Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Dihedral angle
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
==Mathematical background== When the two intersecting planes are described in terms of [[Cartesian coordinates]] by the two equations :<math> a_1 x + b_1 y + c_1 z + d_1 = 0 </math> :<math>a_2 x + b_2 y + c_2 z + d_2 = 0 </math> the dihedral angle, <math>\varphi</math> between them is given by: :<math>\cos \varphi = \frac{\left\vert a_1 a_2 + b_1 b_2 + c_1 c_2 \right\vert}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}</math> and satisfies <math>0\le \varphi \le \pi/2.</math> It can easily be observed that the angle is independent of <math>d_1</math> and <math>d_2 </math>. Alternatively, if {{math|'''n'''<sub>A</sub>}} and {{math|'''n'''<sub>B</sub>}} are [[normal vector]] to the planes, one has :<math>\cos \varphi = \frac{ \left\vert\mathbf{n}_\mathrm{A} \cdot \mathbf{n}_\mathrm{B}\right\vert}{|\mathbf{n}_\mathrm{A} | |\mathbf{n}_\mathrm{B}|}</math> where {{math|'''n'''<sub>A</sub> Β· '''n'''<sub>B</sub>}} is the [[dot product]] of the vectors and {{math|{{abs|'''n'''<sub>A</sub>}} {{abs|'''n'''<sub>B</sub>}}}} is the product of their lengths.<ref>{{cite web |title=Angle Between Two Planes |url=https://math.tutorvista.com/geometry/angle-between-two-planes.html |website=TutorVista.com |access-date=2018-07-06 |archive-date=2020-10-28 |archive-url=https://web.archive.org/web/20201028133313/https://math.tutorvista.com/geometry/angle-between-two-planes.html |url-status=dead }}</ref> <!-- [[File:Spherical bond dihedral angle.png|thumb|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 '''b'''<sub>1</sub> and '''b'''<sub>2</sub> and through '''b'''<sub>2</sub> and '''b'''<sub>3</sub>, respectively.]] --> 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 value]]s can be and should be avoided when considering the dihedral angle of two [[half plane]]s whose boundaries are the same line. In this case, the half planes can be described by a point {{mvar|P}} of their intersection, and three vectors {{math|'''b'''<sub>0</sub>}}, {{math|'''b'''<sub>1</sub>}} and {{math|'''b'''<sub>2</sub>}} such that {{math|''P'' + '''b'''<sub>0</sub>}}, {{math|''P'' + '''b'''<sub>1</sub>}} and {{math|''P'' + '''b'''<sub>2</sub>}} 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 :<math> \cos\varphi = \frac{ (\mathbf{b}_0 \times \mathbf{b}_1) \cdot (\mathbf{b}_0 \times \mathbf{b}_2)}{|\mathbf{b}_0 \times \mathbf{b}_1| |\mathbf{b}_0 \times \mathbf{b}_2|}</math>, and satisfies <math>0\le\varphi <\pi.</math> In this case, switching the two half-planes gives the same result, and so does replacing <math>\mathbf b_0</math> with <math>-\mathbf b_0.</math> In chemistry (see below), we define a dihedral angle such that replacing <math>\mathbf b_0</math> with <math>-\mathbf b_0</math> changes the sign of the angle, which can be between {{math|β{{pi}}}} and {{math|{{pi}}}}.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)