By Luther Pfahler Eisenhart
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Additional resources for A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix Editions)
Then when c the evol\te C in the plane of the curve. The other evolutes lie upon the right is MINIMAL CUEVES 47 cylinder formed of the normals to the plane at points of Co, and cut the elements of the cylinder under the constant angle 00 c, and consequently are helices. Hence we have the theorem : The evolutes of a plane curve are the helices traced on the right cylinder whose base is the plane evolute. Conversely, every cylindrical helix is the evolute of an infinity of plane curves. EXAMPLES 1.
Determine f(u) so that the curve x What is the form of the curve ? = a cosw, y = a sin w, z =f(u) shall be Fundamental theorem. Let C^ and Cz be s, and let points each with the s same values of upon correspond. We assume, 13. Intrinsic equations. two curves defined in terms of their respective arcs furthermore, that at corresponding points the radii of first curva and also the radii of second curvature. ture have the same value, We shall show that Cl and Cz are congruent. By a motion in space the points of the two curves for which = s can be made to coincide in such a way that the tangents, principal normals, and binomials to them at the point coincide also.
The only other curve which has more than one conjugate is a circular helix, for since p and T are constant, A/C can be given any value whatever both the given helix and the circular helices conjugate to it are traced on circular cylinders with the same axis. ; Tangent surface of a curve. 20. For the further discussion of the properties of curves it is necessary to introduce certain curves and surfaces which can be associated with them. However, in con sidering these surfaces we limit our discussion to those properties which have to do with the associated curves, and leave other con siderations to their proper places in later chapters.
A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix Editions) by Luther Pfahler Eisenhart
Categories: Geometry And Topology