Comprehensive Introduction to Differential Geometry: Sold by Michael Spivak

By Michael Spivak

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By Michael Spivak

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Read or Download Comprehensive Introduction to Differential Geometry: Sold Only As Individual Volumes See Isbns 0914098845/0914098853 (Volumes 1 and 2) PDF

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Extra info for Comprehensive Introduction to Differential Geometry: Sold Only As Individual Volumes See Isbns 0914098845/0914098853 (Volumes 1 and 2)

Example text

BC and same straight B C ,... will cut line^ viz. the inter section of the planes of the two figures. be shown that M if is a point lying on the a straight line in the plane lying then the M\ passes through corresponding straight line a will also pass through M. Eut this is evidently the case, since the It is to straight line two a- or , and if <r, <z, straight lines a and a are the intersections of the same projecting plane with the two planes a- and a- and conse a, and a meet in a point, quently the three straight lines , <r</, that viz.

Two 11. The two figures may equally well be generated by the simultaneous motion of a pair of corresponding straight lines a a If a revolve about a fixed point A, then a will always , . pass through the corresponding point Similarly, if a A . envelop a curve, then a will envelop the The corresponding curve. lines a and , in corresponding two curves at positions, touch the corresponding points and again, to the tangents to the first curve from a point A correspond the tangents to / g ; fiT~ v / / / fl _ r/ f|\ / the second from the corresponding Two corresponding curves point / ^\\ \\V\ A / p \c\A\B \\\ y \\\j \\ /* ^Jc D / /< / <* _ .

2 a-, and in this straight line take two points Project the triangle A l B1 Cl from Sl and the triangle A 2 B2 C2 from ,. The points A l A 2 0, S29 S lie in the same plane; therefore A l and S2 A 2 meet one another (in A suppose) similarly S1 B 1 and S2 B2 (in B suppose) and S-^C^ and S2 C2 (in C suppose). lines , , , . , , ; <S\ * The planes a and a are to be regarded as distinct from each other. CENTRAL PROJECTION; FIGURES IN PERSPECTIVE. 8 [17 triangle ABC is in perspective both with A^Bf^ and with A^B2 C2 The straight lines BC, Bf^, B2 C2 intersect in pairs and therefore meet in one and the same point A*.

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