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:
“Yours of the 24th, asking “the best mode of obtaining a thorough knowledge of the law” is received. The mode is very simple, though laborious, and tedious. It is only to get the books, and read, and study them carefully.... Work, work, work, is the main thing.”
—Abraham Lincoln (1809–1865)
“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)
“The years-heired feature that can
In curve and voice and eye
Despise the human span
Of durance—that is I;
The eternal thing in man,
That heeds no call to die.”
—Thomas Hardy (1840–1928)
“The human species, according to the best theory I can form of it, is composed of two distinct races, the men who borrow and the men who lend.”
—Charles Lamb (1775–1834)