Differential geometry of curves

Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the synthetic approach. Differential geometry takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus. One of the most important tools used to analyze a curve is the Frenet frame, a moving frame that provides a coordinate system at each point of the curve that is 'best adapted' to the curve near that point. The theory of curves is much simpler and narrower in scope than the theory of surfaces and its higher-dimensional generalizations because a regular curve in a Euclidean space has no intrinsic geometry. Any regular curve may be parametrized by the arc length (the natural parametrization). From the point of view of a theoretical point particle on the curve that does not know anything about the ambient space, all curves would appear the same. Different space curves are only distinguished by how they bend and twist. Quantitatively, this is measured by the differential-geometric invariants called the curvature and the torsion of a curve. The fundamental theorem of curves asserts that the knowledge of these invariants completely determines the curve. Let (i) n ∈ N {displaystyle nin mathbb {N} } , (ii) r ∈ { N ∪ ∞ } {displaystyle rin {mathbb {N} cup infty }} , and (iii) I {displaystyle I} be a non-empty interval of real numbers. Then a vector-valued function of class C r {displaystyle C^{r}} (i.e., the component functions of γ {displaystyle gamma } are r {displaystyle r} -times continuously differentiable) is called a parametric C r {displaystyle C^{r}} -curve or a C r {displaystyle C^{r}} -parametrization. Note that γ [ I ] ⊆ R n {displaystyle gamma subseteq mathbb {R} ^{n}} is called the image of the parametric curve. It is important to distinguish between a parametric curve γ {displaystyle gamma } and its image γ [ I ] {displaystyle gamma } because a given subset of R n {displaystyle mathbb {R} ^{n}} can be the image of several distinct parametric curves. The parameter t {displaystyle t} in γ ( t ) {displaystyle gamma (t)} can be thought of as representing time, and γ {displaystyle gamma } the trajectory of a moving particle in space. When I {displaystyle I} is a closed interval [ a , b ] {displaystyle } , γ ( a ) {displaystyle gamma (a)} is called the starting point and γ ( b ) {displaystyle gamma (b)} is the endpoint of γ {displaystyle gamma } . If the starting and the end points coincide, i.e. γ ( a ) = γ ( b ) {displaystyle gamma (a)=gamma (b)} , then γ {displaystyle gamma } is called a closed or a loop. Furthermore, γ {displaystyle gamma } is called a closed parametric C r {displaystyle C^{r}} -curve if and only if γ ( k ) ( a ) = γ ( k ) ( b ) {displaystyle {gamma ^{(k)}}(a)={gamma ^{(k)}}(b)} for all k ∈ N ≤ r {displaystyle kin mathbb {N} _{leq r}} . If γ | ( a , b ) : ( a , b ) → R n {displaystyle gamma |_{(a,b)}:(a,b) o mathbb {R} ^{n}} is injective, then γ {displaystyle gamma } is simple. If each component function of γ : I → R n {displaystyle gamma :I o mathbb {R} ^{n}} can be expressed as a power series, then γ {displaystyle gamma } is analytic (i.e. being of class C ω {displaystyle C^{omega }} ). − γ {displaystyle -gamma } is written for the parametric curve that is traversed in the direction opposite to that of γ {displaystyle gamma } . γ {displaystyle gamma } is regular of order m {displaystyle m} (where m ≤ r {displaystyle mleq r} ) if and only if for any t ∈ I {displaystyle tin I} ,