By Luther Pfahler Eisenhart
Read Online or Download A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix Editions) PDF
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Additional resources for A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix Editions)
8. Principal normal and binormal. finity of Evidently there are an in normals to a curve at a point. Two of these are of par the normal, which lies in the osculating plane at the point, called the principal normal; and the normal, which is perpendicular to this plane, called the binormal. ticular interest : If the direction-cosines of the we have from X : / binormal be denoted by X, />t, z>, (35) : v = (y z"- z : y") (z z x"~ : z") (* / -y x"). z ~ P dxd y ~ ds 3 seen by differentiating 2x"*= 1 dyd^x ds* with respect to s.
6 2 = - i0i- From (77) P , for a and we get a 4- and k- 6; 0, we put a s = GO, = So where i 1, 2. = 1, 2. The solutions of this equation Q = - 0i, R = e S = - 1, so that i r , W==Vc + 2 When a, /3, 7 the foregoing values are substituted in (73), and the resulting values of in (61), we get C x- (80) Vc2 + From the la a constant ang c And -v\ - Ccoslds, J 1 y= Xi Si - Vc 2 + 1 Jfstafdt, g = = Vc 2 + 1 ie expressions we find that the tangent to the curve makes the direction of the elements of the cylinder.
In equations (102) has a positive or negative value, the point lies on the portion of the tangent drawn in the According as t TANGENT SUBFACE OF A CUKVE 43 in the opposite direction. It positive direction from the curve or is now our purpose to get an idea of the form of the surface in the neighborhood of the curve. u, 6 pr The for f M = = 0, plane f at which s seen that for points of it is the parameters Q sign. , 2/o , eliminating t From also a point of F. from the M The point Q of (7, the above expression cuts the surface in a curve F.