the hook · definitional crisis

What is “the same curve”?

Drawing the parabola $y = x²$ on a board, you’d say this is a curve. Below you’ll see the same picture drawn three different ways — same paint, different motions. Which one is the curve? Pick an answer in your head before pressing play. Then play.

A module that opens with a definitional crisis instead of an application: the same picture, drawn three different ways. The fix redefines what “a curve” even is — and that redefinition is the tool downstream applications (graphics, physics, animation) consume.

tool spec

what: A function $γ : [0, 1] → ℝ²$ . Not its picture, not its image — the function. Three layers of “same”: same image (loosest), same up to monotone reparametrization (geometric), equal as functions (strictest).
applies when: The question is motion, not just shape. Bezier curves (designer drags handles, computer evaluates γ(t)), trajectories (γ(t) = position at time t), animations (motion path is a parametrization, not the path image), arc-length integrals.
breaks when: If only the picture matters — drawing on paper, fitting an outline — the function machinery is overkill; point-set geometry suffices. The distinction also leaks: “is this the same curve?” has no clean answer until you say which of the three layers you mean.

Spike — Same Curve?

u0.000

Click a γ to play. Drag u to scrub the dot manually.

the arc

γ is a function, not a shape

A parametrized curve is a function $γ : [0, 1] → ℝ²$ . The three γ above are three different functions. They happen to share an image — the set ${γ(t) : t ∈ [0, 1]}$ , which is the parabola $y = x²$ on $x ∈ [−1, 1]$ — but the image is a set of points, stripped of order, speed, and visit count. The function γ is none of those things. Two parametrized curves are equal when $γ₁(t) = γ₂(t)$ for every t — the same equality you’d use for any other function. By that test, the three γ above are three different curves that happen to leave the same shadow. But functional equality is only the strictest of three layers; the next step locates where the line should actually be drawn.

Same image, but is it the same curve?

Step 1’s strict test separates γ₁ from γ₂ even though they paint the same picture in the same direction at different paces. That feels too strict — geometrically they ought to be the same. Differential geometry agrees: it identifies γ with $γ ∘ φ$ whenever φ is a monotone bijection $[0, 1] → [0, 1]$ . With $φ(u) = ((2u−1)³ + 1) / 2$ (monotone since $dφ/du = 3(2u−1)² ≥ 0$ ), one checks $γ₁ ∘ φ = γ₂$ exactly: γ₁ and γ₂ are the same curve at different paces. γ₃ has no such partner — it visits each interior image point twice, and no monotone φ can take a once-visit schedule to a twice-visit one. So between strict equality (too tight) and image equality (too loose) sits reparametrization equivalence — and that is what most working mathematicians mean by “the same curve.”

Where this shows up — same γ, two pillars

The same map γ(t) can be a designer’s stroke, a projectile’s path, or a clock’s motion. Once “curve” stops meaning shape and starts meaning function-of-time, the same machinery splits cleanly across pillars.

graphics : γ is design — a stroke a user wants to see.
physics  : γ is motion — a body's position at time t.

Bezier curves — the curve a designer pulls is literally a parametrized γ : [0, 1] → ℝ². Dragging a handle changes γ; the picture changes because γ has changed. The whole de Casteljau evaluator is “given t and the control points, compute γ(t).” Same tool, used as drawing.

Projectile motion — γ(t) = (v₀ cos θ · t, v₀ sin θ · t − ½ g t²) is the position at time t. Here time is the parameter and the image is the trajectory through space. Same tool, used as motion.

The pendulum clock — the bob’s tip traces a γ over time as θ(t) swings. The image is an arc; the function is the schedule that says when the bob is where. The escapement only works because that schedule is periodic in t — a property of γ, not of its image.

The split arc 2 made — image vs. parametrization — is exactly what lets a designer’s drag and a physicist’s pendulum both be γ. The image is what each pillar shows; γ is what each pillar manipulates.

Three layers of “same”:

same image — the picture: γ₁ ~ γ₂ ~ γ₃
same up to reparametrization — the geometric curve: γ₁ ~ γ₂; γ₃ stands alone
equal as functions — the parametrized curve: all three differ

The picture is the weakest layer; the function is the strictest. Geometry lives in between. School math conflates all three under the single word curve; that is why “is this the same curve?” has no clean answer — the word was quietly carrying three different objects. The widget you played with above is one diagram showing all three at once.

exercises · 손으로 풀기

1read the widget · trail density

Play γ₂ and look at the trail. Where are the points most tightly packed — near the origin, or near the endpoints? Why? (Trail spacing is constant in time, not in arc length.)

2construct · same image, different scheduleno calculator

Write a γ : [0, 1] → ℝ² whose image is exactly $y = x²$ on $x ∈ [−1, 1]$ , but which traces that parabola three times as t goes 0 → 1. Give a closed-form expression. (Hint: the round trip γ₃ traces it twice using cosine.)

3the evil one · 'extensional equality'

A junior says: “Two functions with the same outputs are the same function — that’s extensional equality. Three γ producing the same image must be the same curve.” Find the bug in one sentence. Then state the correct equality test for parametrized curves.

glossary · used on this page · 4

definitional crisis·정의의 위기

A moment when a single everyday word turns out to be carrying multiple distinct objects, so a question that depended on it has no clean answer until those objects are named separately. Lemma uses such moments as motivating questions when no external application is needed.

parametrized curve·매개변수 곡선

A function γ : I → ℝⁿ from an interval I to a space. Equality is functional: γ₁ = γ₂ iff γ₁(t) = γ₂(t) for every t ∈ I. The picture you draw — its image — is a consequence, not the curve itself.

image (of γ)·상 (γ의 상)

For γ : I → ℝⁿ, the image is the set {γ(t) : t ∈ I} ⊂ ℝⁿ. It records _where_ γ goes, not _when_ or _how often_. The image is what you see on paper; the curve γ is the function that drew it.

⚠ Different from γ itself. The image is a *set of points* — order, speed, and visit count are gone. Many γ can share an image; recovering γ from its image alone is impossible. In casual math speech, "the curve" usually means this set; in differential geometry, "curve" means γ.

reparametrization·재매개화

A monotone bijection φ : I → I. If γ : I → ℝⁿ is a parametrized curve, γ ∘ φ traces the same image in the same direction, possibly at a different pace (and reversed if φ is decreasing). Two parametrized curves are "the same up to reparametrization" when one is γ ∘ φ of the other — the working definition of "the same curve" in differential geometry.