Curves

Note: by \R^n, we indicate an Euclidean geometrical space, i.e. space of (geometric) points with an origin and equipped with a normed vector space.

1- Parametrized smooth curves

Definition: a parametrized smooth curve is a function \gamma:I\sub\R\to\R^n where I=(a,b) is an open interval in extended \R and \gamma is a smooth function. 
Note that a parameterized curve (always smooth) is referred to the function \gamma(t)=(\gamma_1(t), \dots,\gamma_n(t))\in \R^n, not to the function image \gamma(I), which is a set of points in \R^n.
By a curve we mean parametrized smooth curve unless otherwise stated. A curve in \R^n means a curve (function) whose image is in \R^n.

Definition: a parametrized curve \tilde{\gamma}:(\tilde{t_1},\tilde{t_2})\to\R^n is a reparametrization of the curve \gamma:(t_1,t_2)\to\R^n if there is a smooth bijection \phi:( \tilde{t_1},\tilde{t_2} )\to(t_1,t_2) such that the inverse map \phi^{-1}:( t_1,t_2)\to (\tilde{t_1},\tilde{t_2}) is also smooth and \gamma(\phi(\tilde{t}))=\tilde{\gamma}(\tilde{t}) \quad \forall \tilde{t} \in (\tilde{t_1},\tilde{t_2})\sub \R. Note that t = \phi(\tilde{t}). This indicates that \gamma is also a reparametrization of \tilde\gamma. Also note that \phi(t) is monotonic. If decreasing, we have t_2 = \phi(\tilde{t_1}) \gt t_1 = \phi(\tilde{t_2})\text{ for } \tilde{t_1}\lt \tilde{t_2}. This indicates that reparametrization can change the relative direction of a curve.

Curves with different reparametrizations have the same image and hence the same geometrical properties.

Note: \tilde{\gamma}(\tilde{t}):=\gamma(\phi(\tilde{t}))=(\gamma_1(\phi(\tilde{t})),\dots,\gamma_n(\phi(\tilde{t}))=(\tilde{\gamma_1}(\tilde{t}),\dots,\tilde{\gamma_n}(\tilde{t}))

Arc length and unit speed curves

For any parameter t, \dot\gamma(t):=\frac{d}{dt}\gamma(t).

The arc length function is defined as:

s(t)=\int_{t_0}^t\|\dot{\gamma}(u)\|du\quad \in \R

That can be negative or positive depending on t is larger or smaller than t.

From a physical point of view, the trajectory of a moving point can be represented by a curve, therefore, the speed of a moving point along a curve (defined as the rate of change of its distance along the curve ) is:

\frac{ds(t)}{dt}=\|\dot{\gamma}(t)\|\in\R

A unit-speed curve is by definition is a curve with \|\dot{\gamma}(t)\|=1, \forall t\in I, i.e. \dot{\gamma}(t) is a unit vector. In applications for the sake of geometry, not physics, the speed of a curve is not important, therefore, considering a unit-speed parametrization of the curve is shown to simplify formulas and proofs. For example the following useful proposition is often useful in proofs:

Proposition: let n(t)\in\R^n\text{ with }\|n(t)\|=1,\forall t\in I be a smooth function of t, then n(t)\cdot\dot{n}(t)=0 meaning that \dot{n}(t) is either perpendicular to n(t) or zero for all t. Proof by considering n(t)\cdot n(t)=1. In particular, the proposition is true for \dot{\gamma}(t)\cdot \ddot{\gamma}(t) when the curve is unit-speed.

Definition: A point P:=\gamma(t) of a curve [latext]\gamma[/latex] is called a regular point iff \dot\gamma(t)\neq 0; otherwise a singular point. A regular curve is a curve iff all of its points are regular.

Propositions: A reparametrization of a regular curve is regular and the arc length function of a regular curve is smooth. A regular curve has a unit-speed reparametrization, and a unit-speed curve is a regular curve. The arc length (s) is the only unit-speed reparametrization of a regular curve (actually u=\pm s + c for some constant c).

From above, the tangent vector (and speed of an object moving along a regular curve; it’s moving, not stationary) does not vanish at any point.

2- Curvature and torsion of a curve

The curvature is a measure of an extent to which a curve (geometric object) is not contained in a straight line. This impose zero curvature for straight lines. Torsion quantifies the extent to which a curve is not contained in a plane. Hence, plane curves have zero torsion.

To find the measure of curvature, a plane curve is considered. The results can easily be extended to curves in =R^3 and higher dimensional curves (by extending the definition).

Let \gamma(t)\in \R^2 be a unit-speed curve, i.e. t\equiv s and \dot\gamma(t)=n(t). For any parameter value t corresponding to a point P=\gamma(t), any changes \Delta t in the parameter gives another point of the curve P_{\Delta}:=\gamma(t+\Delta t). The distance between latter point and the tangent line at the former point is (the distance between a line and a point), is defined as the shortest distance between the line and the point. This is the length of the perpendicular line segment from the point to the tangent line. Therefore, a measure of distance is (\gamma(t+\Delta t)-\gamma(t))\cdot \hat n where \hat n is the unit vector perpendicular to the tangent line (\dot\gamma(t)\cdot \hat n=0). This is a signed distance. By Taylor’s theorem:

\gamma(t+\Delta t)= \gamma(t) + \dot\gamma(t)\Delta t + \frac{1}{2}\ddot\gamma(t)(\Delta t)^2 + O(\Delta t)^2

Therefore, (\gamma(t+\Delta t)- \gamma(t))\cdot \hat n \approx \frac{1}{2}(\Delta t)^2\ddot\gamma(t)\cdot\hat n. Because \gamma(t) is a unit-speed curve we have \dot\gamma \perp \ddot\gamma; which indicates \ddot\gamma \parallel \hat n. Therefore:

(\gamma(t+\Delta t)- \gamma(t))\cdot \hat n \approx \pm \frac{1}{2}\|\ddot\gamma(t)\|(\Delta t)^2

So the magnitude of the distance, i.e. \frac{1}{2}|\ddot\gamma(t)|(\Delta t)^2 indicates the deviation of the curve from its tangent line (at a particular point). For a fixed deviation in the parameter, and consequently in the point, the quantity \|\ddot\gamma(t)\| expresses the scale of the deviation from the tangent line which is a straight line. This suggests the following definition:

Definition: for a unit-speed curve \gamma(t)\in \R^n, its curvature is defined as \kappa(t):=\|\ddot\gamma(t)\|.

For any regular curve (which always can be reparametrized to a unit-speed curve), the curvature at a point is the magnitude of the rate of the change of the unit tangent vector at that point. For a unit speed curve, its simply \|\ddot\gamma(t)\| . For a general regular curve in \R^3 though:

\kappa(t):=\frac{\|\ddot\gamma(t)\times\dot\gamma(t)\|}{\|\dot\gamma(t)\|^3}

The direction of \ddot\gamma(t)

The direction of \ddot \gamma is in the sense of the direction of the turning of the curve when tracked in its forward direction. The forward direction of a curve is the sense of moving from one point to the next point. The next point of a point \gamma(t) is \gamma(t+\delta t) for positive or negative \delta.

The curvature of a straight line is zero and \dot\gamma\cdot\ddot\gamma = 0 \iff \ddot\gamma=0 though \dot\gamma=c s.t. c\not= 0 being a constant.