Parametrized Curves and Surfaces

date2026-07-15document_iddoc_d1778feb062a3b745cc75ee5f576b970description正則曲線と正則曲面パッチ、再パラメータ化、向き、弧長、面積要素を体系的に導入する講義である。prerequisites多変数関数と偏微分 / 接平面・連鎖律・Jacobian / 多重積分と変数変換type講義content_typelecturestatusactive

mathmultivariable-calculusparameterizationregular-curvesurface-patchlecture

data/lecture/math/multivariable-calculus/tangent-planes-chain-rule-and-jacobian.lecture.n.md data/lecture/math/multivariable-calculus/multiple-integrals-and-change-of-variables.lecture.n.md

1Introduction

This lecture uses parameters to describe curves and surfaces and defines tangent directions, orientation, arc length, and surface area elements. In the subsequent lectures on line and surface integrals, these constructions convert geometric objects into one- or two-variable integrals.

The same curve or surface can have several parameterizations. We must therefore distinguish the geometric quantities that are invariant under reparameterization from those that depend on orientation.

2Regular Curves

Let $I = [a, b]$ and let

r : I \to R^{n}, r (t) = (r_{1} (t), \dots, r_{n} (t))

be of class $C^{1}$ , meaning that each component is continuously differentiable. The image $r (I)$ is a parametrized curve.

If $r^{'} (t) \neq 0$ for every $t \in I$ , then $r$ is a regular parameterization. The derivative

r^{'} (t) = (r_{1^{'}} (t), \dots, r_{n^{'}} (t))

is a tangent vector to the curve. Regularity ensures that the motion does not stop in the interior of the parameter interval and that a tangent direction is defined.

2.1Example 1: The Unit Circle

Let

r (t) = (\cos t, \sin t), 0 [PARSE ERROR: Undefined("Command(\"le\")")] t [PARSE ERROR: Undefined("Command(\"le\")")] 2 π .

Then

r^{'} (t) = (- \sin t, \cos t), ∥ r^{'} (t) ∥ = 1.

Thus the parameterization is regular, and increasing $t$ gives the counterclockwise orientation of the unit circle. The endpoints $t = 0$ and $t = 2 π$ represent the same point, but no point is repeated in the interior.

3Reparameterization and Orientation

Let $ψ : [α, β] \to [a, b]$ be a $C^{1}$ bijection with $ψ^{'} (s) \neq 0$ . Then

ρ (s) = r (ψ (s))

is a reparameterization with the same curve image. By the chain rule,

ρ^{'} (s) = r^{'} (ψ (s)) ψ^{'} (s) .

It preserves orientation when $ψ^{'} (s) > 0$ and reverses orientation when $ψ^{'} (s) < 0$ .

Changing the speed of a parameter does not change the curve image or its arc length. In contrast, the work line integral defined in the next lecture changes sign when orientation is reversed. Thus, the same point set and the same oriented curve are different concepts.

4Arc Length

The arc length of a $C^{1}$ curve $r : [a, b] \to R^{n}$ is

L (r) = \int_{a}^{b} ∥ r^{'} (t) ∥ d t .

The quantity $∥ r^{'} (t) ∥$ is the speed with respect to $t$ . Under an orientation-preserving reparameterization, the one-variable change-of-variables formula shows that arc length is invariant. An orientation-reversing reparameterization gives the same length because of the absolute value.

For the unit circle, $∥ r^{'} (t) ∥ = 1$ , so

L = \int_{0}^{2 π} 1 d t = 2 π .

A piecewise $C^{1}$ curve is a continuous curve for which the parameter interval can be divided into finitely many subintervals on each of which the curve is of class $C^{1}$ . Its arc length is the sum of the lengths of these curve segments. The line integrals in the next lecture are likewise defined on each segment and then summed.

5Regular Surface Patches

5.1The Cross Product

For $a = (a_{1}, a_{2}, a_{3})$ and $b = (b_{1}, b_{2}, b_{3})$ , define

a \times b = (a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1}) .

This vector is perpendicular to both $a$ and $b$ , and its norm $∥ a \times b ∥$ is the area of the parallelogram spanned by them. Moreover, $a \times b = 0$ if and only if $a$ and $b$ are linearly dependent. The cross product is linear in each variable; this property is called bilinearity. The component formula also gives the anticommutative law $b \times a = - a \times b$ .

Let $D \subset R^{2}$ be a bounded Jordan region, and let

r : D \to R^{3}, r (u, v) = (x (u, v), y (u, v), z (u, v))

be of class $C^{1}$ on a neighborhood of $\overline{D}$ . The partial derivatives

r_{u} = \frac{\partial r}{\partial u}, r_{v} = \frac{\partial r}{\partial v}

are tangent vectors in the $u$ - and $v$ -directions.

r_{u} \times r_{v} \neq 0

throughout the interior of $D$ and $r$ is one-to-one there, then its image is a regular surface patch. The nonzero cross-product condition is equivalent to linear independence of the tangent vectors and determines the tangent plane

r (u_{0}, v_{0}) + s r_{u} (u_{0}, v_{0}) + t r_{v} (u_{0}, v_{0}) .

Interior injectivity prevents the same portion of a surface from being counted repeatedly. Because the boundary of a bounded Jordan region has area zero, overlap confined to the boundary contributes no area. An interior multiple covering is counted according to its multiplicity in an integral.

6The Surface Area Element

Under the linear approximation, a small rectangle in the parameter plane maps to a parallelogram with edges $r_{u} d u$ and $r_{v} d v$ . Its area is

∥ r_{u} \times r_{v} ∥ d u d v .

Therefore the surface area element is

d S = ∥ r_{u} \times r_{v} ∥ d u d v .

The cross product itself is normal to the tangent plane, and its magnitude is the local area-scaling factor.

6.1Example 2: A Graph Surface

Let $f$ be of class $C^{1}$ and parameterize $z = f (x, y)$ by

r (x, y) = (x, y, f (x, y)) .

Then

r_{x} = (1, 0, f_{x}), r_{y} = (0, 1, f_{y}),

and

r_{x} \times r_{y} = (- f_{x}, - f_{y}, 1) .

Hence

d S = \sqrt{1 + f_{x^{2}} + f_{y^{2}}} d x d y .

For the plane $z = 0$ , both partial derivatives vanish, so this reduces to $d S = d x d y$ .

7Surface Orientation and Reparameterization

The vector $r_{u} \times r_{v}$ selects one normal direction. Interchanging $u$ and $v$ reverses the sign of the cross product and selects the opposite orientation. A surface is orientable if a continuous normal direction can be selected over the entire surface.

Let $E, D \subset R^{2}$ be parameter domains and let $φ : E \to D$ be a $C^{1}$ diffeomorphism. Define the reparameterization $ρ = r \circ φ$ . The chain rule and bilinearity of the cross product give

ρ_{s} \times ρ_{t} = \det D φ (r_{u} \times r_{v}) \circ φ .

Thus $\det D φ > 0$ preserves orientation, whereas $\det D φ < 0$ reverses it. The surface area element uses an absolute value and is independent of orientation, but flux uses a normal direction and changes sign when orientation is reversed.

8Computational Procedure

State the parameter domain and the image of the curve or surface.
Check $r^{'} \neq 0$ for a curve or $r_{u} \times r_{v} \neq 0$ for a surface.
Check that there is no unintended multiple covering in the interior.
When orientation is relevant, state the direction of increasing parameter or the selected normal direction.
Use $∥ r^{'} ∥$ for arc length and $∥ r_{u} \times r_{v} ∥$ for surface area.

9Cautions

At a point where the tangent vector vanishes, a tangent direction is not defined by a regular curve parameterization.
At a point where the two surface tangent vectors are parallel, the regular surface-patch condition fails.
Two parameterizations with the same curve image but opposite orientations define different oriented curves.
If a surface parameterization covers an interior portion more than once, a surface integral counts that portion with multiplicity.

10Related Lectures

data/lecture/math/vector-calculus/line-integrals-and-conservative-fields.lecture.n.md data/lecture/math/vector-calculus/surface-integrals-and-flux.lecture.n.md