Instruct: Given a question, retrieve relevant passages that can help answer the question.
Query: What is a ray in geometry?
Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the geodesics of the geometry. An example is provided by the picture of the complement of the Borromean rings. The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views of horoball neighborhoods of the link. By thickening the link in a standard way, the horoball neighborhoods of the link components are obtained. Even though the boundary of a neighborhood is a torus, when viewed from inside the link complement, it looks like a sphere. Each link component shows up as infinitely many spheres (of one color) as there are infinitely many light rays from the observer to the link component. The fundamental parallelogram (which is indicated in the picture), tiles both vertically and horizontally and shows how to extend the pattern of spheres infinitely.
Symmetry (from Greek συμμετρία "symmetria" "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so in this article they are discussed together.
A polarizer or polariser is an optical filter that lets light waves of a specific polarization pass through while blocking light waves of other polarizations. It can filter a beam of light of undefined or mixed polarization into a beam of well-defined polarization, that is polarized light. The common types of polarizers are linear polarizers and circular polarizers. Polarizers are used in many optical techniques and instruments, and polarizing filters find applications in photography and LCD technology. Polarizers can also be made for other types of electromagnetic waves besides light, such as radio waves, microwaves, and X-rays.
Light or other electromagnetic radiation from many sources, such as the sun, flames, and incandescent lamps, consists of short wave trains with an equal mixture of polarizations; this is called "unpolarized light". Polarized light can be produced by passing unpolarized light through a polarizer, which allows waves of only one polarization to pass through. The most common optical materials (such as glass) are isotropic and do not affect the polarization of light passing through them; however, some materials—those that exhibit birefringence, dichroism, or optical activity—can change the polarization of light. Some of these are used to make polarizing filters. Light is also partially polarized when it reflects from a surface.
The pitch is rectangular in shape. The longer sides are called touchlines. The other opposing sides are called the goal lines. The two goal lines must be between wide, and be the same length. The two touchlines be between long, and be the same length. All lines on the ground must be equally wide, not to exceed . The corners of the pitch are marked by corner flags.
Polarization (also polarisation) is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave. A simple example of a polarized transverse wave is vibrations traveling along a taut string "(see image)"; for example, in a musical instrument like a guitar string. Depending on how the string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string. In contrast, in longitudinal waves, such as sound waves in a liquid or gas, the displacement of the particles in the oscillation is always in the direction of propagation, so these waves do not exhibit polarization. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves, gravitational waves, and transverse sound waves (shear waves) in solids. In some types of transverse waves, the wave displacement is limited to a single direction, so these also do not exhibit polarization; for example, in surface waves in liquids (gravity waves), the wave displacement of the particles is always in a vertical plane.
In mathematics, a square root of a number "a" is a number "y" such that ; in other words, a number "y" whose "square" (the result of multiplying the number by itself, or ) is "a". For example, 4 and −4 are square roots of 16 because .
Every nonnegative real number "a" has a unique nonnegative square root, called the "principal square root", which is denoted by , where √ is called the "radical sign" or "radix". For example, the principal square root of 9 is 3, which is denoted by = 3, because and 3 is nonnegative. The term (or number) whose square root is being considered is known as the "radicand". The radicand is the number or expression underneath the radical sign, in this example 9.
It was Gauss who coined the term "non-Euclidean geometry". He was referring to his own, unpublished work, which today we call "hyperbolic geometry". Several authors still consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. In 1871, Felix Klein, by adapting a metric discussed by Arthur Cayley in 1852, was able to bring metric properties into a projective setting and was thus able to unify the treatments of hyperbolic, euclidean and elliptic geometry under the umbrella of projective geometry. Klein is responsible for the terms "hyperbolic" and "elliptic" (in his system he called Euclidean geometry "parabolic", a term which has not survived the test of time and is used today only in a few disciplines.) His influence has led to the common usage of the term "non-Euclidean geometry" to mean either "hyperbolic" or "elliptic" geometry.
In general relativity, light follows the curvature of spacetime, hence when light passes around a massive object, it is bent. This means that the light from an object on the other side will be bent towards an observer's eye, just like an ordinary lens. In General Relativity the speed of light depends on the gravitational potential (aka the metric) and this bending can be viewed as a consequence of the light traveling along a gradient in light speed.
Light rays are the boundary between the future, the spacelike, and the past regions. The gravitational attraction can be viewed as the motion of undisturbed objects in a background curved "geometry" or alternatively as the response of objects to a "force" in a flat geometry. The angle of deflection is:
Viète's formula for , the ratio of a circle's circumference to its diameter, is
Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary; the points for these numbers lie on the vertical axis of the complex plane. A complex number whose imaginary part is zero can be viewed as a real number; its point lies on the horizontal axis of the complex plane. Complex numbers can also be represented in polar form, which associates each complex number with its distance from the origin (its magnitude) and with a particular angle known as the argument of this complex number.
An important aspect of light's nature is its frequency. The frequency of a wave is its rate of oscillation and is measured in hertz, the SI unit of frequency, where one hertz is equal to one oscillation per second. Light usually has multiple frequencies that sum to form the resultant wave. Different frequencies undergo different angles of refraction, a phenomenon known as dispersion.
Geocoder (Ruby) is a geocoding library for Ruby. Geocoding helps to enhance webpages by presenting location relevant information to the user. When used with Rails, Geocoder adds geocoding functionality such as finding coordinates with street addresses or vice versa in addition to distance calculations for ActiveRecord objects. Since the functionality does not rely on proprietary database functions, finding different geocoded objects in an area works out-of-the-box for databases like MySQL, PostgreSQL and SQLite.
Electromagnetic radiation of wavelengths other than those of visible light were discovered in the early 19th century. The discovery of infrared radiation is ascribed to astronomer William Herschel, who published his results in 1800 before the Royal Society of London. Herschel used a glass prism to refract light from the Sun and detected invisible rays that caused heating beyond the red part of the spectrum, through an increase in the temperature recorded with a thermometer. These "calorific rays" were later termed infrared.
An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted , and to the angle between two vectors and by means of the formula
Measurement of light or photometry is generally concerned with the amount of useful light falling on a surface and the amount of light emerging from a lamp or other source, along with the colors that can be rendered by this light. The human eye responds differently to light from different parts of the visible spectrum, therefore photometric measurements must take the luminosity function into account when measuring the amount of useful light. The basic SI unit of measurement is the candela (cd), which describes the luminous intensity, all other photometric units are derived from the candela. Luminance for instance is a measure of the density of luminous intensity in a given direction. It describes the amount of light that passes through or is emitted from a particular area, and falls within a given solid angle. The SI unit for luminance is candela per square metre (cd/m). The CGS unit of luminance is the stilb, which is equal to one candela per square centimetre or 10 kcd/m. The amount of useful light emitted from a source or the luminous flux is measured in lumen (lm).
Light is measured with two main alternative sets of units: radiometry consists of measurements of light power at all wavelengths, while photometry measures light with wavelength weighted with respect to a standardised model of human brightness perception. Photometry is useful, for example, to quantify Illumination (lighting) intended for human use. The SI units for both systems are summarised in the following tables.
Red is the color at the end of the visible spectrum of light, next to orange and opposite violet. It has a dominant wavelength of approximately 625–740 nanometres. It is a primary color in the RGB color model and the CMYK color model, and is the complementary color of cyan. Reds range from the brilliant yellow-tinged scarlet and vermillion to bluish-red crimson, and vary in shade from the pale red pink to the dark red burgundy. The red sky at sunset results from Rayleigh scattering, while the red color of the Grand Canyon and other geological features is caused by hematite or red ochre, both forms of iron oxide. Iron oxide also gives the red color to the planet Mars. The red colour of blood comes from protein hemoglobin, while ripe strawberries, red apples and reddish autumn leaves are colored by anthocyanins.
Among irrational numbers are the ratio  of a circle's circumference to its diameter, Euler's number "e", the golden ratio "φ", and the square root of two; in fact all square roots of natural numbers, other than of perfect squares, are irrational.
The following example demonstrates two facts (describing what "father" is) and one rule (describing the "meaning" of "sibling"). These three sentences use symbols (father, zeus, hermes, perseus and sibling) and some abstract variables (X, Y and Z). The "mother" relationship has been omitted for clarity.
In Ruby, symbols can be created with a literal form, or by converting a string.
They can be used as an identifier or an interned string. Two symbols with the same contents will always refer to the same object.
It is considered a best practice to use symbols as keys to an associative array in Ruby.
Rally is a form of motorsport that takes place on public or private roads with modified production or specially built road-legal cars. It is distinguished by running not on a circuit, but instead in a point-to-point format in which participants and their co-drivers drive between set control points (special stages), leaving at regular intervals from one or more start points. Rallies may be won by pure speed within the stages or alternatively by driving to a predetermined ideal journey time within the stages.
Given distinct points "A" and "B", they determine a unique ray with initial point "A". As two points define a unique line, this ray consists of all the points between "A" and "B" (including "A" and "B") and all the points "C" on the line through "A" and "B" such that "B" is between "A" and "C". This is, at times, also expressed as the set of all points "C" such that "A" is not between "B" and "C". A point "D", on the line determined by "A" and "B" but not in the ray with initial point "A" determined by "B", will determine another ray with initial point "A". With respect to the "AB" ray, the "AD" ray is called the "opposite ray".
Thus, we would say that two different points, "A" and "B", define a line and a decomposition of this line into the disjoint union of an open segment and two rays, "BC" and "AD" (the point "D" is not drawn in the diagram, but is to the left of "A" on the line "AB"). These are not opposite rays since they have different initial points.
In ray casting the geometry which has been modeled is parsed pixel by pixel, line by line, from the point of view outward, as if casting rays out from the point of view. Where an object is intersected, the color value at the point may be evaluated using several methods. In the simplest, the color value of the object at the point of intersection becomes the value of that pixel. The color may be determined from a texture-map. A more sophisticated method is to modify the colour value by an illumination factor, but without calculating the relationship to a simulated light source. To reduce artifacts, a number of rays in slightly different directions may be averaged.
In topology, a ray in a space "X" is a continuous embedding R → "X". It is used to define the important concept of end of the space.
The "ray" A/B (read as "the ray from A away from B") is the set of points P such that [PAB].