Touch

Touch is a property of two lines or a line and a surface to have a common tangent line at some point or a property of two surfaces to have a common tangent plane at some point .

The point at which two geometric shapes have a touch is called a touch point or a point of contact .

Touch Order

The order of contact is a characteristic of the proximity of two lines , a line and a surface or two surfaces in the vicinity of their common point .

Assume for two curves $\gamma _{1}=\{\mathbf {r} _{1}(s)|s\in \mathbb {R} \}$ ${\ displaystyle \ gamma _ {1} = \ {\ mathbf {r} _ {1} (s) | s \ in \ mathbb {R} \}}$ and $\gamma _{2}=\{\mathbf {r} _{2}(s)|s\in \mathbb {R} \}$ ${\ displaystyle \ gamma _ {2} = \ {\ mathbf {r} _ {2} (s) | s \ in \ mathbb {R} \}}$ natural parameterization is given. They say that curves have a point $P$ ${\ displaystyle P}$ touch order $m$ ${\ displaystyle m}$ if point $P$ ${\ displaystyle P}$ belongs to both of them and their first $m$ ${\ displaystyle m}$ derivatives ${\frac {d^{m}\mathbf {r} _{1,2}(s)}{ds^{m}}}$ ${\ displaystyle {\ frac {d ^ {m} \ mathbf {r} _ {1,2} (s)} {ds ^ {m}}}}$ at the point $P$ ${\ displaystyle P}$ match up. In other words, the distance between $\mathbf {r} _{1}(s)$ ${\ displaystyle \ mathbf {r} _ {1} (s)}$ and $\mathbf {r} _{2}(s)$ ${\ displaystyle \ mathbf {r} _ {2} (s)}$ there is $o(s^{m})$ ${\ displaystyle o (s ^ {m})}$ .

Related Definitions

Tangent to a curve $\gamma$ ${\ displaystyle \ gamma}$ at the point $P$ ${\ displaystyle P}$ - a straight line having $\gamma$ ${\ displaystyle \ gamma}$ at the point $P$ ${\ displaystyle P}$ first order touch.
Curvature Radius $\gamma$ ${\ displaystyle \ gamma}$ at the point $P$ ${\ displaystyle P}$ Is the radius of a circle having a curve $\gamma$ ${\ displaystyle \ gamma}$ at the point $P$ ${\ displaystyle P}$ second order touch.

Touch

Touch Order

Related Definitions

See also

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