# An elementary treatise on curve tracing by Percival Frost

By Percival Frost

This obtainable therapy covers orders of small amounts, types of parabolic curves at an enormous distance, types of curves in the community of the beginning, and sorts of branches whose tangents on the foundation are the coordinate axes. extra themes comprise asymptotes, analytical triangle, singular issues, extra. 1960 variation.

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Esto se debe a que toda recta en el espacio est´a completamente contenida en un plano. Ahora nos preguntamos si M es una superficie cualquiera, ¿Cu´ales son las rectas que minimizan la distancia entre dos puntos sobre una superficie?. Las curvas geod´esicas son aquellas que responden nuestra pregunta. 6: Las geod´esicas minimizan la distancia en una superficie. c T. 1. Sea M una superficie. Sea α una curva regular que vive sobre M . Diremos que α es una curva geod´ esica si κg = 0. 2. Sea α una curva no plana.

Guardia Geod´ esicas 48 Y por el Teorema Fundamental del C´alculo dl ∂H ∂H (0) =< , > dt ∂t ∂s d t=0 − < c ∂H ∂ 2 H , > ∂t ∂s2 t=0 ds. 56 en t = 0 ∂H (s, 0) = b1 (s)X1 (β(s)) + b2 (s) + X2 (β(s)) = λ(s)S(s). 58 en t = 0 ∂H (s, 0) = a1 (s)X1 (β(s)) + a2 (s)X2 (β(s)) = T(s). 59) Por lo tanto < ∂H ∂H , > ∂t ∂s t=0 =< λ(s)S(s), T(s) >= λ < S(s), T(s) >= 0. 59 respecto a s ∂ 2H d (s, 0) = (T(s)) = T (s). 54 dl (0) = − dt d < λ(s)S(s), T (s) > ds. 63) c T. 62 se convierte en dl (0) = − dt d λ(s)κg (s)ds.

31 c T. Guardia Geod´ esicas 32 Dejamos al lector verificar que la matriz G(p) es sim´etrica, invertible y que det G(p) = X1 (p) × X2 (p) 2 . En el cap´ıtulo 1 definimos los vectores T(s), N(s) y B(s) para una curva regular parametrizada por longitud de arco. A continuaci´on definiremos los vectores an´alogos para una superficie. 3. Sea M una superficie, y X : D ⊆ R2 local. Sea α una curva regular parametrizada por longitud de arco que vive en M . 10 si X es una carta local de la superficie M y α es una curva regular que vive sobre X.