By Luther Pfahler Eisenhart
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The aim of this booklet is to illustrate that complicated numbers and geometry may be mixed jointly superbly. This leads to effortless proofs and common generalizations of many theorems in aircraft geometry, akin to the Napoleon theorem, the Ptolemy-Euler theorem, the Simson theorem, and the Morley theorem.
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Extra info for A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix Editions)
Since the point 1 condition that the point l ( 0, TJ M = d% = 0. And since M tends to move at axis, Brj = 0. Now equations (82) reduce to have d% this l }Ll-, ds p (96) From the second we ds see that k is right angles to 5--*. ds r vO, > a constant. Moreover, M if co denotes makes with the tangent at M, the angle which the tangent at l of first and third these equations, we have, from the tan co 8? = = -^r or Sf sin cos co co (97) kp T (k p) sin co k We have seen ( 11) that according as r is positive or negative, near the osculating plane to a curve at a point cuts the below or above the osculating plane at M.
Curve 4. y = The locus is Find the radii of a cos 2 it, z construction. 5. of the center of curvature of a curve is an evolute only when the plane. and second curvature of the curve x = a sin u cos w, is spherical, and give a geometrical first = asinw. Show that the curve Find its evolutes. Derive the properties of Bertrand curves ( 10) without the use of the moving trihedral. 6. Find the involutes and evolutes of the twisted cubic. 7. Determine whether there is a curve whose bmormals are the binormals of a second curve.
The coordinates of the point MI of the curve with reference to the axes at are a, 0, (i) 0. In this case equations (82) reduce to ^-l^ds~ ds~ a P ^ds~ M CURVES IN SPACE 34 Hence denotes the length of arc of C\ from the point corresponding to if Si s = on C, we have of the tangent to Ci with reference to the and the direction-cosines moving axes iven b are given by 7 Hence the tangent to Ci Va2 4- p2 a2 + is parallel to the osculating plane at the corresponding p 2 point of C. i By means of (40) the binormal we derive the following expressions for the direction-cosines of : r (a 2 + r (a 2 2 p )^ + ft2 2 p )^ > + ?