To determine if a curve is regular, you need to check if its tangent vector is non-zero at every point along its path.
Understanding Regular Curves
A curve, often represented parametrically by a position vector function $\mathbf{x}(t)$, is considered regular if its tangent vector, $\mathbf{x}'(t)$, is never the zero vector for any value of $t$ in its domain. This condition ensures that the curve is smooth and does not have any sharp corners, cusps, or points where it momentarily stops.
The position vector $\mathbf{x}(t)$ describes the location of a point on the curve at parameter $t$. The derivative of this position vector, $\mathbf{x}'(t)$, is the tangent vector. This vector points in the direction of the curve's instantaneous motion and its magnitude represents the speed.
Key Condition for Regularity
The fundamental requirement for a curve to be regular is:
$\mathbf{x}'(t) \neq \mathbf{0}$ for all $t$ in the curve's parameter domain.
This means that for every point on the curve, there is a well-defined, non-zero tangent direction.
Why is a Non-Zero Tangent Vector Important?
- Smoothness: A non-zero tangent vector implies that the curve is continuously differentiable and smooth. There are no abrupt changes in direction.
- Well-Defined Tangent Line: At every point, you can define a unique tangent line to the curve.
- No Singularities: Points where $\mathbf{x}'(t) = \mathbf{0}$ are called singular points. These points often correspond to geometric features like cusps or self-intersections where the curve "pauses" or its direction becomes ambiguous.
How to Check for Regularity: A Step-by-Step Guide
- Parametrize the Curve: Ensure the curve is defined by a vector function $\mathbf{x}(t) = (x(t), y(t), z(t))$ for some parameter $t$ in an interval $I$.
- Compute the Tangent Vector: Differentiate each component of the position vector with respect to $t$ to find the tangent vector $\mathbf{x}'(t) = (x'(t), y'(t), z'(t))$.
- Check for Zero Vector: Determine if there are any values of $t$ in the domain $I$ for which $\mathbf{x}'(t) = \mathbf{0}$. This means simultaneously checking if $x'(t) = 0$, $y'(t) = 0$, and $z'(t) = 0$.
- Conclude:
- If $\mathbf{x}'(t) \neq \mathbf{0}$ for all $t \in I$, the curve is regular.
- If there exists at least one $t_0 \in I$ such that $\mathbf{x}'(t_0) = \mathbf{0}$, the curve is not regular (it has a singularity at $t_0$).
Examples and Practical Insights
Let's look at some examples to clarify the concept:
Example 1: A Regular Curve (Circle)
Consider a circle in the plane parametrized by $\mathbf{x}(t) = (\cos t, \sin t)$ for $t \in [0, 2\pi)$.
- Position Vector: $\mathbf{x}(t) = (\cos t, \sin t)$
- Tangent Vector: $\mathbf{x}'(t) = (-\sin t, \cos t)$
- Check for Zero Vector:
- $-\sin t = 0 \implies t = 0, \pi$
- $\cos t = 0 \implies t = \pi/2, 3\pi/2$
There is no value of $t$ for which both components are simultaneously zero. The magnitude of the tangent vector is $||\mathbf{x}'(t)|| = \sqrt{(-\sin t)^2 + (\cos t)^2} = \sqrt{\sin^2 t + \cos^2 t} = \sqrt{1} = 1$. Since the magnitude is always 1 (never 0), the tangent vector is never the zero vector.
- Conclusion: The circle is a regular curve.
Example 2: A Non-Regular Curve (Cusp)
Consider a curve (a cusp) parametrized by $\mathbf{x}(t) = (t^3, t^2)$ for $t \in \mathbb{R}$.
- Position Vector: $\mathbf{x}(t) = (t^3, t^2)$
- Tangent Vector: $\mathbf{x}'(t) = (3t^2, 2t)$
- Check for Zero Vector:
- $3t^2 = 0 \implies t = 0$
- $2t = 0 \implies t = 0$
At $t=0$, both components of the tangent vector are zero, so $\mathbf{x}'(0) = (0,0) = \mathbf{0}$.
- Conclusion: This curve is not regular because it has a singular point at $t=0$, which corresponds to the cusp at the origin $(0,0)$.
Regular vs. Non-Regular Curves
| Feature | Regular Curve | Non-Regular Curve |
|---|---|---|
| Tangent Vector | $\mathbf{x}'(t) \neq \mathbf{0}$ for all $t$ | $\mathbf{x}'(t_0) = \mathbf{0}$ for some $t_0$ |
| Smoothness | Smooth, no sharp corners or cusps | May have cusps, corners, or self-intersections |
| Tangent Line | Well-defined at every point | Undefined or ambiguous at singular points |
| Geometric Shape | Generally "well-behaved," predictable flow | Can exhibit "pauses" or abrupt changes in direction |
| Example | Circle, straight line, ellipse | Cusp, certain self-intersecting curves |
A regular curve is essential in many areas of mathematics and physics, particularly in differential geometry and mechanics, as it guarantees a smooth trajectory and predictable motion.
