Talk:Dihedral angle

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The red arrows in the figures on this page are misleading. As vectors, the arrows should be on the opposite end. -- JackSnoeyink 04:00, 18 April 2007 (UTC)

Problems with first algorithm[edit]

As well as being longer and more complex than the method given below it the algorithm can fail with a divide by zero error, if the 'arbitrary vector' is such that either Va or Vb is zero. The way to avoid that is put the whole thing in a while loop until the product of their magnitudes is non-zero, but that would make it even less clear. And even if not exactly zero it can be very inaccurate for vectors close to zero, due rounding errors. I.e. it's a generally poor algorithm.

Any objections to removing it and tidying up the cross product approach ?--JohnBlackburnewordsdeeds 10:54, 6 March 2010 (UTC)

This seems to be the sort of thing that would be better explained in plain words (preferably with a picture) than with code. At any rate, it is distilled in the observation that in order to find a normal vector, one can take a basis of tangent vectors to the plane, and apply Gram-Schmidt to any extension of this basis to a basis of the Euclidean space. The relevant implementation details would be in the Gram-Schmidt algorithm itself, and the selection of the vector that completes the bases. This has the advantage over the cross product approach because it works in any dimension. The pseudocode itself is so problematic that it should be removed immediately. I was going to comment more about this, but I think the issues are sufficiently obvious and numerous that they will not escape detection. Sławomir Biały (talk) 12:19, 7 March 2010 (UTC)
I've gone ahead and taken a shot at writing out the more serious problems with the section. Please feel free to expand on this! Sławomir Biały (talk) 12:59, 7 March 2010 (UTC)

Problem with the Figure 4: The backbone dihedral angles of proteins.[edit]

Looks like the Phi dihedral angle is drawn incorrectly (it should be counted in the opposite direction). — Preceding unsigned comment added by (talk) 10:30, 1 February 2013 (UTC)

atan2 formula does not give the correct sign in some cases[edit]

The formula

\varphi = \operatorname{atan2} \left( |[\mathbf{b}_1 \times \mathbf{b}_2]\times [\mathbf{b}_2 \times \mathbf{b}_3]|, [\mathbf{b}_1 \times \mathbf{b}_2] \cdot [\mathbf{b}_2 \times \mathbf{b}_3] \right),

does not give correct results because the first argument is always positive. Using this formula, \varphi is always positive. I believe that the correct formula is

\varphi = \operatorname{atan2} \left( \left([\mathbf{b}_1 \times \mathbf{b}_2]\times [\mathbf{b}_2 \times \mathbf{b}_3]\right) \cdot \frac{\mathbf{b}_2}{|\mathbf{b}_2|}, [\mathbf{b}_1 \times \mathbf{b}_2] \cdot [\mathbf{b}_2 \times \mathbf{b}_3] \right).

As a citation, I propose formulae (3) and (4) in Blondel and Karplus, 1995 (DOI: 10.1002/(SICI)1096-987X(19960715)17:9<1132::AID-JCC5>3.0.CO;2-T). The above formula is a slight manipulation of the sin and cos formulae cited. — Preceding unsigned comment added by John Jumper (talkcontribs) 03:52, 1 November 2013 (UTC)

I'm not sure. A dihedral angle, defined as the angle between two planes should be between 0 and 180 degrees, so atan2(abs(y),x) should work fine. I don't know under what conditions you want the answer as >180. Perhaps the intro should say the angle is between 0 and 180? Tom Ruen (talk) 06:56, 2 November 2013 (UTC)
If you define the dihedral angle between two planes as the angle you need to rotate the first plane so the normals are parallel and unidirectional, then the angle might be more than 180 (or, more elegantly, between -180 and 180). This is exactly the definition used in biochemistry to report covalent bond torsion angles. --RasF (talk) 12:02, 8 November 2013 (UTC)
I apologize for the late reply. As used in biochemistry, dihedral angle really refers to the angle between the vectors r2->r1 and r3->r4, after both vectors have been projected onto the plane whose normal is given by the vector r2->r3. Since this angle is in the plane, it is defined on [-180,180]. As an example of the use of this definition, see Ramachandran plot, noting that both axes on the figures are on [-180,180]. From a more mathematical perspective, the dihedral angle defined on (-180,180) is a continuous function of the 4 points, so long as 3 consecutive points are not collinear. Reversing the order of the points negates the dihedral angle. At the very least, I would note the difference in convention (I started this edit because I copied this formula into a piece of code where I expected the biochemistry definition and wasted an hour tracking down the bug). That the biochemist follow the signed dihedral convention is documented by the Blondel and Karplus paper from 1995. John Jumper (talk) 08:56, 20 November 2013 (UTC)
I also just implemented a small script with a Computational Chemistry background for which I needed the dihedral angle. And I can confirm that the current atan2()-formula did not work the way I needed. The replacement suggested here worked fine. (Tobias) 10:05, 25 November 2013 (UTC) — Preceding unsigned comment added by (talk)

Dihedral angles of the regular polytopes[edit]


  • {3,3}: 70°32′ = arcsec(3)
  • {3,4}: 109°28′ = π − arcsec(3)
  • {4,3}: 90° = π/2
  • {3,5}: 138°11′ = π − arcsin(2/3)
  • {5,3): 116°34′ = π − arctan(2)
  • {5/2,5}: 116°34′ = π − arctan(2)
  • {5,5/2}: 63°26′ = arctan(2)
  • {5/2,3}: 63°26′ = arctan(2)
  • {3,5/2}: 41°49′ = arcsin(2/3)


  • {3,3,3}: 75°31′ = arcsec(4)
  • {3,3,4}: 120° = 2π/3
  • {4,3,3}: 90° = π/2
  • {3,4,3}: 120° = 2π/3
  • {3,3,5}: 164°29′ = 4π/3 − arcsec(4)
  • {5,3,3}: 144° = 4π/5
  • {5/2,5,3}: 144° = 4π/5
  • {3,5,5/2}: 120° = 2π/3
  • {5,5/2,5}: 144° = 4π/5
  • {5/2,3,5}: 144° = 4π/5
  • {5,3,5/2}: 72° = 2π/5
  • {5/2,5,5/2}: 72° = 2π/5
  • {3,5/2,5}: 120° = 2π/3
  • {5,5/2,3}: 72° = 2π/5
  • {5/2,3,3}: 72° = 2π/5
  • {3,3,5/2}: 44°29′ = 2π/3 − arcsec(4)


  • αn (n-simplex): arcsec(n)
  • βn (n-orthoplex): π − 2 arccsc(√n)
  • γn (n-cube): 90° = π/2

Ref: Coxeter, Regular Polytopes (3rd ed.), pp.292–5. Double sharp (talk) 20:49, 16 March 2015 (UTC)

Mathematical accuracy[edit]

I have corrected the lead, which contained several errors. Firstly one must not confuse between angles of planes and angles of half planes. Secondly the sign of a dihedral angle cannot defined, even in the most constrained case of the originated angle of two half planes. Moreover, as a plane has two opposite normal vectors, the dot product of the unit normal vectors is only defined up to its sign. Therefore, one must take the absolute value of the dot product for getting the cosine of the dihedral angle.

The remainder of the article still requires to be corrected accordingly.

I have a doubt: I believe that "dihedral angle" generally refers to the angle of two half planes, and in the case of planes one simply talk of "angle of planes". This has to be checked in the literature. In any case, I have created Angle of planes that redirects here. D.Lazard (talk) 15:06, 14 October 2015 (UTC)

Thanks for your edit, (but I did edited a bit further.) I think it still needs much more. I did some checking. mathworld [1] thinks the main definition is a angle between planes, while James and James "mathematics dictionary" (1992) defines it as " the union of a line and the two half planes which have this line as a common edge " (a rather nice description, quite similar to the one of the 2d angle: an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle). I think in the lead there should be only one definition other definitions could go to to the section "alternative definitions" I will later edit it to the james and james version. (if you agree). I also do think it needs more editing (I moved image of the angle as angle between vectors down) but also it still needs a bit more. I did post an help at Wikipedia:Talk:WikiProject Mathematics#Dihedral angle (sorry did not manage to link correctly) maybe more would join us. WillemienH (talk) 16:20, 14 October 2015 (UTC)
Just to toss in my two cents worth. I see at least three different traditions merging in this article, and they are not always compatible. There is the traditional synthetic treatment which defines a dihedral angle as an angle determined by two (intersecting) planes. The half-plane description follows from this, but runs into the difficulty of deciding which half-planes are to be used. The more modern linear algebra/computational approach that D. Lazard is espousing starts more naturally with half-planes so that this ambiguity does not become an issue. Then there is the biochemical approach which is algorithmically based from what I can tell from the above remarks. One place where this makes a difference is in the measure of the dihedral angles. The restrictions that D. Lazard put in make no sense from the synthetic point of view, and are not the conventions used by the biochemists. As we go forward with this article I suggest that we carefully distinguish these viewpoints and back them up with good citations (and please, no dictionaries) from each of the areas. Bill Cherowitzo (talk) 04:06, 15 October 2015 (UTC)
Thanks for your two cents, I don't just copy dictionary definitions, I look at them (edit them where needed) and the one of James and James was just very nice. (Clear , general and succinct) In the lead I also would like to have the calculation of one Dihedral Angle in an irregular Tetrahedron given all its sides , but I only found not the succinct formula I would like to have and deducting it would be WP:OR. I would move the rest from the lead to other sections, so there is only one definition and one calculation in the lead
I could not follow the biochemical approach, to me it is not clear at all , is it really about the same angle (as described in James and james , if so what are the faces and edges? (I could think of something that the middle bond is part of the edge , and the other bonds are on the faces, but is this correct or just my interpretation? )WillemienH (talk) 06:59, 15 October 2015 (UTC)
I believe that the secret to understanding the biochemical approach is that the four atoms involved form an ordered set of points. The first three determine one half-plane and the last three determine the other. Then, using the two vectors determined by each triple they calculate the normal to the half-plane by taking the cross product of the vectors, which, due to the ordering of the points, has a single direction (essentially using the right hand rule to determine the direction). The dihedral angle is then the angle between the two normals, and because these have a direction, the angle can be assigned a sign. In the introduction we currently say that you can't assign a sign to a dihedral angle and this is true because you can't determine the direction of the normals used to calculate the angle (this needs to be better explained in the lead), but what the biochemists have done is to use the order of the points to pick a specific direction for each normal. Also, in their description, they refer to the angle between the projected vectors as being the dihedral angle, but this is the same angle as the angle between the normals of those projected lines, which are the normals of the half-planes.
The definition you've chosen more properly comes from analytic geometry. Solid geometry more often refers to a synthetic approach and in this case that would mean the angle determined by two planes. My objection to dictionary definitions of mathematical terms is that they are never written by specialists and are really only tertiary sources, not necessarily trustworthy. I just avoid them whenever I can. Bill Cherowitzo (talk) 18:17, 15 October 2015 (UTC)

Improper dihedral angle[edit]

I have commented out this section as it is, in my opinion, rubbish. The very name dihedral implies two planes. See polyhedron for related etymology. Original text:

An "improper" dihedral angle is a similar geometric analysis of four atoms, but typically involves a central atom with three others attached to it rather than the standard arrangement of all four of them bonded sequentially each to the next. One of the vectors is the bond from the central atom to one of its attachments. The other two vectors are pairs of the attachments, and thus together represent the plane of the attachments. Improper dihedral angles are useful for analyzing the planarity of the central atom: as the angle deviates from zero, the central atom moves out of the plane defined by the three attached to it.[1]

User:Petergans|Petergans]] (talk) 11:02, 3 November 2015 (UTC)

IUPAC definition[edit]

IUPAC definition of a torsion angle In the Compendium of Chemical Terminology (IUPAC Gold Book) from the International Union of Pure and Applied Chemistry (IUPAC) a torsion angle is defined as:

In a chain of atoms A-B-C-D, the dihedral angle between the plane containing the atoms A,B,C and that containing B,C,D. In a Newman projection the torsion angle is the angle (having an absolute value between 0° and 180°) between bonds to two specified (fiducial) groups, one from the atom nearer (proximal) to the observer and the other from the further (distal) atom.

The torsion angle between groups A and D is then considered to be positive if the bond A-B is rotated in a clockwise direction through less than 180° in order that it may eclipse the bond C-D: a negative torsion angle requires rotation in the opposite sense.

Stereochemical arrangements corresponding to torsion angles between 0° and ±90° are called syn (s), those corresponding to torsion angles between ±90° and 180° anti (a).

Similarly, arrangements corresponding to torsion 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).

syn/anti peri/clinal

The two types of terms can be combined so as to define four ranges of torsion 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 macromolecular usage the symbols T, C, G+, G-, A+ and A- are recommended (ap, sp, +sc, -sc, +ac and -ac respectively). [2]

I have removed the reinstatements (above) by User:WillemienH for two reasons:

Petergans (talk) 10:20, 5 November 2015 (UTC)

  1. ^ CHARMM parmfile.doc definition of "IMPH" energy parameter
  2. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "torsion angle".
Not sure what you mean by unsourced. There was a link to --> GoldBookRef|title=torsion angle|file=T06406| accessdate = 2015-10-25 <-- where it is comming from. Torsion angle does redirect to this page. My problem is more that the description dihedral angle#Dihedral angles of four atoms, the IUPAC definition and the dihedral angle#Angle between three vectors all differ subtely. While in general they will give similar results, directions and instructions differ.WillemienH (talk) 09:36, 6 November 2015 (UTC)
There is an issue of semantics. To me the name "dihedral" means the angle between two (gk. di) planes (gk. hedra). I think that the three vector approach was used historically to obtain the dihrdral angle between two adjaced faces of polyhedra. I'm not sure about it as I have not been able find a source for verification. If you can source it, please feel free to re-instate in the section on polyhedra. Petergans (talk) 12:05, 15 November 2015 (UTC)