic color is specified by the syntax N V/ . For example,

3920-458: The range of the samples, unlike linear interpolation. For example, the interpolant above has a local maximum at x ≈ 1.566, f ( x ) ≈ 1.003 and a local minimum at x ≈ 4.708, f ( x ) ≈ −1.003. However, these maxima and minima may exceed the theoretical range of the function; for example, a function that is always positive may have an interpolant with negative values, and whose inverse therefore contains false vertical asymptotes . More generally,

3990-538: The relationship between hue, value, and chroma was not understood. Albert Munsell, an artist and professor of art at the Massachusetts Normal Art School (now Massachusetts College of Art and Design , or MassArt), wanted to create a "rational way to describe color" that would use decimal notation instead of color names (which he felt were "foolish" and "misleading"), which he could use to teach his students about color. He first started work on

4060-453: The seven points: Substituting x = 2.5, we find that f (2.5) = ~0.59678. Generally, if we have n data points, there is exactly one polynomial of degree at most n −1 going through all the data points. The interpolation error is proportional to the distance between the data points to the power n . Furthermore, the interpolant is a polynomial and thus infinitely differentiable. So, we see that polynomial interpolation overcomes most of

4130-499: The shape of the resulting curve, especially for very high or low values of the independent variable, may be contrary to commonsense; that is, to what is known about the experimental system which has generated the data points. These disadvantages can be reduced by using spline interpolation or restricting attention to Chebyshev polynomials . Linear interpolation uses a linear function for each of intervals [ x k , x k+1 ]. Spline interpolation uses low-degree polynomials in each of

4200-521: The solid to depict all conceivable colors in an organized three-dimensional structure. Different color theorists have each designed unique color solids. Many are in the shape of a sphere , whereas others are warped three-dimensional ellipsoid figures—these variations being designed to express some aspect of the relationship of the colors more clearly. The color spheres conceived by Phillip Otto Runge and Johannes Itten are typical examples and prototypes for many other color solid schematics. As in

4270-651: The solution of differential equations . TFC achieves this by constructing a constrained functional (a function of a free function), that inherently satisfies given constraints regardless of the expression of the free function. This simplifies solving various types of equations and significantly improves the efficiency and accuracy of methods like Physics-Informed Neural Networks (PINNs). TFC offers advantages over traditional methods like Lagrange multipliers and spectral methods by directly addressing constraints analytically and avoiding iterative procedures, although it cannot currently handle inequality constraints. Interpolation

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4340-511: The spacing of colors along these dimensions by taking measurements of human visual responses. In each dimension, Munsell colors are as close to perceptually uniform as he could make them, which makes the resulting shape quite irregular. As Munsell explains: Desire to fit a chosen contour, such as the pyramid, cone, cylinder or cube, coupled with a lack of proper tests, has led to many distorted statements of color relations, and it becomes evident, when physical measurement of pigment values and chromas

4410-648: The straight line in the CIE xy chromaticity diagram (the " line of purples "), leading to magenta or purple-like colors. In optimal color solids, the colors of the visible spectrum are theoretically black, because their emission or reflection spectrum is 1 (100%) in only one wavelength, and 0 in all of the other infinite visible wavelengths that there are, meaning that they have a lightness of 0 with respect to white, and will also have 0 chroma, but, of course, 100% of spectral purity. In short: In optimal color solids, spectral colors are equivalent to black (0% lightness, 0 chroma), but have full spectral purity (they are located in

4480-469: The system in 1898 and published it in full form in A Color Notation in 1905. The original embodiment of the system (the 1905 Atlas) had some deficiencies as a physical representation of the theoretical system. These were improved significantly in the 1929 Munsell Book of Color and through an extensive series of experiments carried out by the Optical Society of America in the 1940s resulting in

4550-446: The values of a function for a limited number of values of the independent variable . It is often required to interpolate ; that is, estimate the value of that function for an intermediate value of the independent variable. A closely related problem is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from

4620-455: The visible spectrum), generate the "spectrum cone". The black point (coordinates (0, 0, 0)) of the optimal color solid (and only the black point) is tangent to the "spectrum cone", and the white point ((1, 1, 1)) (only the white point) is tangent to the "inverted spectrum cone", with the "inverted spectrum cone" being symmetrical to the "spectrum cone" with respect to the middle gray point ((0.5, 0.5, 0.5)). This means that, in linear color spaces,

4690-421: The visible spectrum, ergo, it will approach black. In perceptually uniform color spaces, the lightness of the full colors varies from around 30% in the violetish blue hues, to around 90% in the yellowish hues. The chroma of each maximum chroma point also varies depending on the hue; in optimal color solids plotted in perceptually uniform color spaces, semichromes like red, green, blue, violet, and magenta have

4760-478: The volume or gamut of a screen, printer, the human eye, etc, because it gives information about the dimension of lightness, whilst the commonly used chromaticity diagram lacks this dimension of color. Artists and art critics find the color solid to be a useful means of organizing the three variables of color—hue, lightness (or value), and saturation (or chroma), as modelled in the HCL and HSL color models —in

4830-402: The yellow semichrome does, approaching white, until it is reached when the full spectrum is emitted or reflected. The described process is called "cumulation". Cumulation can be started at either end of the visible spectrum (we just described cumulation starting from the red end of the spectrum, generating the "warm" sharp edge), cumulation starting at the violet end of the spectrum will generate

4900-418: Was developed during the 18th and 19th centuries. Several different shapes for such a solid were proposed, including: a double triangular pyramid by Tobias Mayer in 1758, a single triangular pyramid by Johann Heinrich Lambert in 1772, a sphere by Philipp Otto Runge in 1810, a hemisphere by Michel Eugène Chevreul in 1839, a cone by Hermann von Helmholtz in 1860, a tilted cube by William Benson in 1868, and

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