In aerospace engineering, the dihedral is the angle between the two wings; see dihedral. Aerospace engineering is the branch of Engineering behind the design construction and science of Aircraft and Spacecraft. Dihedral is the upward angle from horizontal of the wings or tailpane of a Fixed-wing aircraft or the wing of a Bird. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called Dihedral is the upward angle from horizontal of the wings or tailpane of a Fixed-wing aircraft or the wing of a Bird.

In geometry, the angle between two planes is called their dihedral or torsion angle. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called The term torsion may refer the following In geometry Torsion of curves Torsion tensor in differential geometry

Figure 1: Dihedral angle of three vectors, defined as an exterior spherical angle. The longer and shorter black segments are arcs of the great circles passing through and and through and , respectively.

Figure 2: Dihedral angle defined by three bond vectors (shown in red, green and blue) connecting four atoms.

Figure 3: Dihedral angle defined by three bond vectors (shown in red, green and blue) connecting four atoms. From this perspective, the second bond vector (green) is coming out of the page.

Figure 4: The backbone dihedral angles of proteins. Proteins are large Organic compounds made of Amino acids arranged in a linear chain and joined together by Peptide bonds between the Carboxyl

The dihedral angle of two planes can be seen by looking at the planes "edge on", i. e. , along their line of intersection. The dihedral angle φ_AB between two planes denoted A and B is the angle between their two normal unit vectors and

A dihedral angle can be signed; for example, the dihedral angle φ_AB can be defined as the angle through which plane A must be rotated (about their common line of intersection) to align it with plane B. A negative number is a Number that is less than zero, such as −2 A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation Thus, φ_AB = − φ_BA. For precision, one should specify the angle or its supplement, since both rotations will cause the planes to coincide. A pair of Angles is supplementary if their measurements add up to 180 degrees If the two supplementary angles are adjacent (i

Alternative definitions

Since a plane can be defined in several ways (e. g. , by vectors or points in them, or by their normal vectors), there are several equivalent definitions of a dihedral angle.

Any plane can be defined by two non-collinear vectors lying in that plane; taking their cross product and normalizing yields the normal vector to the plane. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which Thus, a dihedral angle can be defined by four, pairwise non-collinear vectors.

We may also define the dihedral angle of three non-collinear vectors , and (shown in red, green and blue, respectively, in Figure 1). The vectors and define the first plane, whereas and define the second plane. The dihedral angle corresponds to an exterior spherical angle (Figure 1), which is a well-defined, signed quantity. A spherical angle is a particular Dihedral angle; it is the angle between two intersecting arcs on a Sphere, and is measured by the angle between the planes containing

where the two-argument atan2 takes care of the sign. In Computing, atan2 is a two-argument function that makes it easy to find the angle round the origin of a point

Dihedral angles in polyhedra

Every polyhedron, regular and irregular, convex and concave, has a dihedral angle at every edge. What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions

A dihedral angle (also called the face angle) is the internal angle at which two adjacent faces meet. An angle of zero degrees means the face normal vectors are antiparallel and the faces overlap each other (Implying part of a degenerate polyhedron). An angle of 180 degrees means the faces are parallel (like a tiling). This table shows the 11 convex uniform tilings of the Euclidean plane, and their dual tilings 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 4 Kepler-Poinsot solids, the two quasiregular solids, and two quasiregular dual solids. In Geometry, a Platonic solid is a convex Regular polyhedron. The Kepler-Poinsot polyhedra is a popular name for the regular star polyhedra.

See Table of polyhedron dihedral angles. The Dihedral angles for the Edge-transitive polyhedra are

Dihedral angles of four atoms

To a good approximation, the bond lengths and bond angles of most molecules do not change between synthesis and degradation. Hence, the structure of a molecule can be defined with high precision by the dihedral angles between three successive chemical bond vectors (Figure 2). The dihedral angle φ varies only the distance between the first and fourth atoms; the other interatomic distances are constrained by the chemical bond lengths and bond angles.

To visualize the dihedral angle of four atoms, it's helpful to look down the second bond vector (Figure 3). The first atom is at 6 o'clock, the fourth atom is at roughly 2 o'clock and the second and third atoms are located in the center. The second bond vector is coming out of the page. The dihedral angle φ is the counterclockwise angle made by the vectors (red) and (blue). When the fourth atom eclipses the first atom, the dihedral angle is zero; when the atoms are exactly opposite (as in Figure 2), the dihedral angle is 180°.

Dihedral angles of biological molecules

The backbone dihedral angles of proteins are called φ (phi, involving the backbone atoms C'-N-C^α-C'), ψ (psi, involving the backbone atoms N-C^α-C'-N) and ω (omega, involving the backbone atoms C^α-C'-N-C^α). Proteins are large Organic compounds made of Amino acids arranged in a linear chain and joined together by Peptide bonds between the Carboxyl Thus, φ controls the C'-C' distance, ψ controls the N-N distance and ω controls the C^α-C^α distance.

The planarity of the peptide bond usually restricts ω to be 180° (the typical trans case) or 0° (the rare cis case). A peptide bond is a Chemical bond formed between two Molecules when the Carboxyl group of one molecule reacts with the The distance between the C^α atoms in the trans and cis isomers is approximately 3. Trans-2-butenesvg|right|thumb|Trans-2-butene]] In Chemistry, cis-trans isomerism or geometric isomerism or configuration isomerism is a form of 8 and 2. 9 Å, respectively. The cis isomer is mainly observed in Xaa-Pro peptide bonds (where Xaa is any amino acid). Proline (abbreviated as Pro or P) is an α- Amino acid, one of the twenty DNA -encoded amino acids A peptide bond is a Chemical bond formed between two Molecules when the Carboxyl group of one molecule reacts with the In Chemistry, an amino acid is a Molecule containing both Amine and Carboxyl Functional groups In Biochemistry, this

The sidechain dihedral angles of proteins are denoted as χ₁-χ₅, depending on the distance up the sidechain. Proteins are large Organic compounds made of Amino acids arranged in a linear chain and joined together by Peptide bonds between the Carboxyl The χ₁ dihedral angle is defined by atoms N-C^α-C^β-C^γ, the χ₂ dihedral angle is defined by atoms C^α-C^β-C^γ-C^δ, and so on.

The sidechain dihedral angles tend to cluster near 180°, 60°, and -60°, which are called the trans, gauche⁺, and gauche^- conformations. The choice of sidechain dihedral angles is affected by the neighbouring backbone and sidechain dihedrals; for example, the gauche⁺ conformation is rarely followed by the gauche⁺ conformation (and vice versa) because of the increased likelihood of atomic collisions.

Dihedral angles have also been defined by the IUPAC for other molecules, such as the nucleic acids (DNA and RNA) and for polysaccharides. The International Union of Pure and Applied Chemistry ( IUPAC) (aɪjuːpæk or ay-yoo-pec) is an international Non-governmental organization A nucleic acid is a Macromolecule composed of chains of monomeric Nucleotides In Biochemistry these Molecules carry Genetic information Deoxyribonucleic acid ( DNA) is a Nucleic acid that contains the genetic instructions used in the development and functioning of all known Ribonucleic acid ( RNA) is a Nucleic acid that consists of a long chain of Nucleotide units Polysaccharides are relatively complex Carbohydrates They are Polymers made up of many Monosaccharides joined together by Glycosidic bonds

See also

• Ramachandran plot
• Flory convention

External links

• Analysis of the 5 Regular Polyhedra gives a step-by-step derivation of these exact values. A Ramachandran plot (also known as a Ramachandran map or a Ramachandran diagram) developed by Gopalasamudram Narayana Ramachandran, is a way to visualize The Flory convention for defining the variables involved on modeling the position vectors of atoms in macromolecules it is often necessary to convert from Cartesian coordinates
• Olshevsky, George, Dihedral angle at Glossary for Hyperspace. George Olshevsky (born 1946 is a freelance editor, Writer, Publisher, Paleontologist, and Mathematician living in San Diego