Math 2.0

Differential geometry Definition

Tangent plane and normal vector of a surface

Builds on Parametrised surfacePartial derivativeGradient

Definition

Let r ⁣:D→ℝ3r \colon D \to \mathbb{R}^3 be a parametrised surface that is regular at the point (u0,v0)(u_0, v_0) of DD, and let p=r(u0,v0)p = r(u_0, v_0). The tangent plane at pp is the plane {p+s ru(u0,v0)+t rv(u0,v0):s,t∈ℝ}\{ p + s \, r_u(u_0, v_0) + t \, r_v(u_0, v_0) : s, t \in \mathbb{R} \}. The vector N(u0,v0)=ru(u0,v0)×rv(u0,v0)N(u_0, v_0) = r_u(u_0, v_0) \times r_v(u_0, v_0) is the normal vector of rr at (u0,v0)(u_0, v_0), and n=N/∥N∥n = N / \|N\| is the unit normal vector.

Remarks

The vector NN is perpendicular to rur_u and rvr_v, so it is perpendicular to the tangent plane. The other unit normal at pp is −n-n. For the graph z=f(x,y)z = f(x, y), the normal is (−fx,−fy,1)(-f_x, -f_y, 1), which points upward. If the surface is the level set g(x,y,z)=cg(x, y, z) = c of a C1C^1 function gg with ∇g(p)≠0\nabla g(p) \ne 0, then the gradient ∇g(p)\nabla g(p) is a normal vector at pp. For the unit sphere with r(φ,θ)=(sin⁡φcos⁡θ,sin⁡φsin⁡θ,cos⁡φ)r(\varphi, \theta) = (\sin \varphi \cos \theta, \sin \varphi \sin \theta, \cos \varphi), one gets rφ×rθ=sin⁡φ r(φ,θ)r_\varphi \times r_\theta = \sin \varphi \, r(\varphi, \theta), which points outward for 0<φ<π0 < \varphi < \pi.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionGradient, definition, no evidence levelGradientdefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementParametrised curve, definition, no evidence levelParametrised curvedefinitionParametrised surface, definition, no evidence levelParametrised surfacedefinitionTangent plane and normal vector of a surface, definition, no evidence levelTangent plane and nor…definition
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  1. Euclidean space object
  2. Function concept
  3. Fréchet derivative definition
  4. Function of several variables concept
  5. Vector-valued function concept
  6. Limit of a function of several variables concept
  7. Partial derivative concept
  8. Continuity in R^n definition
  9. Differentiable function of several variables definition
  10. Partial derivative definition
  11. Gradient definition
  12. Differentiable on an open set definition
  13. Higher partial derivatives definition
  14. Iterated derivative definition
  15. C^k function characterised by continuous derivatives statement
  16. Parametrised curve definition
  17. Parametrised surface definition
  18. Tangent plane and normal vector of a surface definition

Where this leads

34 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Parametric_surface, Tangent plane and normal vector
  2. Web https://en.wikipedia.org/wiki/Normal_(geometry), Normal to general surfaces in 3D space
  3. Web https://en.wikipedia.org/wiki/Normal_(geometry), Orientation
About this page
Tag0BJDQ2HBH1
URIm2://0BJDQ2HBH1
Kinddefinition
WritingW0 stub
MSC53A05
DefinesTangent plane and normal vector of a surface
UsesParametrised surface, Partial derivative, Gradient

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