词条 | Conchoid of Dürer | |||
释义 |
The conchoid of Dürer, also called Dürer's shell curve, is a variant of a conchoid or plane algebraic curve, named after Albrecht Dürer. It is not a true conchoid. ConstructionSuppose two perpendicular lines are given, with intersection point O. For concreteness we may assume that these are the coordinate axes and that O is the origin, that is (0, 0). Let points {{math|1=Q = (q, 0)}} and {{math|1=R = (0, r)}} move on the axes in such a way that {{math|1=q + r = b}}, a constant. On the line {{math|QR}}, extended as necessary, mark points {{math|P}} and {{mvar|P'}} at a fixed distance {{mvar|a}} from {{math|Q}}. The locus of the points {{math|P}} and {{mvar|P'}} is Dürer's conchoid.[1] EquationThe equation of the conchoid in Cartesian form is In parametric form the equation is given by where the parameter {{mvar|t}} is measured in radians.[2] PropertiesThe curve has two components, asymptotic to the lines .[3] Each component is a rational curve. If a > b there is a loop, if a = b there is a cusp at (0,a). Special cases include:
The envelope of straight lines used in the construction form a parabola (as seen in Durer's original diagram above) and therefore the curve is a point-glissette formed by a line and one of its points sliding respectively against a parabola and one of its tangents.[4] HistoryIt was first described by the German painter and mathematician Albrecht Dürer (1471–1528) in his book Underweysung der Messung (Instruction in Measurement with Compass and Straightedge p. 38), calling it Ein muschellini (Conchoid or Shell). Dürer only drew one branch of the curve. See also
References1. ^{{citation | first=J. Dennis|last= Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | page=157}} 2. ^{{cite web|url=https://www.desmos.com/calculator/cc8tpciqr8|title=Dürer's Conchoid}} beware that the constants {{mvar|a}} and {{mvar|b}} are interchanged in this source 3. ^{{citation|first=Henry E.|last=Fettis|title=The Geometry of Dürer's Conchoid|journal=Crux Mathematicorum|volume=9|issue=2|year=1983|url=https://cms.math.ca/crux/backfile/Crux_v9no2_Feb.pdf|issn=0705-0348}} 4. ^{{citation|first=E. H.|last=Lockwood|title=A Book of Curves|publisher=Cambridge University Press|year=2007|orig-year=1967|page=164|isbn=9780521044448}} External links{{MathWorld|title=Dürer's Conchoid|id=DuerersConchoid}}{{DEFAULTSORT:Conchoid of Durer}} 2 : Algebraic curves|Albrecht Dürer |
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