spike · 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.

Not an external application. A spike — testing whether Lemma's grammar can carry a definitional crisis the way it carries Bitcoin Pizza or float underflow.

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."

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.