Normal (geometry)

 ( Surfaces ) 

In geometry, a normal is an object (e.g. a line, ray, or Euclidean vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the infinite straight line perpendicular to the tangent line to the curve at the point.

A normal vector is a vector perpendicular to a given object at a particular point. A normal unit vector is called a unit normal vector or normal direction. A curvature vector is a normal vector whose length is the curvature of the object. Multiplying a normal vector by results in the opposite vector, which may be used for indicating sides (e.g., interior or exterior).

In three-dimensional space, a surface normal, or simply normal, to a surface at point is a vector perpendicular to the tangent plane of the surface at . The vector field of normal directions to a surface is known as Gauss map. The word "normal" is also used as an adjective: a line normal to a Euclidean plane, the normal component of a force, etc. The concept of normality generalizes to orthogonality ().

The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at point $P$ is the set of vectors which are orthogonal to the tangent space at $P.$ Normal vectors are of special interest in the case of smooth curves and smooth surfaces.

The normal is often used in 3D computer graphics (notice the singular, as only one normal will be defined) to determine a surface's orientation toward a light source for flat shading, or the orientation of each of the surface's corners (vertices) to mimic a curved surface with Phong shading.

The foot of a normal at a point of interest Q (analogous to the foot of a perpendicular) can be defined at the point P on the surface where the normal vector contains Q. The normal distance of a point Q to a curve or to a surface is the Euclidean distance between Q and its foot P.

---

Normal to space curves The normal direction to a space curve is:

$\backslash mathbf\{N\}\; =\; R\; \backslash frac\{\backslash mathrm\{d\}\; \backslash mathbf\{T\}\; \}\{\; \backslash mathrm\{d\}\; s\; \}$

where $R\; =\; \backslash kappa^\{-1\}$ is the radius of curvature (reciprocal curvature); $\backslash mathbf\{T\}$ is the tangent vector, in terms of the curve position $\backslash mathbf\{r\}$ and arc-length $s$:

$\backslash mathbf\{T\}\; =\; \backslash frac\{\backslash mathrm\{d\}\; \backslash mathbf\{r\}\; \}\{\; \backslash mathrm\{d\}\; s\; \}$

---

Normal to planes and polygons For a Convex set polygon (such as a triangle), a surface normal can be calculated as the vector cross product of two (non-parallel) edges of the polygon.

For a plane given by the general form plane equation $ax\; +\; by\; +\; cz\; +\; d\; =\; 0,$ the vector $\backslash mathbf\; n\; =\; (a,\; b,\; c)$ is a normal.

For a plane whose equation is given in parametric form $$\backslash mathbf\{r\}(s,t)\; =\; \backslash mathbf\{r\}\_0\; +\; s\; \backslash mathbf\{p\}\; +\; t\; \backslash mathbf\{q\},$$ where $\backslash mathbf\{r\}\_0$ is a point on the plane and $\backslash mathbf\{p\},\; \backslash mathbf\{q\}$ are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both $\backslash mathbf\{p\}$ and $\backslash mathbf\{q\},$ which can be found as the cross product $\backslash mathbf\{n\}=\backslash mathbf\{p\}\backslash times\backslash mathbf\{q\}.$

---

Normal to general surfaces in 3D space If a (possibly non-flat) surface $S$ in 3D space $\backslash R^3$ is parameterized by a system of curvilinear coordinates $\backslash mathbf\{r\}(s,\; t)\; =\; (x(s,\; t),\; y(s,\; t),\; z(s,\; t)),$ with $s$ and $t$ real number variables, then a normal to S is by definition a normal to a tangent plane, given by the cross product of the partial derivatives $$\backslash mathbf\{n\}=\backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; s\}\; \backslash times\; \backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; t\}.$$

If a surface $S$ is given implicitly as the set of points $(x,\; y,\; z)$ satisfying $F(x,\; y,\; z)\; =\; 0,$ then a normal at a point $(x,\; y,\; z)$ on the surface is given by the gradient $$\backslash mathbf\{n\}\; =\; \backslash nabla\; F(x,\; y,\; z).$$ since the gradient at any point is perpendicular to the level set $S.$

For a surface $S$ in $\backslash R^3$ given as the graph of a function $z\; =\; f(x,\; y),$ an upward-pointing normal can be found either from the parametrization $\backslash mathbf\{r\}(x,y)=(x,y,f(x,y)),$ giving $$\backslash mathbf\{n\}\; =\; \backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; x\}\; \backslash times\; \backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; y\}\; =\; \backslash left(1,0,\backslash tfrac\{\backslash partial\; f\}\{\backslash partial\; x\}\backslash right)\; \backslash times\; \backslash left(0,1,\backslash tfrac\{\backslash partial\; f\}\{\backslash partial\; y\}\backslash right)\; =\; \backslash left(-\backslash tfrac\{\backslash partial\; f\}\{\backslash partial\; x\},\; -\backslash tfrac\{\backslash partial\; f\}\{\backslash partial\; y\},1\backslash right);$$ or more simply from its implicit form $F(x,\; y,\; z)\; =\; z-f(x,y)\; =\; 0,$ giving $\backslash mathbf\{n\}\; =\; \backslash nabla\; F(x,\; y,\; z)\; =\; \backslash left(-\backslash tfrac\{\backslash partial\; f\}\{\backslash partial\; x\},\; -\backslash tfrac\{\backslash partial\; f\}\{\backslash partial\; y\},\; 1\; \backslash right).$ Since a surface does not have a tangent plane at a singular point, it has no well-defined normal at that point: for example, the vertex of a cone. In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous.

---

Orientation The normal to a (hyper)surface is usually scaled to have Unit vector, but it does not have a unique direction, since its opposite is also a unit normal. For a surface which is the topological boundary of a set in three dimensions, one can distinguish between two normal orientations, the inward-pointing normal and outer-pointing normal. For an Orientability, the normal is usually determined by the right-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 a pseudovector.

---

Transforming normals 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 $\backslash mathbf\{M\},$ we can determine the matrix $\backslash mathbf\{W\}$ that transforms a vector $\backslash mathbf\{n\}$ perpendicular to the tangent plane $\backslash mathbf\{t\}$ into a vector $\backslash mathbf\{n\}^\{\backslash prime\}$ perpendicular to the transformed tangent plane $\backslash mathbf\{Mt\},$ by the following logic:

Write n′ as $\backslash mathbf\{Wn\}.$ We must find $\backslash mathbf\{W\}.$

Choosing $\backslash mathbf\{W\}$ such that $W^\backslash mathrm\{T\}\; M\; =\; I,$ or $W\; =\; (M^\{-1\})^\backslash mathrm\{T\},$ will satisfy the above equation, giving a $W\; \backslash mathbb\; n$ perpendicular to $M\; \backslash mathbb\; t,$ or an $\backslash mathbf\{n\}^\{\backslash prime\}$ perpendicular to $\backslash mathbf\{t\}^\{\backslash prime\},$ 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.

---

Hypersurfaces in n-dimensional space For an $(n-1)$-dimensional hyperplane in $n$-dimensional space $\backslash R^n$ given by its parametric representation $$\backslash mathbf\{r\}\backslash left(t\_1,\; \backslash ldots,\; t\_\{n-1\}\backslash right)\; =\; \backslash mathbf\{p\}\_0\; +\; t\_1\; \backslash mathbf\{v\}\_1\; +\; \backslash cdots\; +\; t\_\{n-1\}\backslash mathbf\{v\}\_\{n-1\},$$ where $\backslash mathbf\{p\}\_0$ is a point on the hyperplane and $\backslash mathbf\{v\}\_i$ for $i\; =\; 1,\; \backslash ldots,\; n\; -\; 1$ are linearly independent vectors pointing along the hyperplane, a normal to the hyperplane is any vector $\backslash mathbf\; n$ in the null 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 $\backslash mathbf\{n\}\; =\; \backslash left(a\_1,\; \backslash ldots,\; a\_n\backslash right)$ is a normal.

The definition of a normal to a surface in three-dimensional space can be extended to $(n\; -\; 1)$-dimensional in . A hypersurface may be Local property defined implicitly as the set of points $(x\_1,\; x\_2,\; \backslash ldots,\; x\_n)$ satisfying an equation , where $F$ is a given Scalar field. If $F$ is continuously differentiable then the hypersurface is a differentiable manifold in the neighbourhood of the points where the gradient is not zero. At these points a normal vector is given by the gradient: $$\backslash mathbb\; n\; =\; \backslash nabla\; F\backslash left(x\_1,\; x\_2,\; \backslash ldots,\; x\_n\backslash right)\; =\; \backslash left(\; \backslash tfrac\{\backslash partial\; F\}\{\backslash partial\; x\_1\},\; \backslash tfrac\{\backslash partial\; F\}\{\backslash partial\; x\_2\},\; \backslash ldots,\; \backslash tfrac\{\backslash partial\; F\}\{\backslash partial\; x\_n\}\; \backslash right)\backslash ,.$$

The normal line is the one-dimensional subspace with basis $\backslash \{\backslash mathbf\{n\}\backslash \}.$

A vector that is normal to the space spanned by the linearly independent vectors and falls within the space spanned by the linearly independent vectors is given by the column of the matrix , where the matrix is the juxtaposition of the column vectors. (Proof: so each of is perpendicular to the last column of .) This formula works even when is less than the dimension of the Euclidean The formula simplifies to when .

---

Varieties defined by implicit equations in n-dimensional space A analytic variety defined by implicit equations in the $n$-dimensional space $\backslash R^n$ is the set of the common zeros of a finite set of differentiable functions in $n$ variables $$f\_1\backslash left(x\_1,\; \backslash ldots,\; x\_n\backslash right),\; \backslash ldots,\; f\_k\backslash left(x\_1,\; \backslash ldots,\; x\_n\backslash right).$$ The Jacobian matrix of the variety is the $k\; \backslash times\; n$ matrix whose $i$-th row is the gradient of $f\_i.$ By the implicit function theorem, the variety is a manifold in the neighborhood of a point where the Jacobian matrix has rank $k.$ At such a point $P,$ the normal vector space is the vector space generated by the values at $P$ of the gradient vectors of the $f\_i.$

In other words, a variety is defined as the intersection of $k$ hypersurfaces, and the normal vector space at a point is the vector space generated by the normal vectors of the hypersurfaces at the point.

The normal (affine) space at a point $P$ of the variety is the affine subspace passing through $P$ and generated by the normal vector space at $P.$

These definitions may be extended to the points where the variety is not a manifold.

---

Example Let V be the variety defined in the 3-dimensional space by the equations $$x\backslash ,y\; =\; 0,\; \backslash quad\; z\; =\; 0.$$ This variety is the union of the $x$-axis and the $y$-axis.

At a point $(a,\; 0,\; 0),$ where $a\; \backslash neq\; 0,$ the rows of the Jacobian matrix are $(0,\; 0,\; 1)$ and $(0,\; a,\; 0).$ Thus the normal affine space is the plane of equation $x\; =\; a.$ Similarly, if $b\; \backslash neq\; 0,$ the normal plane at $(0,\; b,\; 0)$ is the plane of equation $y\; =\; b.$

At the point $(0,\; 0,\; 0)$ the rows of the Jacobian matrix are $(0,\; 0,\; 1)$ and $(0,\; 0,\; 0).$ Thus the normal vector space and the normal affine space have dimension 1 and the normal affine space is the $z$-axis.

---

Uses

• Surface normals are useful in defining of .
• Surface normals are commonly used in 3D computer graphics for lighting calculations (see Lambert's cosine law), often adjusted by normal mapping.
• Render layers containing surface normal information may be used in digital compositing to change the apparent lighting of rendered elements.
• In computer vision, the shapes of 3D objects are estimated from surface normals using photometric stereo.
• The normal vector may be obtained as the gradient of the signed distance function.

---

Normal in geometric optics The is the outward-pointing ray perpendicular to the surface of an optical medium at a given point. In reflection of light, the angle of incidence and the angle of reflection are respectively the angle between the normal and the incident ray (on the plane of incidence) and the angle between the normal and the reflected ray.

---

See also

---

External links

• An explanation of normal vectors from Microsoft's MSDN
• Clear pseudocode for calculating a surface normal from either a triangle or polygon.

Post Comment