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Curves of Double Curvature

curve, plane, tangent, intersection, normal and equations

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CURVES OF DOUBLE CURVATURE In French, courbes pouches; in German, Curven doppelter Krummung). A curve whose points do not all lie in a plane is variously called space curve, twisted curve, tortuous curve or curve of double curvature. The sig nificance of the last-named designation be comes apparent in the description of the ele ments which enter the theory of the curve, There are two ways of regarding the curve: first, as given immediately and in all its ex tent by the intersection of two surfaces — a purely geometric conception; second, as arising kinematically through the continuous move ment of a point. The first conception leads to the analytical formulation of a curve as the locus of points whose Cartesian co-ordinates satisfy two equations, the equations of the in tersecting surfaces (1) y, 3". 2)=0.

The second conception leads to the expression of the Cartesian co-ordinates x, y, s as func tions of a variable magnitude u called the parameter: (2) x =-- #(u), y = x(u), a ---- Cu), a form of representation called the parametric representation of the curve. To the continuous succession of values of u corresponds the con tinuous succession of points of the curve.

It is known that not every curve in space is the complete intersection of two surfaces, and therefore (1) and (2) are not always equiva lent forms. In fact, when the three equations (2), defining a curve, are transformed to the two equations (1), the latter often furnish curve branches in addition to the original curve. The parametric representation is, in many im portant respects, preferable to the first form, and such equations and formulas as are here after used will have reference to this form.

The subject matter may be conveniently di vided into four parts: first, the determining elements of an infinitely small portion of the curve in the neighborhood of anypoint P of the curve; second, the character of the curve as a whole and the associated curves and sur faces; third, special curves and their derivation from given properties; fourth, classification, miscellaneous matters and the literature of the subject.

1. The Elements of the Curve at a Point. —The consideration of two points, of three points and of four points of the curve conducts immediately to fundamental elements.

The tangent at P is the limiting position as sumed by the right line through P and a neigh boring point Pi of the curve as Pi moves along the curve toward coincidence with P. It is convenient to say that the tangent at P is the right line through P and Pi, P, being infinitely near or consecutive to P, but it must be remem bered that this and similar expressions which will be used further on are only abbreviations and figures of speech.

The single infinity of lines through P per pendicular to the tangent are normals to the curve and define the normal plane of the curve at P. Every plane through the tangent is a tangent plane at P. A plane passing through P and two neighboring points of the curve assumes in general a limiting position at P as the two neighboring points move along the curve toward coincidence with P. This is the plane of osculation at P and may be described as the plane determined by three consecutive points P, P,, P. It is the tangent plane at P which has the closest contact with the curve.

The plane through the tangent perpendicular to the plane of osculation is called the plane of rectification at P. The normal plane, the plane of osculation and the plane of rectification are mutually at right angles and constitute the principal planes of the curve at P. Their in tersections are three lines through P: the tan gent, which is the intersection of the plane of osculation and the plane of rectification; the principal normal, which is the intersection of the normal plane and the plane of osculation; the binormal, the intersection of the normal plane and the plane of rectification. The three lines form a configuration called the principal triedral at P.

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