Curves in differential geometry - Definition and Overview

This page covers mathematical example of curves in differential geometry.

Constant curve

Given a point p₀ in R³

<math> \mathbf{\gamma}:t \mapsto \mathbf{p_0} = \begin{pmatrix}

x_0\\
y_0\\
z_0\\

\end{pmatrix}\qquad (t \in I)<math>

defines the constant curve, a parametric curve of class C^∞. The image of the constant curve is the single point p. The curve is closed and analytic but not simple.

Line

A slightly more complex example is the line. A parametric definition of a line through the points p₀ and p₁ (p₀ ≠ p₁ and p₀,p₁ ∈ R³) is given by

<math> \mathbf{\gamma}:t \mapsto \mathbf{p_0} + t(\mathbf{p_1} - \mathbf{p_0})= \begin{pmatrix}

x_0 + t (x_1 - x_0)\\
y_0 + t (y_1 - y_0) \\
z_0 + t (z_1 - z_0) \\

\end{pmatrix} \qquad (t \in I) <math>

The image of the curve is a line. Note that

<math> \mathbf{\gamma}:t \mapsto \mathbf{p_0} + t^3 (\mathbf{p_1} - \mathbf{p_0})= \begin{pmatrix}

x_0 + t^3 (x_1 - x_0)\\
y_0 + t^3 (y_1 - y_0) \\
z_0 + t^3 (z_1 - z_0) \\

\end{pmatrix} \qquad (t \in I) <math>

is a different curve but the image of both curves is the same line.

Helix

Give r, b in R

<math> \mathbf{\gamma}:t \mapsto \begin{pmatrix}

r cos (\omega t)\\
r sin (\omega t)\\
bt\\

\end{pmatrix}\qquad (t \in I)<math>

defines the helix.