Space Curves

The central idea

A space curve is the image of a vector-valued function $\mathbf r:I\to\mathbb R^3$. Its parameter interval, orientation, and repeated tracing are part of the description. At a regular point $\mathbf r'(t)\ne0$, the derivative gives a tangent direction; component relations or projections help identify the curve's geometry.

Definitions, hypotheses, and notation

A curve can intersect itself even when its parametrization is regular; distinct parameter values may map to the same point with different tangent directions. Conversely, a zero velocity at one parameter may arise from a poor parametrization of an otherwise smooth geometric curve. Local and global properties should therefore be separated.

Arc length can supply a special parameter $s$ with unit speed. Under unit-speed parametrization, $\mathbf T=\mathbf r'$ and curvature is $\|\mathbf r''\|$, simplifying geometric analysis. Constructing arc-length parameters explicitly is not always elementary.

Conceptual meaning

Three coordinate functions evolve together to trace a one-dimensional path in space. Projections onto coordinate planes reveal shadows of the curve, while the parameter orders points and can represent time.

A dependable method and decision rule

1. State the parameter interval and calculate several anchor points.
2. Eliminate the parameter between pairs of coordinates when helpful.
3. Inspect projections onto the $xy$, $xz$, and $yz$ planes.
4. Compute velocity to determine orientation and regularity.
5. Use tangent, length, or curvature formulas only on appropriate regular pieces.

Fully worked example

Common mistakes and why they fail