📂Geometry

Definition of a Curve

Definitions ¹

1. The mapping $\alpha : (a,b) \to \mathbb{R}^{3}$ is referred to as a Curve.
2. A point $t = t_{0}$ at the curve where $\alpha^{\prime} = \dfrac{d \alpha}{d t} = \mathbf{0}$ is called a Singular Point.
3. A curve $\alpha \in C^{k}$ for some $k \in \mathbb{N}$ where at all $t \in (a,b)$, $\displaystyle {{ d \alpha } \over { d t }} \ne \mathbf{0}$ is known as a Regular Curve. In other words, a regular curve is one without singular points.
4. The derivative $\alpha^{\prime}(t_{0})$ of curve $\alpha$ at $t=t_{0}$, when $t = t_{0}$, is called the velocity vector of $\alpha$, and the derivative $\alpha^{\prime}$ of $\alpha$ is referred to as the velocity vector field of $\alpha$. Therefore, a regular curve is a curve whose velocity does not become $\mathbf{0}$, which means, physically, that its direction of motion never reverses.
5. The magnitude $\left|\alpha^{\prime}(t_{0}) \right|$ of the velocity of $\alpha$ at $t = t_{0}$ is called speed.
6. A curve with $\left| \alpha^{\prime} \right| = 1$ is called a Unit Speed Curve.

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• $C^{k}$ is $k$ times differentiable and its derivative is a set of continuous functions.

Explanation

$$ \alpha (t) := \left( \alpha_{1} (t) , \alpha_{2} (t) , \alpha_{3} (t) \right) $$

In geometry, the object of interest is the figure, and note that in the definition, this figure has been represented as a function of the parameter $t$. This has allowed us to utilize many mathematical tools to study figures, especially in differential geometry where calculus is extensively used.

A singular point is simply a point that is bent or has come to a stop. Depending on the perspective, a bent point can have two directions. Such points are difficult to handle and are not dealt with in undergraduate differential geometry. A ‘stopping point’ is explained with an example.

For some $k \in \mathbb{N}$, that $\alpha \in C^{k}$ implies it is differentiable at least once, and how many times it can be differentiated is not very important. Generally, if it’s differentiable at least $k=1$ times, it’s simply considered Smooth.

Example

Straight Line

$$ l(t) := \left( t, t, t \right) $$ According to the definition, there’s no reason why the straight line $l : \mathbb{R} \to \mathbb{R}^{3}$ shouldn’t be considered a curve. In Korean, because the character for curve 굽을 곡曲 has the implication of bending, it may cause confusion, so it might be better to just read it according to its English pronunciation, curve.

Helix

$$ \zeta (t) := \left( \cos t , \sin t , t \right) $$

According to $0 \to t \to \infty$, a helix is drawn as follows.

Irregular Curve

$$ \beta (t) := \left( t^{2} , t^{3} , t^{4} \right) $$ If the above curve $\beta$ is differentiated, then $$ \beta^{\prime} (t) := \left( 2t , 3t^{2} , 4t^{3} \right) $$ thus at $t = 0$, $\displaystyle {{ d \beta } \over { d t }} (0) = \mathbf{0}$ occurs. This singular point is not bent but literally comes to a stop following $t$ at $t=0$. Therefore, if the domain of $\beta$ is $\mathbb{R}$, it’s not a regular curve. However, if its domain does not include $0$, for example, $(0,\infty)$, then it is a regular curve. Note that curves can be regular or not, depending on their domain.

Code

The following is the code to create the gif seen in the Helix example with Julia.

using Plots

ζ(t) = (cos(t), sin(t), t)

anim = @animate for T ∈ 0.1:0.1:10
    t = 0:0.1:(T*π)
    helix = plot(ζ.(t), camera = (45,45), legend = :none)
    xlims!(-2,2); ylims!(-2,2); zlims!(0,40)
end
gif(anim, "helix.gif")