Types of angles |
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2D angles |
Spherical |
2D angle pairs |
Adjacent |
3D angles |
Solid |
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the union of a line and two half-planes that have this line as a common edge. In higher dimensions, a dihedral angle represents the angle between two hyperplanes. The planes of a flying machine are said to be at positive dihedral angle when both starboard and port main planes (commonly called "wings") are upwardly inclined to the lateral axis; when downwardly inclined they are said to be at a negative dihedral angle.
When the two intersecting planes are described in terms of Cartesian coordinates by the two equations
the dihedral angle, between them is given by:
and satisfies It can easily be observed that the angle is independent of and .
Alternatively, if nA and nB are normal vector to the planes, one has
where nA · nB is the dot product of the vectors and |nA||nB| is the product of their lengths. [1]
The absolute value is required in above formulas, as the planes are not changed when changing all coefficient signs in one equation, or replacing one normal vector by its opposite.
However the absolute values can be and should be avoided when considering the dihedral angle of two half planes whose boundaries are the same line. In this case, the half planes can be described by a point P of their intersection, and three vectors b0, b1 and b2 such that P + b0, P + b1 and P + b2 belong respectively to the intersection line, the first half plane, and the second half plane. The dihedral angle of these two half planes is defined by
and satisfies In this case, switching the two half-planes gives the same result, and so does replacing with In chemistry (see below), we define a dihedral angle such that replacing with changes the sign of the angle, which can be between −π and π.
In some scientific areas such as polymer physics, one may consider a chain of points and links between consecutive points. If the points are sequentially numbered and located at positions r1, r2, r3, etc. then bond vectors are defined by u1=r2−r1, u2=r3−r2, and ui=ri+1−ri, more generally. [2] This is the case for kinematic chains or amino acids in a protein structure. In these cases, one is often interested in the half-planes defined by three consecutive points, and the dihedral angle between two consecutive such half-planes. If u1, u2 and u3 are three consecutive bond vectors, the intersection of the half-planes is oriented, which allows defining a dihedral angle that belongs to the interval (−π, π]. This dihedral angle is defined by [3]
or, using the function atan2,
This dihedral angle does not depend on the orientation of the chain (order in which the point are considered) — reversing this ordering consists of replacing each vector by its opposite vector, and exchanging the indices 1 and 3. Both operations do not change the cosine, but change the sign of the sine. Thus, together, they do not change the angle.
A simpler formula for the same dihedral angle is the following (the proof is given below)
or equivalently,
This can be deduced from previous formulas by using the vector quadruple product formula, and the fact that a scalar triple product is zero if it contains twice the same vector:
Given the definition of the cross product, this means that is the angle in the clockwise direction of the fourth atom compared to the first atom, while looking down the axis from the second atom to the third. Special cases (one may say the usual cases) are , and , which are called the trans, gauche+, and gauche− conformations.
Configuration names according to dihedral angle | synn- Butane in the gauche− conformation (−60°) Newman projection | synn- Butane sawhorse projection |
In stereochemistry, a torsion angle is defined as a particular example of a dihedral angle, describing the geometric relation of two parts of a molecule joined by a chemical bond. [4] [5] Every set of three non-colinear atoms of a molecule defines a half-plane. As explained above, when two such half-planes intersect (i.e., a set of four consecutively-bonded atoms), the angle between them is a dihedral angle. Dihedral angles are used to specify the molecular conformation. [6] Stereochemical arrangements corresponding to angles between 0° and ±90° are called syn (s), those corresponding to angles between ±90° and 180° anti (a). Similarly, arrangements corresponding to angles between 30° and 150° or between −30° and −150° are called clinal (c) and those between 0° and ±30° or ±150° and 180° are called periplanar (p).
The two types of terms can be combined so as to define four ranges of angle; 0° to ±30° synperiplanar (sp); 30° to 90° and −30° to −90° synclinal (sc); 90° to 150° and −90° to −150° anticlinal (ac); ±150° to 180° antiperiplanar (ap). The synperiplanar conformation is also known as the syn- or cis-conformation; antiperiplanar as anti or trans; and synclinal as gauche or skew.
For example, with n-butane two planes can be specified in terms of the two central carbon atoms and either of the methyl carbon atoms. The syn-conformation shown above, with a dihedral angle of 60° is less stable than the anti-conformation with a dihedral angle of 180°.
For macromolecular usage the symbols T, C, G+, G−, A+ and A− are recommended (ap, sp, +sc, −sc, +ac and −ac respectively).
A Ramachandran plot (also known as a Ramachandran diagram or a [φ,ψ] plot), originally developed in 1963 by G. N. Ramachandran, C. Ramakrishnan, and V. Sasisekharan, [7] is a way to visualize energetically allowed regions for backbone dihedral angles ψ against φ of amino acid residues in protein structure. In a protein chain three dihedral angles are defined:
The figure at right illustrates the location of each of these angles (but it does not show correctly the way they are defined). [8]
The planarity of the peptide bond usually restricts ω to be 180° (the typical trans case) or 0° (the rare cis case). The distance between the Cα atoms in the trans and cis isomers is approximately 3.8 and 2.9 Å, respectively. The vast majority of the peptide bonds in proteins are trans, though the peptide bond to the nitrogen of proline has an increased prevalence of cis compared to other amino-acid pairs. [9]
The side chain dihedral angles are designated with χn (chi-n). [10] They tend to cluster near 180°, 60°, and −60°, which are called the trans, gauche−, and gauche+ conformations. The stability of certain sidechain dihedral angles is affected by the values φ and ψ. [11] For instance, there are direct steric interactions between the Cγ of the side chain in the gauche+ rotamer and the backbone nitrogen of the next residue when ψ is near -60°. [12] This is evident from statistical distributions in backbone-dependent rotamer libraries.
Every polyhedron has a dihedral angle at every edge describing the relationship of the two faces that share that edge. This dihedral angle, also called the face angle, is measured as the internal angle with respect to the polyhedron. An angle of 0° means the face normal vectors are antiparallel and the faces overlap each other, which implies that it is part of a degenerate polyhedron. An angle of 180° means the faces are parallel, as in a tiling. An angle greater than 180° exists on concave portions of a polyhedron.
Every dihedral angle in an edge-transitive polyhedron has the same value. This includes the 5 Platonic solids, the 13 Catalan solids, the 4 Kepler–Poinsot polyhedra, the two quasiregular solids, and two quasiregular dual solids.
Given 3 faces of a polyhedron which meet at a common vertex P and have edges AP, BP and CP, the cosine of the dihedral angle between the faces containing APC and BPC is: [13]
This can be deduced from the spherical law of cosines, but can also be found by other means [14] .
In physics, the cross section is a measure of the probability that a specific process will take place in a collision of two particles. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians.
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a given point in space is specified by three real numbers: the radial distancer along the radial line connecting the point to the fixed point of origin; the polar angleθ between the radial line and a given polar axis; and the azimuthal angleφ as the angle of rotation of the radial line around the polar axis. (See graphic regarding the "physics convention".) Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle) is called the reference plane (sometimes fundamental plane).
In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra.
In mathematics, an n-sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional, and the sphere 2-dimensional, because the surfaces themselves are 1- and 2-dimensional respectively, not because they exist as shapes in 1- and 2-dimensional space. As such, the -sphere is the setting for -dimensional spherical geometry.
The Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).
A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Common forms of the Fourier series below.
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.
In the mathematical field of differential geometry, a metric tensor is an additional structure on a manifold M that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point p of M is a bilinear form defined on the tangent space at p, and a metric field on M consists of a metric tensor at each point p of M that varies smoothly with p.
Particle velocity is the velocity of a particle in a medium as it transmits a wave. The SI unit of particle velocity is the metre per second (m/s). In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse wave as with the vibration of a taut string.
The reciprocal lattice is a term associated with solids with translational symmetry, and plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid. It emerges from the Fourier transform of the lattice associated with the arrangement of the atoms. The direct lattice or real lattice is a periodic function in physical space, such as a crystal system. The reciprocal lattice exists in the mathematical space of spatial frequencies, known as reciprocal space or k space, which is the dual of physical space considered as a vector space, and the reciprocal lattice is the sublattice of that space that is dual to the direct lattice.
In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z).
A parametric surface is a surface in the Euclidean space which is defined by a parametric equation with two parameters . Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space rather than just the real line.
In crystallography and solid state physics, the Laue equations relate incoming waves to outgoing waves in the process of elastic scattering, where the photon energy or light temporal frequency does not change upon scattering by a crystal lattice. They are named after physicist Max von Laue (1879–1960).
In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.
In spherical astronomy, the parallactic angle is the angle between the great circle through a celestial object and the zenith, and the hour circle of the object. It is usually denoted q. In the triangle zenith—object—celestial pole, the parallactic angle will be the position angle of the zenith at the celestial object. Despite its name, this angle is unrelated with parallax. The parallactic angle is zero or 180° when the object crosses the meridian.
In classical mechanics, the central-force problem is to determine the motion of a particle in a single central potential field. A central force is a force that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center. In a few important cases, the problem can be solved analytically, i.e., in terms of well-studied functions such as trigonometric functions.
In physics, a sinusoidal plane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane. It is also called a monochromatic plane wave, with constant frequency.