Main Theorem of Curve Theory
Given n functions
with
then there exists a unique (up to transformations using the Euclidean group) Cn+1-curve γ which is regular of order n and has the following properties
where the set
is the Frenet frame for the curve.
By additionally providing a start t0 in I, a starting point p0 in Rn and an initial positive orthonormal Frenet frame {e1, ..., en-1} with
we can eliminate the Euclidean transformations and get unique curve γ.
Read more about this topic: Differential Geometry Of Curves
Famous quotes containing the words main, theorem, curve and/or theory:
“Whoever considers morality the main objective of human existence, seems to me like a person who defines the purpose of a clock as not going wrong. The first objective for a clock, is, however, that it does run; not going wrong is an additional regulative function. If not a watch’s greatest accomplishment were not going wrong, unwound watches might be the best.”
—Franz Grillparzer (1791–1872)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)
“Nothing ever prepares a couple for having a baby, especially the first one. And even baby number two or three, the surprises and challenges, the cosmic curve balls, keep on coming. We can’t believe how much children change everything—the time we rise and the time we go to bed; the way we fight and the way we get along. Even when, and if, we make love.”
—Susan Lapinski (20th century)
“In the theory of gender I began from zero. There is no masculine power or privilege I did not covet. But slowly, step by step, decade by decade, I was forced to acknowledge that even a woman of abnormal will cannot escape her hormonal identity.”
—Camille Paglia (b. 1947)