Chapter 1: Parametric Curves

Q: What is a parametric curve?

A smooth map γ: I → R^n from an interval I to R^n (n ≥ 2). As the parameter t varies, γ(t) traces out a curve in space. It can also handle curves that are hard to write in the form y = f(x), such as circles, self-intersecting curves, and closed curves. In differential geometry, plane curves (n = 2) and space curves (n = 3) are the most commonly studied.

Q: What is a regular curve?

A parametric curve whose velocity vector γ'(t) is nonzero at every parameter value t (C^1 or smoother). Regularity makes the tangent direction uniquely determined at each parameter value and rules out cusps and stationary points. Smoothness is an assumption of the definition, not a consequence of regularity (γ(t) = (t^3, t^2) is smooth but not regular).

Q: What is reparametrization?

Replacing the parameter of a curve by means of a smooth bijection φ: J → I with φ'(s) ≠ 0 for all s, giving the new parametric curve γ̃(s) = γ(φ(s)). Geometrically the same curve is traversed, but the rate of traversal (the speed) changes. If φ' > 0 the orientation is preserved; if φ' < 0 it is reversed.

Curve Definitions, Regular Curves, and Reparametrization

Introduction (first- to second-year university)

Introduction

In differential geometry, curves are expressed using a "parameter." Choosing a value of a single variable (the parameter) determines one point on the curve, so the properties of the curve can be studied using the tools of calculus.

Definition of a Curve

Definition: Parametric Curve

A smooth map from an interval $I \subset \mathbb{R}$ to $\mathbb{R}^n$ ($n \geq 2$),

$$\gamma: I \to \mathbb{R}^n, \quad t \mapsto \gamma(t) = (\gamma_1(t), \gamma_2(t), \ldots, \gamma_n(t))$$

is called a parametric curve. Here "smooth" means of class $C^\infty$ (continuously differentiable any number of times). We take $I$ to be an open interval; when a closed interval $[a, b]$ is used, one-sided derivatives are taken at the endpoints. In differential geometry, the cases $n = 2$ (plane curves) and $n = 3$ (space curves) are most commonly studied.

A parametric curve: a point t in the interval [a, b] corresponds to a point γ(t) on the curve, with the tangent vector γ'(t) drawn

Figure 1: A parametric curve — each point $t$ in the interval $[a, b]$ corresponds to a point $\gamma(t)$ on the curve. The red arrow represents the tangent vector $\gamma'(t)$.

Examples: Circle, Parabola, and Helix

Example 1: Unit Circle

$$\gamma(t) = (\cos t, \sin t), \quad t \in [0, 2\pi]$$

The parameter $t$ corresponds to the angle (in radians).

Parametric representation of the unit circle γ(t) = (cos t, sin t), where the parameter t corresponds to the angle

Figure 2: Parametric representation of the unit circle — the parameter $t$ corresponds to the angle, and the point $\gamma(t)$ moves counterclockwise around the circle.

Example 2: Parabola

$$\gamma(t) = (t, t^2), \quad t \in \mathbb{R}$$

This is a parametric representation of the graph $y = x^2$.

Parametric representation of the parabola γ(t) = (t, t²), with the vertex at t=0 and the tangent vector drawn at t=1

Figure 3: Parametric representation of the parabola — the graph $y = x^2$ is expressed as $\gamma(t) = (t, t^2)$. The red arrow shows the tangent vector $\gamma'(1) = (1, 2)$ at $t = 1$.

Example 3: Helix

$$\gamma(t) = (a\cos t, a\sin t, bt), \quad t \in \mathbb{R}$$

For $a > 0$ and $b \neq 0$, this represents a helix winding around a cylinder. It rises as $t$ increases if $b > 0$ and descends if $b \lt 0$.

Three-dimensional view of the helix γ(t) = (a cos t, a sin t, bt): a spiral curve winding around a cylinder, with the tangent vector γ'(t)

Figure 4: Helix — for $b > 0$, the spiral curve $\gamma(t) = (a\cos t, a\sin t, bt)$ rises around the $z$-axis (it descends if $b \lt 0$). The red arrow represents the tangent vector $\gamma'(t)$.

Regular Curves

Definition: Regular Curve

A parametric curve $\gamma: I \to \mathbb{R}^n$ is said to be regular if for all $t \in I$,

$$\gamma'(t) \neq \mathbf{0}$$

The vector $\gamma'(t)$ is called the velocity vector. (What the discussion of regularity needs is only that $\gamma$ be of class $C^1$ (at least once continuously differentiable), and the same definition is used for $C^1$ curves.)

The heart of regularity is the condition $\gamma'(t) \neq \mathbf{0}$. Smoothness (differentiability) alone does not rule out cusps: the example $\gamma(t) = (t^3, t^2)$ below is of class $C^\infty$ yet has a cusp at the origin. Imposing $\gamma'(t) \neq \mathbf{0}$ rules out cusps and stationary points, so the tangent direction is uniquely determined at each parameter value $t$.

All three examples of the previous section are regular. For the unit circle, $\gamma'(t) = (-\sin t, \cos t)$ and $|\gamma'(t)| = 1$; for the parabola, $\gamma'(t) = (1, 2t)$ and $|\gamma'(t)| = \sqrt{1 + 4t^2} \geq 1$; for the helix, $\gamma'(t) = (-a\sin t, a\cos t, b)$ and $|\gamma'(t)| = \sqrt{a^2 + b^2} > 0$ (a constant).

Meaning of the Velocity Vector

When $\gamma'(t) \neq \mathbf{0}$, the vector $\gamma'(t) = \left(\dfrac{d\gamma_1}{dt}, \dfrac{d\gamma_2}{dt}, \ldots, \dfrac{d\gamma_n}{dt}\right)$ represents the tangent direction of the curve at parameter $t$ (where $\gamma'(t) = \mathbf{0}$ it gives no direction). If the parameter $t$ is interpreted as time, then $\gamma'(t)$ corresponds to the velocity of a particle, and regularity means "the particle is always in motion."

Example: A Non-Regular Curve

$$\gamma(t) = (t^3, t^2), \quad t \in \mathbb{R}$$

Since $\gamma'(t) = (3t^2, 2t)$, we have $\gamma'(0) = (0, 0)$ at $t = 0$. Therefore the curve is not regular at $t = 0$.

This curve has a cusp at the origin.

The non-regular curve γ(t) = (t³, t²) with a cusp at the origin where γ'(0) = (0, 0)

Figure 5: A non-regular curve $\gamma(t) = (t^3, t^2)$ — the curve has a cusp at the origin. At $t = 0$, $\gamma'(0) = \mathbf{0}$, so the velocity vector gives no tangent direction.

Reparametrization

The same curve can be expressed using different parameters.

Definition: Reparametrization

Take a smooth bijection $\phi: J \to I$ from an interval $J$ onto $I$ with $\phi'(s) \neq 0$ for all $s \in J$. Then $\tilde{\gamma} = \gamma \circ \phi$ is called a reparametrization of $\gamma$.

$$\tilde{\gamma}(s) = \gamma(\phi(s))$$

For example, given the unit circle $\gamma(t) = (\cos t, \sin t)$ ($t \in (0, 2\pi)$), setting $\phi(s) = s^2$ on $J = (0, \sqrt{2\pi})$ yields

$$\tilde{\gamma}(s) = \gamma(s^2) = (\cos s^2, \sin s^2)$$

which traces the same circle but at a different speed. While $\gamma$ traverses the circle at constant speed, $\tilde{\gamma}$ moves slowly when $s$ is small and accelerates as $s$ increases. Geometrically it is the same curve, but the "rate of traversal" differs. (Since $\phi'(s) = 2s$ vanishes only at $s = 0$, any interval not containing $s = 0$ will do. Taking $J = (-\sqrt{2\pi}, 0)$ also gives a reparametrization, this time with the orientation reversed.)

Comparison of reparametrizations: left, evenly spaced t gives evenly spaced nodes; right, evenly spaced s traversed with t = s² gives nodes clustered near the start with spacing widening along the way

Figure 6: Comparison of reparametrizations — the same unit circle traversed with parameter $t$ (left) and with the parameter $s$ of the reparametrization by $\phi(s) = s^2$ (right). Evenly spaced values of $t$ produce evenly spaced nodes on the circle, whereas evenly spaced values of $s$ cluster nodes near the starting point, with spacing increasing along the way.

Proposition: Velocity under Reparametrization

If $\tilde{\gamma} = \gamma \circ \phi$, then by the chain rule,

$$\tilde{\gamma}'(s) = \gamma'(\phi(s)) \cdot \phi'(s)$$

Therefore, if $\gamma$ is regular and $\phi'(s) \neq 0$ for all $s \in J$, then $\tilde{\gamma}$ is also regular.

When $\phi' > 0$, the reparametrization is said to be "orientation-preserving"; when $\phi' < 0$, it is "orientation-reversing."

Summary

In this article, a parametric curve is defined as a smooth map from an interval into a space (the curve as a figure is its image).
A regular curve is one whose velocity vector is nonzero at every parameter value ($C^1$ or smoother).
The same curve can be expressed with different parameters (reparametrization).
In differential geometry, regular curves are the fundamental objects of study.

References

Frequently Asked Questions

Q1: What is a parametric curve?

A smooth map $\gamma: I \to \mathbb{R}^n$ from an interval $I$ to $\mathbb{R}^n$ ($n \geq 2$). As the parameter $t$ varies, $\gamma(t)$ traces out a curve in space. It can also handle curves that are hard to write in the form $y = f(x)$, such as circles, self-intersecting curves, and closed curves. In differential geometry, plane curves ($n = 2$) and space curves ($n = 3$) are the most commonly studied.

Q2: What is a regular curve?

A parametric curve whose velocity vector $\gamma'(t)$ is nonzero at every parameter value $t$ ($C^1$ or smoother). Regularity makes the tangent direction uniquely determined at each parameter value and rules out cusps and stationary points. Smoothness is an assumption of the definition, not a consequence of regularity ($\gamma(t) = (t^3, t^2)$ is smooth but not regular).

Q3: What is reparametrization?

Replacing the parameter of a curve by means of a smooth bijection $\phi: J \to I$ with $\phi'(s) \neq 0$ for all $s$, giving the new parametric curve $\tilde{\gamma}(s) = \gamma(\phi(s))$. Geometrically the same curve is traversed, but the rate of traversal (the speed) changes. If $\phi' > 0$ the orientation is preserved; if $\phi' \lt 0$ it is reversed.