LECTURE 01 / Curves & parametrization

A curve and its clock

The same geometric path can have many parametrizations.

3D EXPLORATION

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r(t)=(\cos t,\sin t,t)

Curve r(t)Tangent r′(t)Current point

Sampled geometry; formulas establish the claims. Field and normal arrows use scaled lengths.

Parameter t≈ 0.60

-3.143.14

r(t) = (0.825, 0.565, 0.6) · speed = √2

TRY THIS

Trace the helix and compare equal changes in the parameter with changes in position. Explain what changes under $t=u^3$ .

What to notice

Geometry is the image; parametrization is the way the image is visited.

MATHEMATICAL NOTES

\begin{gathered}\mathbf r(t)=(\cos t,\sin t,t)\\\mathbf s(u)=\mathbf r(\phi(u))\end{gathered}

Definition & meaning

A parametrized curve is a continuous map from an interval into $\mathbb R^2$ or $\mathbb R^3$ ; its image is the geometric curve. A vector-valued map is continuous exactly when its components are continuous. A parametrization records order, speed and possible repetitions that its image alone does not record.

PRECISE HYPOTHESES

To cover the same complete image, $\phi:J\to I$ must be continuous and onto the original parameter interval. For differentiable reparametrizations, $\mathbf s'(u)=\mathbf r'(\phi(u))\phi'(u)$ .

Why this works

The image identity follows from surjectivity: every parameter $t\in I$ equals $\phi(u)$ for some $u$ . The componentwise one-variable chain rule gives the derivative formula. An increasing clock preserves traversal direction; a decreasing clock reverses it.

Concepts in this exploration 7

vector-valued functions
componentwise continuity
curve as continuous image
parametrization
reparametrization
orientation and speed
space-filling curve caveat

Course reference: MH2100 Lecture 01.
Original explanations; lecture files are not redistributed.

MH2100 · Calculus Atlas Geometry for intuition. Arguments for certainty.