Curves and Surfaces | bearish….

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.