Subnormal

Graphic representation of a sub-normal

The subnormal is a term from analytical geometry . If a curve is differentiable at one point , the subnormal is the distance between the point on the abscissa and the zero point of the normal . In the adjacent figure, the observed curve is shown in red and the normal in green. The projection of the normal on the abscissa is called the sub-normal (yellow). ${\displaystyle x_{0}}$ ${\displaystyle x_{0}}$ ${\displaystyle x_{0}}$

About the equation of normals

${\displaystyle n(x)={\frac {-1}{f'(x_{0})}}\cdot (x-x_{0})+f(x_{0})}$

one arrives at the zero and thus at the subnormal . ${\displaystyle x_{s}=f(x_{0})\cdot f'(x_{0})+x_{0}}$ ${\displaystyle |f(x_{0})\cdot f'(x_{0})|}$

The magnitude of the subnormal of the function is for all normals , since: ${\displaystyle e}$ ${\displaystyle f(x)=e^{x}}$ ${\displaystyle e^{2x}}$

${\displaystyle |f(x_{0})\cdot f'(x_{0})|=|e^{x_{0}}\cdot e^{x_{0}}|=e^{2x_{0}}}$.

For parabolas that are symmetrical about the -axis, the subnormal has the same length at all points . With the function equation ${\displaystyle x}$ ${\displaystyle x_{0}}$

${\displaystyle f(x)=\pm {\sqrt {2a(x-c)}}}$,

where and are fixed parameters, the result is the length . ${\displaystyle a\neq 0}$ ${\displaystyle c}$ ${\displaystyle |f(x_{0})\cdot f'(x_{0})|=|a|}$

As an analogy to the tangent, there is the subtangent .

Web links

Wiktionary: Subnormal  - explanations of meanings, word origins, synonyms, translations