Скачать или смотреть Contact (mathematics) | Wikipedia audio article

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00:00:16 1 Applications
00:00:33 2 Contact between curves
00:00:50 2.1 Contact between a curve and a circle
00:01:07 2.2 Bi-tangents in econometrics



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SUMMARY
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In mathematics, two functions have a contact of order k if, at a point P, they have the same value and k equal derivatives. This is an equivalence relation, whose equivalence classes are generally called jets. The point of osculation is also called the double cusp. Contact is a geometric notion; it can be defined algebraically as a valuation.
One speaks also of curves and geometric objects having k-th order contact at a point: this is also called osculation (i.e. kissing), generalising the property of being tangent. (Here the derivatives are considered with respect to arc length.) An osculating curve from a given family of curves is a curve that has the highest possible order of contact with a given curve at a given point; for instance a tangent line is an osculating curve from the family of lines, and has first-order contact with the given curve; an osculating circle is an osculating curve from the family of circles, and has second-order contact (same tangent angle and curvature), etc.