Normal Schmormal: My occasionally helpful guide to parenting kids with special needs (Down syndrome, autism, ADHD, neurodivergence)

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n = ∂ r ∂ s × ∂ r ∂ t . {\displaystyle \mathbf {n} ={\frac {\partial \mathbf {r} }{\partial s}}\times {\frac {\partial \mathbf {r} }{\partial t}}.} 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. n = ∂ r ∂ x × ∂ r ∂ y = ( 1 , 0 , ∂ f ∂ x ) × ( 0 , 1 , ∂ f ∂ y ) = ( − ∂ f ∂ x , − ∂ f ∂ y , 1 ) ; {\displaystyle \mathbf {n} ={\frac {\partial \mathbf {r} }{\partial x}}\times {\frac {\partial \mathbf {r} }{\partial y}}=\left(1,0,{\tfrac {\partial f}{\partial x}}\right)\times \left(0,1,{\tfrac {\partial f}{\partial y}}\right)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right);} This article is about the normal to 3D surfaces. For the normal to 3D curves, see Frenet–Serret formulas. A polygon and its two normal vectors A normal to a surface at a point is the same as a normal to the tangent plane to the surface at the same point.

r ( s , t ) = r 0 + s p + t q , {\displaystyle \mathbf {r} (s,t)=\mathbf {r} _{0}+s\mathbf {p} +t\mathbf {q} ,} in this section we only use the upper 3 × 3 {\displaystyle 3\times 3} matrix, as translation is irrelevant to the calculationA normal vector may have length one (in which case it is a unit normal vector) or its length may represent the curvature of the object (a curvature vector). If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a pseudovector. In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point.

Specifically, given a 3×3 transformation matrix M , {\displaystyle \mathbf {M} ,} we can determine the matrix W {\displaystyle \mathbf {W} } that transforms a vector n {\displaystyle \mathbf {n} } perpendicular to the tangent plane t {\displaystyle \mathbf {t} } into a vector n ′ {\displaystyle \mathbf {n} or more simply from its implicit form F ( x , y , z ) = z − f ( x , y ) = 0 , {\displaystyle F(x,y,z)=z-f(x,y)=0,} giving n = ∇ F ( x , y , z ) = ( − ∂ f ∂ x , − ∂ f ∂ y , 1 ) . {\displaystyle \mathbf {n} =\nabla F(x,y,z)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right).} The normal distance of a point Q to a curve or to a surface is the Euclidean distance between Q and its foot P.For a plane given by the equation a x + b y + c z + d = 0 , {\displaystyle ax+by+cz+d=0,} the vector n = ( a , b , c ) {\displaystyle \mathbf {n} =(a,b,c)} is a normal. defined by implicit equations in n-dimensional space Toggle Varieties defined by implicit equations in n-dimensional space subsection where r 0 {\displaystyle \mathbf {r} _{0}} is a point on the plane and p , q {\displaystyle \mathbf {p} ,\mathbf {q} } are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both p {\displaystyle \mathbf {p} } and q , {\displaystyle \mathbf {q} ,} which can be found as the cross product n = p × q . {\displaystyle \mathbf {n} =\mathbf {p} \times \mathbf {q} .} English–Arabic English–Bengali English–Catalan English–Czech English–Danish English–Hindi English–Korean English–Malay English–Marathi English–Russian English–Tamil English–Telugu English–Thai English–Turkish English–Ukrainian English–Vietnamese