angle between two planes
Let π1 and π2 be two planes in the three-dimensional Euclidean space ℝ3. The angle θ between these planes is defined by means of the normal vectors 𝒏1 and 𝒏2 of π1 and π2 through the relationship
cosθ=|⟨𝒏1,𝒏2⟩∥𝒏1∥∥𝒏2∥|, |
where the numerator is the inner product of 𝒏1 and 𝒏2 and the denominator is product of the lengths of 𝒏1 and 𝒏2. The formula implies that the angle θ satisfies
0≤θ≤π2. |
The quotient in the formula remains unchanged as one multiplies the normal vectors by some non-zero real numbers, so that the cosine is independent of the lengths of the chosen vectors. Therefore, there is no ambiguity in this definition.
Generalization. The above definition can be generalized, at least locally, to a pair of intersecting differentiable surfaces in ℝ3. Given two differentiable surfaces S1 and S2 and a point p∈S1∩S2, the angle between S1 and S2 at p is defined to be the angle between the tangent planes
Tp(S1) and Tp(S2).
Title | angle between two planes |
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Canonical name | AngleBetweenTwoPlanes |
Date of creation | 2013-03-22 16:19:41 |
Last modified on | 2013-03-22 16:19:41 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 9 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 51N20 |
Synonym | angle between planes |
Related topic | AngleBetweenTwoLines |
Related topic | AngleBetweenLineAndPlane |
Defines | angle between differentiable surfaces |