Ingeometry, anormal is anobject (e.g. aline,ray, orvector) that isperpendicular to a given object. For example, thenormal line to aplane curve at a givenpoint is the infinite straight line perpendicular to thetangent line to the curve at the point.
Anormal vector is avector perpendicular to a given object at a particular point.A normalvector of length one is called aunit normal vector ornormal direction. Acurvature vector is a normal vector whose length is thecurvature of the object. Multiplying a normal vector by−1 results in theopposite vector, which may be used for indicating sides (e.g., interior or exterior) or orientation (e.g., clockwise vs. counterclockwise,right handed vs. left handed).
Inthree-dimensional space, asurface normal, or simplynormal, to asurface at pointP is a vector perpendicular to thetangent plane of the surface atP. Thevector field of normal directions to a surface is known asGauss map. The word "normal" is also used as an adjective: a linenormal to aplane, thenormal component of aforce, etc. The concept of normality generalizes toorthogonality (right angles).
The concept has been generalized todifferentiable manifolds of arbitrary dimension embedded in aEuclidean space. Thenormal vector space ornormal space of a manifold at point is the set of vectors which are orthogonal to thetangent space atNormal vectors are of special interest in the case ofsmooth curves andsmooth surfaces.
The normal is often used in3D computer graphics (notice the singular, as only one normal will be defined) to determine a surface's orientation toward alight source forflat shading, or the orientation of each of the surface's corners (vertices) to mimic a curved surface withPhong shading.
Thefoot of a normal at a point of interestQ (analogous to thefoot of a perpendicular) can be defined at the pointP on the surface where the normal vector containsQ.Thenormal distance of a pointQ to a curve or to a surface is theEuclidean distance betweenQ and its footP.
The normal direction to aspace curve is:
where is theradius of curvature (reciprocalcurvature); is thetangent vector, in terms of the curve position and arc-length:
For aconvexpolygon (such as atriangle), a surface normal can be calculated as the vectorcross product of two (non-parallel) edges of the polygon.
For aplane given by the general formplane equation the vector is a normal.
For a plane whose equation is given in parametric formwhere is a point on the plane and are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both and which can be found as thecross product
If a (possibly non-flat) surface in 3D space isparameterized by a system ofcurvilinear coordinates with andreal variables, then a normal toS is by definition a normal to a tangent plane, given by the cross product of thepartial derivatives
If a surface is givenimplicitly as the set of points satisfying then a normal at a point on the surface is given by thegradientsincethe gradient at any point is perpendicular to the level set
For a surface in given as the graph of a function an upward-pointing normal can be found either from the parametrization givingor more simply from its implicit form giving Since a surface does not have a tangent plane at asingular point, it has no well-defined normal at that point: for example, the vertex of acone. In general, it is possible to define a normal almost everywhere for a surface that isLipschitz continuous.
The normal to a (hyper)surface is usually scaled to haveunit length, but it does not have a unique direction, since its opposite is also a unit normal. For a surface which is thetopological boundary of a set in three dimensions, one can distinguish between twonormal orientations, theinward-pointing normal andouter-pointing normal. For anoriented surface, the normal is usually determined by theright-hand rule or its analog in higher dimensions.
If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is apseudovector.
When applying a transform to a surface it is often useful to derive normals for the resulting surface from the original normals.
Specifically, given a 3×3 transformation matrix we can determine the matrix that transforms a vector perpendicular to the tangent plane into a vector perpendicular to the transformed tangent plane by the following logic:
Writen′ as We must find
Choosing such that or will satisfy the above equation, giving a perpendicular to or an perpendicular to as required.
Therefore, one should use the inverse transpose of the linear transformation when transforming surface normals. The inverse transpose is equal to the original matrix if the matrix is orthonormal, that is, purely rotational with no scaling or shearing.
For an-dimensionalhyperplane in-dimensional space given by its parametric representationwhere is a point on the hyperplane and for are linearly independent vectors pointing along the hyperplane, a normal to the hyperplane is any vector in thenull space of the matrix meaning. That is, any vector orthogonal to all in-plane vectors is by definition a surface normal. Alternatively, if the hyperplane is defined as the solution set of a single linear equation, then the vector is a normal.
The definition of a normal to a surface in three-dimensional space can be extended to-dimensionalhypersurfaces in. A hypersurface may belocally defined implicitly as the set of points satisfying an equation, where is a givenscalar function. If iscontinuously differentiable then the hypersurface is adifferentiable manifold in theneighbourhood of the points where thegradient is not zero. At these points a normal vector is given by the gradient:
Thenormal line is the one-dimensional subspace with basis
A vector that is normal to the space spanned by the linearly independent vectorsv1, ...,vr−1 and falls within ther-dimensional space spanned by the linearly independent vectorsv1, ...,vr is given by ther-th column of the matrixΛ =V(VTV)−1, where the matrixV = (v1, ...,vr) is the juxtaposition of ther column vectors. (Proof:Λ isV times a matrix so each column ofΛ is a linear combination of the columns ofV. Furthermore,VTΛ =I, so each column ofV other than the last is perpendicular to the last column ofΛ.) This formula works even whenr is less than the dimension of the Euclideanspacen. The formula simplifies toΛ = (VT)−1 whenr =n.
Adifferential variety defined by implicit equations in the-dimensional space is the set of the common zeros of a finite set of differentiable functions in variablesTheJacobian matrix of the variety is the matrix whose-th row is the gradient of By theimplicit function theorem, the variety is amanifold in the neighborhood of a point where the Jacobian matrix has rank At such a point thenormal vector space is the vector space generated by the values at of the gradient vectors of the
In other words, a variety is defined as the intersection of hypersurfaces, and the normal vector space at a point is the vector space generated by the normal vectors of the hypersurfaces at the point.
Thenormal (affine) space at a point of the variety is theaffine subspace passing through and generated by the normal vector space at
These definitions may be extendedverbatim to the points where the variety is not a manifold.
LetV be the variety defined in the 3-dimensional space by the equationsThis variety is the union of the-axis and the-axis.
At a point where the rows of the Jacobian matrix are and Thus the normal affine space is the plane of equation Similarly, if thenormal plane at is the plane of equation
At the point the rows of the Jacobian matrix are and Thus the normal vector space and the normal affine space have dimension 1 and the normal affine space is the-axis.
Thenormal ray is the outward-pointing rayperpendicular to the surface of anoptical medium at a given point.[2] Inreflection of light, theangle of incidence and theangle of reflection are respectively the angle between the normal and theincident ray (on theplane of incidence) and the angle between the normal and thereflected ray.
